As the universe operates anything but intuitively logical, it is a crucial skill to be able to identify a provable reality vs a mere mathematical solution. This is what makes the multiverse idea and string theory's hidden dimensions just fantasy. However, block time is proven i.e. having no freewill is proven via the constant speed of light. Yet how many people believe they have no freewill..
i think the trivial take of the paradox is that thinking that it only contradicts with itself but the deeper problem is that it contradicts with the assumption that every statement can only be either true or false
The halting problem is a nicer way of making this issue much more tangible in my opinion. If you have a computer program and it really does halt then you could imagine simulating it all the way out to the halting state in finite time. It seems like there must be a fact of the matter about whether it would eventually halt or not. But Turing showed this question is computationally undecidable in general through a pretty simple proof, and this also implies both Tarski's undefinability theorem and the incompleteness theorems.
Not only that. From the Halting problem follows Rice's Theorem, which informally can be read as "for a sufficiently complex program, you can't predict what it will do just looking at the source code." And "sufficiently complex" is really, really simple. So right there you can prove things like "your compiler will always risk outputting code you'll never actually run" and "your tests will always risk not hitting the conditions that demonstrate you have a mistake in the code." It's an incredibly powerful statement for computer science.
@@tricky778 No, it doesn't need even an unbounded amount of memory or time, let alone an infinite amount. You can figure out if a bounded machine halts, but you can't do it merely by looking at the code of the program without actually simulating it. So, basically, run it for long enough that it's either looping or halted, and then you know. But you still wind up having to run the machine to determine the answer.
@@darrennew8211 no, you don't have to simulate the machine if it has bounded memory, although you might argue there are some program snippets that can be written for some iterations of which the analysis is equivalent to simulation. I don't think that qualifies to be called simulating the machine because the latter phrase implies writing an emulator and analysing the state to synthesise a prover rather than finding that you can transform a subprover into a emulator.
For me, Godel's incompleteness theorem is made less scary/problematic when you realise that language itself can only construct a *countably* infinite set of ideas, even though we assume an *uncountably* infinite set of numbers regularly without issues. It's not so much trying to create a paradox as it is saying that there will always be indeterminate things which are not worth discussing (unless if you find it fun) because we can't know anything with the tools we have. It's like trying to discuss what lies "within" a black hole. It's a meaningless question, albeit fun to think about, because we have no idea what actually happens to matter that dense. I see Godel's incompleteness in new light after reading Kant's Critique of Reason. Interested to see how you will animate Godel numbering, it's a fascinating proof and I feel like you will actually do a good job of demonstrating it.
Thank you! We're a big fan of Kant here, and know exactly what you mean. Also, it's interesting you bring up the uncountability aspect to the argument, since it was the uncountability of real numbers which was demonstrated by Cantor's original diagonalization argument, and it was that argument in turn which inspired Richard's Paradox, which in turn inspired Gödel's Theorem. All these things definitely seem connected in one way or another, but it's teasing out what exactly these connections are which are the difficult part.
@@dialectphilosophy @dialectphilosophy you said accelaration is not the solution of twin paradox then what's the real solution of twin paradox?you stopped making video on it and never told the real solution of twin paradox?
I taught a seminar on the first incompleteness theorem presenting Godel's proof, and we eventually reached this conclusion. A proof must terminate, so proofs are countable, but a proposition with a universal quantifier ( for all n, P(n) ), like Godel's statement, is itself a countably infinite conjunction of propositions ( P(1) and P(2) and P(3) and ... ). If one presumes to enumerate the provable conjunctions of propositions, one can construct by diagonalization a conjunction of propositions that cannot be on the list and so cannot be provable even though each proposition is part of a provable conjunction of propositions. At that point, I lost interest in the subject. Is mental masturbation any more useful than the literal kind?
@@restonthewindThere are grounds for asserting the equivalency of (a) a universally quantified claim and (b) a conjunction of claims, as you described, when quantifying over a finite domain of discourse. From this, it does not follow that there are grounds for doing so when that domain is countably infinite -- much less uncountably infinite.
@@Raghuraam143 lol meaning instead of doing the task at hand, I'm watching this 15 minute video. Dialect is one of those once in a while channels that I have alerts on for 👌
The video is generally good, but I find some statements to be quite questionable. 7:06 - "By showing that an absurd self-referential statement could be constructed in almost solely mathematical terms, one could argue that Gödel was cleverly demonstrating that mathematics was merely a language like any other and therefore subject to all of its usual foibles". That is inaccurate (and it's a relatively common misconception around Gödel incompleteness theorems). First of all, Gödel theorems are about formal systems, not about "math" in general. Like all theorems, they have hypothesis, and if those are not satisfied, the results do not follow. Gödel theorems apply only to _some_ formal systems, but not to others, and not to _all_ formal systems. It was proved that, for example, Boolean Algebra (aka, propositional logic) is both complete and consistent formal system. In fact, Boolean Algebra is "perfect" (perfect, in this context, means: complete + consistent + decidable). Euclidean geometry is also a notable exception to Gödel incompleteness theorems. Secondly, it is well-known that mathematics, when viewed as a language (which is quite restrictive view, but let's go along with it), is a _formal_ language, which natural languages (such as English) simply are not. The differences are many, so concluding that "[...] mathematics was merely a language like any other" is just wrong. 7:51 - "The validity of every incompleteness theorem boils down to whether or not you believe the sentence "This statement is unprovable." is truly a valid paradox". This is excessive, to put it mildly. The word "believe", here, is really inappropriate: by using that word, it makes it sound like mathematics is a matter of belief (basically a religion). Would you say: "The validity of 2+2=4 boils down to whether or not you believe in addition"? I hope no. Theorems are, well, theorems: their validity is an objective matter (proof verification, etc.), not a matter of belief. In summary, while I found the video quite enjoyable, I also found it does not manage to go past some misconceptions about Gödel incompleteness theorems. Advanced theorems of mathematical logic are hard to correctly "translate" in laymen terms: analogies, metaphors, and informal descriptions can be used. But we need to be very careful about them, because, if we take them too far, we risk losing key details (as the saying goes, "the devil is in the details").
We addressed the fact that some system of formal logic can be considered complete towards the end of the video, but we think your objection here doesn't really take into account what is meant by the word "formal". Formal just means your language conforms to more specific and agreed-upon rules than non-formal languages, but the implication and meaning of something doesn't change simply because the language is "formal". Indeed, Gödel is often argued to be showing that if you want your math to be expressive and powerful enough to handle certain systems, you have to adopt particular tools of language that lead to incompleteness. This of course, would only be profound if you didn't believe math was a language to begin, and rather treated it as something almost religiously existing of its own accord, which was the case of the Hilbert Program. Mathematics intrinsically is belief. No "proof" is ever objective; it relies on assuming certain axioms. Nothing proves those axioms. Furthermore, you have to assume inference rules, which nothing can prove either. "2 + 2 = 4" is not a universal objective truth, and there are plenty of systems in physics where basic addition isn't useful and 2 + 2 doesn't equal four. (Honestly, in our opinion, a large part of the reverence for mathematics just stems from misplaced religious feeling.) So yes, if you don't believe the Liar's Paradox is significant, it would be extremely inconsistent to then go on and say that you do believe Gödel's Theorem is significant. Now, one can make the case that the Liar's Paradox is significant, but that's a different matter...
@@dialectphilosophy Firstly, thanks for the reply, and I would like to point out that, despite my criticism, I did find the video enjoyable and the quality (animation, special effects, etc.) high. I want to make clear that the criticism is precisely aimed at particular statements, but does not detract from the video as a whole. Breaking down the arguments. 1. Formal language vs non-formal languages - Here I have many things to say, but it would take too long to properly explain why I think formal languages and non-formal languages are fundamentally different. 2. "Mathematics intrinsically is belief". That's an interesting point of view, and I'm sure you are not the only one upholding it. But do you have any evidence to back it up? 3. ""2 + 2 = 4" is not a universal objective truth". That's a remarkable claim. Can you prove it? Going a step back: are there any objective truths? Or even more basic: are there any truths at all? 4. "there are plenty of systems in physics where basic addition isn't useful and 2 + 2 doesn't equal four" - I have some doubts regarding the second part of the claim: can you show a physical system where 2+2 does not equal 4? In fact, are physical systems even relevant to show whether 2+2=4? From Wikipedia (page "Mathematics"): _Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent from any scientific experimentation_ . 5. "No "proof" is ever objective; it relies on assuming certain axioms. Nothing proves those axioms." - Boolean Algebra has literally zero axioms. And yet, it does have theorems. As a side-note, I think we fundamentally disagree on the meaning of "objective". For example, to me, relying on assuming certain axioms doesn't detract from the objectivity of a proof (regardless of what we think of said axioms). 6. "Furthermore, you have to assume inference rules, which nothing can prove either". But do we need to "prove" inference rules? At a basic level, this sounds like the fallacy of arguing against the principle of non-contradiction. We would still dismiss a contradictory argument even if we didn't have a proof of the principle of non-contradiction. What about nonsensical arguments? Would we not reject them, despite (natural language) "implication" having no proof either? At a deeper level, there are formal systems (again, Boolean Algebra) where inference rules aren't truly "rules" at all - they are just definitions (of some particular functions, namely, Boolean operators). If we agree that definitions aren't assumptions, then we can still do inference without assumptions. I understand your final (and probably most important) argument about the link between the Liar Paradox and Gödel theorems. I find that an argument can be reasonably made. Yet, if we start from "mathematics intrinsically is belief", I think the Liar Paradox (and Gödel incompleteness) is the least of your problems.
@@QuantumMechanicYT To attempt to answer point 4, consider physical systems where the usual arithmetic might not apply as straightforwardly. For instance, take clouds in the sky or water droplets on a surface. When two clouds merge, we could say 1 cloud + 1 cloud = 1 cloud. Similarly, when two water droplets combine, 1 drop + 1 drop = 1 drop. In some physical contexts, combining entities doesn’t follow the traditional 1 + 1 = 2 arithmetic but rather a different kind of addition. To describe this mathematically, we can use the concept of a semigroup. A semigroup is a set equipped with an associative binary operation. Consider a set S = {0, 1} with the binary operation defined as follows: 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1 (which interistingly looks identical to an or operation in boolean algebra). This structure is independent of traditional algebra (groups and rings) which also require axioms such as having an identity element, inverses and be commutative. Our semigroup is closed and has binary operation. That's it. Whether it's useful to define this structure is another question. For practical purposes, counting individual water molecules in the drops might be more relevant, where traditional arithmetic would apply (i.e., the sum of molecules from both drops). Then you don't need two separate systems of mathematics, you just need one, but instead you reformulate which questions are worth to be asked. Is it worth to ask whether combining clouds or drops constitutes addition? Is it another operation? Deciding which one of these mathematics is more useful and decisions to make requires judgement and knowledge of other experiments, this is where absolute objectiveness breaks down. It’s crucial to note that mathematics serves as a tool to model and understand the physical world. The "unreasonable effectiveness of mathematics in the natural sciences," as Eugene Wigner put it, suggests that through a process of abstraction and pattern recognition, humanity has developed mathematical frameworks that most effectively describe the physical world. Whether a grand unified theory in physics would reveal new fundamental truths about the universe's objective properties is speculative. Such a theory would rely on its underlying axioms and mathematical structures. Different intelligent beings, such as hypothetical aliens, might develop alternative mathematical systems to describe the same physical phenomena, leading to isomorphic but distinct frameworks. Which one of them is more true? Is it the one which has more bang for its buck, i.e. simpler with more dense explanatory power? Which definitions would be more true of there's alternative ones? (This is where definitions would manifest as just another choice of axioms). On your note about Boolean algebra: Boolean algebra does have a foundational set of axioms and operations like AND, OR, and NOT. You can argue these would exist independently of humans, as they are abstract constructs that model binary logic. However, whether they would be formulated in the same way by other intelligent beings is an open question. I feel like it's an obvious choice to develop the axiom isomorphic to "A is true or not true" for any intelligent species which has a physical brain, since it's simply impossible to imagine a universe where a wall can be both green and not green at the same time. This is where this strong feeling of objectiveness comes from, I believe, and indeed it's really hard to disprove, hence it's a very strong foundation. It's helpful to realize however that it's still a belief. We simply believe that it's impossible to imagine a universe which doesn't follow our common logic, and that's where our misunderstanding comes from.
@@Игор-ь9щ Although I disagree with your points, I do note that your comment was well-written and polite. "When two clouds merge, we could say 1 cloud + 1 cloud = 1 cloud" Why are we assuming that the merging of two physical entities can be described by addition, which is an abstract operator rigorously defined in formal logic? "In some physical contexts, combining entities doesn’t follow the traditional 1 + 1 = 2 arithmetic but rather a different kind of addition" I would say, instead, a different operator. "Is it worth to ask whether combining clouds or drops constitutes addition? Is it another operation?" It is worth, and the answer to the second question is a clear-cut yes. Let "merge(x,y)" denote the merge between x and y. Since merge(x,y) does not equal to "x+y", "merge" is not equivalent to addition. They are very clearly different operators. "Deciding which one of these mathematics is more useful and decisions to make requires judgement and knowledge of other experiments, this is where absolute objectiveness breaks down" Usefulness is application-dependent and ultimately subjective. The metric we are using here is not usefulness, but truthfulness. My reply on point 4 was precisely that, i.e., it was an objection to the claim: C1: "there are physical systems where 2+2 does not equal 4" to which I replied: Q1: "are physical systems even relevant to show whether 2+2=4?" "Boolean algebra does have a foundational set of axioms and operations like AND, OR, and NOT" Actually, Boolean algebra does not have any axioms. "I feel like it's an obvious choice to develop the axiom isomorphic to "A is true or not true" for any intelligent species which has a physical brain, since it's simply impossible to imagine a universe where a wall can be both green and not green at the same time. This is where this strong feeling of objectiveness comes from, I believe, and indeed it's really hard to disprove, hence it's a very strong foundation. It's helpful to realize however that it's still a belief" But the crucial point is that "A + NOT(A) = 1" (please note that it is different from "A is true or not true", that's not what Boolean algebra says) is not a belief, but a theorem of Boolean algebra. Here there is a difference to be noted: people who studied philosophy call it the "principle of non contradiction" (one of the three key principles identified by Aristotle), and the word "principle" literally means that nothing precedes it, it can't be proven, it is a starting point. So, it really seems it is simply a belief, since it is left unproven. But people who studied Boolean algebra call it "theorem of non contradiction", and they use the word "theorem" (not "principle") because it can be proven, and the proof is actually quite easy.
@QuantumMechanicYT Thank you for your thorough reply! You are indeed correct in asserting that the proper formulation is "A + NOT(A) = 1," and I see that Aristotle first defined it as a principle, but later Russell and Whitehead derived it as a theorem. Of course, since it's a theorem in a system-by definition provable from simpler principles-it is no longer a belief. However, my point is different: you must start by accepting that "true" (1) and "false" (0) exist, and these are what I refer to as beliefs. In Principia Mathematica, Russell and Whitehead developed other axioms, but the truth values are not provable within a system that relies on them. They are fundamental assumptions or axioms. Without accepting these basic truth values, the structure of Boolean algebra cannot be established. I doubt you can prove within the system of Boolean algebra that 0 and 1 exist. If you can, then you would have to have started from another set of principles. Furthermore, you have to accept that certain operations you do with 0 and 1 also exist. While we can prove theorems within the system once the axioms are accepted, the existence and definitions of these operations are axiomatic. For example, the operation NOT(A) presupposes that such an operation makes sense and is well-defined within the system. The most satisfying line of reasoning from first principles that I have seen to justify the existence of numbers is by using set theory. The empty set {} is defined as the number 0. Why does it exist? Well, you can semi-butcher Descartes' statement "I think therefore I am" to conclude that at least one thing exists-you. So, you can define the set {you}, and from that derive the number zero-{}. The number one is just the set containing 0: 1={{}}, and each number is the set containing all previous numbers: 2={{}, {{}}}, and so on. You can define normal algebra operations with similar reasoning. The beauty is that the isomorphism between the sets and numbers exists! You can see how the construction naturally contains a sense of "orderedness" (Is 1 < 2? Yes, because the set 1 is contained within the set 2). However, there are several assumptions here! You have to believe that there's a set you can form around objects and accept that it's possible to remove objects from a set. More simply, you have to accept that it makes sense to draw and redraw boundaries (create categories) of objects. This is not clear to me how it can be derived from simpler principles. To follow your argument that usefulness is application-dependent and ultimately subjective, "Are physical systems even relevant to show whether 2+2=4?" My instinct is to say-yes! Since our brain is a physical system and our senses can only perceive what we call a physical world, it ought to contain a sense of physicality in the first principles we accept as true. Judging by this, we can reasonably extrapolate that aliens (in our Universe!) would also develop similar logic and arithmetic, one in which it is true that "A+NOT(A)=1" and "1+1=2," albeit in a different language. It's unimaginable to us that a world exists in which those statements are not true since they follow from our axioms. Axioms are those statements without which thinking is unreasonable-as much as when one accepts that "1=2," one starts living in a paradoxical and useless world. However, is it a universal truth? Universal as in everything that exists and that doesn't exist? Well, in a world that doesn't exist to us, maybe there are entities for which "1=2." We cannot perceive that and certainly cannot perceive their physical world that is ruled by such a statement, however we can conceive it. In that sense, and in that sense only, do I and Dialect (I suspect) assert that "2+2=4" is not an objective truth-it's conceivable that it's not. The question is then to ask what we can be 100% unequivocally sure that exists. Something that is inconceivable to not exist. And that's what philosophers have been battling for thousands of years without a clear answer. "Why are we assuming that the merging of two physical entities can be described by addition, which is an abstract operator rigorously defined in formal logic?" I should have been more clear with my example! Defining a new 'merge' operation in an existing system and a new system in which addition works like merge are conceptually related. I chose to illustrate the second approach. Much of modern physics I feel is built around *kinda the same operations* on *kinda the same stuff* but each separate thing is defined (rigorously) in a separate area, based on formal logic, and we can then look to our world to see where it is isomorphic to. In all cases where you can see the laws of addition being violated, you can cop out by making a new operation! Or defining a new system in which that operation works differently. The merging of two droplets, counting all their molecules before and after, does follow the logic of addition, because of mass conservation! It is hard to argue with what already we have established to be useful, based on what our brain's psychically allows us to imagine as reasonable. Imagine humans that didn't know that molecules existed and only developed systems to look at big systems. No doubt they would have developed separate operations - addition and merge, to describe big objects or had two different systems of arithmetic. In both cases, they would religiously believe that the statement "we need 2 operations or 2 systems to describe our physical world: 1 for counting merging droplets, and 1 for counting a flock of sheep" was true - until later they could prove one operation/system follows from the other, because of molecules. My hypothetical (probably false proposition) in the same sense is then: is it true that 2+2=4 for each physical system? Or should we drop it in favour of a new operation/new system that only *looks* as 2+2=4 in an approximate limit? I feel like that's not an obvious question to answer straight away.
I appreciate how you presented the Nelson-Grelling paradox before moving on to start to tackle these better-known paradoxes. Together, the presentations suggest that the real issue is understanding the nontrivial implications of bivalency. -Tom
I thought from a long time now that incompleteness theorem was some sort of meaningless artifacts... I'm glad to find a video going in that sense... Thank you 😊
I have been thinking about this problem for some time now and one major conclusion that I have come to is that these problems are snowclones of the stone paradox. The stone paradox is a popular variant of an omnipotence paradox. "can god create a stone so heavy that he could not lift it." Either he could not create the stone or he could not lift it. Can I create a Turing machine which cannot be predicted whether it halts or not. Can I create a set that is not contained in any other set. Can I create a statement that cannot be determined if it is true or false. Can I create a mathematical theorem that cannot be proved. There are two principles that are in conflict with one another. One principle is the ability to create any object. (statement, turing machine, set etc.) The other principle is being able to FULLY understand ALL of the properties of these objects. These two principles are in conflict with one another. Either some objects of a particular type cannot be created or some objects cannot be fully understood.
Qabbalistically speaking, I think it is possible that God could occupy both truth states at once and reconcile posterior contradictions on a case by case basis so that in effect what is left is a "hole" of logical divergence in the beginning, which is "stuffed" by manner of endless reconciliations at the tail. This is very intriguing from a Hegelian teleological / Christian eschatological perspective as it may concern the ultimate nature of Jesus Christ as both man and God.
Hegelianism grounds metaphysics in contradiction at the outset, which is indistinguishable (to observers accelerating at less than the speed of light) from nothing short of religious mysticism. I think the actual "solution" to the paradox lies outside of analytical philosophy altogether, in what living biological systems are already able to do without the necessary "ultimate knowing" possible for-itself. In fact, whether ultimate knowing is possible in-itself is also undecidable, for it involves knowledge of what knowing is. Gödel was not replying to Hilbert inasmuch as he was replying to centuries of Hebrew qabbalah on the subject of which him and his transplanted peers were more familiar with than we might give ourselves the permission to know.
It is perfectly possible to ground formal logic on contradiction. We do it all the time in regular logic. Reality is huge, and consists of reconciled contradictions everywhere. The arbitration of contradiction as being something arbitrary was always more profound to me than the arbitrariness of contradiction itself. In other words, reality is better described in terms of exhaustion and mental process (ultimately culminating in the Romantic language of Spirit, well-dressed by the American Transcendentalists as phenomenal-utility) than everything in Western philosophy prior to 1800 (that was only honest with you insofar as it was likewise siphoning off religious mysticism, Jesus-Logic or Jesus-Energy 😂).
It may very well turn out that unprovability and incompleteness theorems are merely that of semantics. Trivial perhaps in the sense of comprehension, yet the terminology it introduces pivotal to our understanding of other emerging phenomena. The narrative construct of reality appears to be strongly at play here. Your work is a subject of deep personal gratitude, it is hard to express the relief it brings when one sees someone else capable of expressing the formerly formless tempests of thought into tractable and tangible form like you have done with this work. 🙏
Thank you for your kind words! We are still in the process of studying Gödel's Theorem, with the hope of bringing a series of videos that will completely walk-through and demystify the proof. But it is certainly insightful to suggest, as you do, that incompleteness may very well be just about semantics at the end of the day, but that semantics is no less important given its pivotal role in how we form theories of our reality.
@@dialectphilosophythat is the most concise and succinct description of my understanding of reality and jotted down effortlessly. You are the @delvetv of my physics! Being on the spectrum makes formulating my thoughts into words challenging, the absence of the visual imagery of general relativity as a flowing river had me stuck. Was it a few things fell into place. It would appear that Claude Shannon is to Einstein what Newton is to Galileo. Each respectively squaring the circle of the other. Notiers theorem cones to the rescue again. Newtonian physics are two-dimensional, and Galileo provided the link on how energy flows from 1D to 2D giving us potential and kinetic energy that is conserved. There exists a similar conservation in space-time with the flow of energy being expressed in the conservation thermal and informational entropy. Shannon providing the link in this case allowing for the interpretation of the 3 expected discontinuities demonstrated in the Heisenberg uncertainty principle, the wave function, and the measuring problem.
Undecidabilty truly exists. The physical meaning of undecidable statements is that their computation never ends. We cannot just look into them and decide if they are true or false, we have to play out (compute) their logic.
This video has helped me get a new understanding. I think of the 'liar paradox' as a natural constructor of inconsistency in a given theory. So for me the three theorems states that their corresponding systems are very solid in the sense we cannot construct a valid proof of a statement that implies a paradox (of the same nature that the liar). The incompleteness therefore seems to be a guardian forbiding access to constructions that can give birth to the inconsistency of the theory.
The problem with comparing the liar's paradox to Gödel's construction to the liar's paradox is that Gödel's construction encodes the formal logic of the system it's written in into the statement. The liar's paradox assumes all of the rules you take for granted regarding English, truth, and more. Imagine if the liar's statement was as long as an English textbook... It wouldn't be such a popular riddle. It's possible with a Gödel statement to reformulate the original rules to include the truth, or even the falsehood of the statement as a given (i.e. as an axiom). The result is a *new* system that requires a new statement to be formulated with the new rules in order to have a "paradox", but then this new statement is different. From the point of view of the liar's paradox, changing the rules of language/truth would alter the meaning of "this statement", since the statement changes when the rules around it change. This is what happens with things like the axiom of choice (in ZF theory). The axiom of choice cannot be proven or disproven, but we can assume its truth or falsehood and to develop two new systems of logic from those assumptions. What Gödel's proof shows is that no matter how carefully you create your system of logic, there are always statements that are like this.
I have disagreements with what was presented here. You seem to have introduced a new category that is "neither true nor false" and then treated it as if it were "true but not false". Take L to be a liar statement such as: "This statement is not true" (so I don't have to nest quotation marks) "L is true because it is indeterminate" is false. And the statement that it was equivocated with ("There are statements within the system which are true but not provable") does not follow. (Given here so you don't have to scroll up and find it in the vid) L is false L is not true L is True At 11:19, the middle statement does follow from the top statement, but the middle statement is more broad. What you essentially did with that middle statement is taking a statement 'A', and deducing the statement 'A or B'. That is valid; its called disjunct introduction. The bottom statement does not follow from the middle statement. Indeterminacy does not mean 'true'. A contradiction is still produced here though because the bottom statement does follow directly from the top statement without the middle statement being there. A statement being false means that the negation of the statement is true. In the context of allowing for indeterminacy, being 'false' is more specific than being 'not true. L is true L is not true L is false or indeterminant At 11:27, the middle statement does follow from the top statement, but to be more explicit, I'd add in the intermediate step of saying "L is false" and then doing disjunct introduction. But there's a problem here. The middle statement also contradicts the top statement. "L is true" contradicts "L is not true". That makes the top statement false because that assumption lead to a contradiction. You can't just do a proof, reach a contradiction halfway through and keep going, and then compare whether some later line contradicts the first assumption to say that there's no contradiction. For a proof to be valid, the entire list of prior assumptions, statements, and possible derivates of them all have to mesh well and not contradict. To make this clear, call P some proposition. "P is indeterminate" contradicts "P is true". "P is indeterminate" contradicts "P is false". Indeterminacy is a category of its own. Being 'not true' means "false or indeterminate". Being 'not false' means "true or indeterminate". Just as being 'false' is more specific than being 'not true' and it implies that the negation of a statement is true, being 'true' is more specific than 'not false' and it implies the negation of the statement is false.
My favorite formulation of this problem involves a grid of knowledge: there is what we KNOW THAT WE KNOW (provable truths), what we KNOW THAT WE DON'T KNOW (provable uncertainties), what we DON'T KNOW THAT WE DON'T KNOW (unprovable uncertainties), and, at the heart of this entire dilemma, WHAT WE DON'T KNOW THAT WE KNOW (unprovable truths). This category of WHAT WE DON'T KNOW THAT WE KNOW, something we must take as true regardless of our proofs for it, is what psychoanalysis, my favorite pseudoscience of pseudosciences, calls the unconscious. If we ask: "Is the answer to this question 'no' ?", then we get the usual paradox. By answering 'no' the answer becomes 'yes', and by answering 'yes' the answer becomes 'no'. The result of the question is not only inconsistent, but consistent in its inconsistency. An universal answer to the question is impossible because here we meet the incompleteness of a language, we fail to meet a sign that represents inconsistency as such (indeterminable, meaningless, undecidable). On the other hand, if we ask: "Is the answer to this question 'yes' ?", then we get another paradox. By answering 'yes' we are immediately correct, but by answering 'no' we are also immediately correct. The result of the question is also an inconsistency, but in this case, it's due to the opposite reason: there is no way for us to answer wrongly. It is unfalsifiable, pseudoscientific, where every sign ends up representing the inconsistency as such. For psychoanalysis, the first question is an index of the superego (the point of incompleteness, of a demand whose answer has no name, which is impossible to meet), and the second question is an index of the Id (the point of inconsistency, of a demand which we can't help but unconsciously meet). They are the points where every language fails. The true danger in not recognizing these paradoxes is that they exist, beyond formal speculation, in our very everyday lives, and influence us constantly. In politics, the paradox appears in the form of double binds: if your political enemy performs an evil action, it's because they are pure evil; but if they perform a good action, it's because they are very good liars, which makes them more, not less, evil. Ideology starts working when all empirical data that could refute it starts working in its favor (any opposing news can be determined as the control of the enemy over the media). Language becomes a formality that only serves to further one's own goals... but that's not an exception to the rule, it's what sustains the rule itself. A system of language/signs can't exist if it doesn't have those inconsistencies/incompleteness. The strengthened liar (essentially an addition of a third option, a sign for inconsistency, of the indeterminate, in the realm of non-true) is to be avoided, in my opinion, because "this statement is not true" simply pushes the problem down a few steps. It still works for the next best thing: "this statement is true" shows a similar inconsistency, because if we claim that it is true, everything's fine (it's true that this statement is true, because the answer is that it's true), but if we claim that it is false, everything's also fine (it's false that this statement is true, because the answer is that it's false). If, then, we allow for "this statement is not false", we can perfectly say that it is indeterminate. But isn't it a different kind of indeterminate than the prior one? "this statement is not true" is indeterminate because any answer you put will be wrong. "this statement is not false" is indeterminate because any answer you put will be correct. One class stands for neither true nor false, and another stands for both true and false. The necessity for these two classes, even if you call them by different names like indeterminate and undecidable, is the true paradox, not something that can be normalized within logic itself.
Interesting thoughts -- thanks for sharing! There are certainly many, like Tarski as quoted in our video, who certainly believed in the profundity of liar statements. Perhaps if he had made an argument more along the lines of yours, his statements would have been more convincing.
The grid-square of "what we don't know that we know" is pretty well demonstrated by Plato in the Meno. This is where Socrates develops the concept of anamnesis, the idea that people forget truths. It seems about as legit or preposterous to me as @straw egg seems to think the unconscious of psychoanalysis is. I don't really have a problem with either of them ultimately because they are just metaphors. They break down at some point. The problem i have with the seeming urgency of Dialect's rhetoric in this video is that it's rushing against (tilting at windmills if i may be so bold) the static/eternalist (Kantian?) nature of formalism. Human language and human decision is temporal and fallible, i.e. subject to revision. Unlike logic, human language (and what is doable with that language) changes over time. The foundationalist project of reducing language to logical principles fails because of the failure of representational semantics (cf. the culmination of that in Wittgenstein's "Tractatus"), NOT because of Gödel's results. Juries have to decide who's guilty, brokers when to sell, brides who to marry and voters who to vote for at a particular moment. What bears on that moment is often beyond the matrix of logical analysis. (Monday morning quarterbacks, notwithstanding) Detective novels and scientists use the word "hunch" when referring to reasons for action. The psychoanalysts may reify it (post action) into the concept of an unconscious. But you have to admit, the specific reasoning in any truly psychoanalytic case only occurs after many hours perhaps years of teasing out a narrative of reasons versus rationales, which takes such a specific course that there is no deterministic outcome, no set of logical, necessary rules. The projection of these kinds of logical paradoxes discussed by Dialect onto human endeavors is the gesture i would object to in any argument about "what the results of the liar's paradox means." I think Dialect in this video is arguing with the way that people have used this result of logical investigation as a metaphor for human affairs, or physical theory, e.g. explanations of the Uncertainty Principle. And that metaphor just breaks down if you examine it closely, just like any metaphor does.
I want to ask about your footnote at 11:37, which I will copy here in full: "The traditional presentation of the Strengthened Liar takes the two mutual assertions listed above to be proof of a further paradox, arguing that a given Liar statement L cannot have the truth values "true" and "indeterminate" at the same time without contradiction. However, the statements "L is true" and "L is indeterminate" have somewhat different contexts; the former asserts the fact that L does not belong [to] the true category, while the latter asserts the fact that L does belong to the indeterminate category. These truth values not being mutually exclusive, there is no contradiction, and the strengthened liar is not actually in any way "strengthened". This reasoning, as will be seen, becomes crucial to the thinking that underpins the metamathematical considerations of the incompleteness theorems of Gödel, Turing, and Tarski." Your second sentence seems wrong to me. Let me unpack my problem with it. You identify two statements, which I will call "P" and "Q". P: "L is true". Q: "L is indeterminate". You then go on to say that P says that L is NOT true. (Specifically, you say that P "asserts the fact that L does not belong [to] the true category"). Actually, P is saying the opposite. P is saying that L IS true. So P definitely does contradict Q. While yes, you could unpack L as "L is not true" and then re-write P as P: "'L is not true' is true." You could do this ad infinitum. You seem to have made an arbitrary choice to only unpack L's self-reference one time, for the sake of your argument. You danced around the contradiction between P and Q.
The OODA LOOP: Observe, Orient, Decide, Act. Incompleteness and uncertainty are hardwired in reality. Nothing is ever complete or fully known, this is why recursion and feedback loops are key in enabling man to survive the reality in which we all share. Excellent post for sure.
A linguistic statement, such as a mathematical formula, asserts (explicitly or implicitly) a specific relationship between two different [categories of] things (a subject and object). A linguistic statement that asserts a "relationship" between itself "and" itself is simply "useless" -- i.e. has no "utility", other than, perhaps, to demonstrate the "incompleteness" of its rules of grammar/syntax as not specifically and explicitly disallowing such statements as "invalid" (or at least "undefined", e.g. 1/0). It's a bit like writing a computer program that does nothing more than print out (or overwrite itself with) its own code -- what's the point / "purpose" / functional utility of that -- other than to expend CPU cycles -- or demonstrate the "trivial" case (e.g. "1 = 1" or "This is a linguistic statement.")? SHAMELESS PLUG: Check out (read carefully -- it's a work in progress -- and thoughtfully) my "Anthropos Cartographer" meta-philosophy (a.k.a. 'Experiential Meta-Mechanics'; a.k.a. 'The Quantum Philosophy'; a.k.a. 'Networkology' ) in the comments at "What is a Worldview" (ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-AVAmY1HvExI.html) Thanks
>A linguistic statement that asserts a "relationship" between itself "and" itself is simply "useless" Watch this: P -> G is unprovable G -> P is true. Two separate statements, not a relationship between a thing and itself, those are two separate statements. You might say, "but but this 'trivial' statement has no bearing on and no consequence on any other statements other than itself, thus its is an isoated incidence" But this is wrong. This relationship of P and G, can be embedded into other logical-relationships as a subgraph minor, and can emerge OUT of other logical-relationships as a subgraph minor in an emergent and non-explicit sense. Thus it will affect an infinite variety of relationships, nontrivially. And it will be unpredictable when this is a subgraph minor of some logical relationships because of emergence. Ie, something as simple as "every even number is the sum of 2 primes" can have this relationship embedded inside it deep within its emergent consequences, you can't tell.
An intuitionist's answer would be if you want to answer whether something is true you need to construct it. Since the Gödel statement explicitly says it is not provable (there is not theorem-number for this statement due to the diagonalization), it cannot be true from an intuitionist standpoint.
As Godel said, the crucial thing about his theorem is that it's 'not a logical paradox but a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics).' The Godel-sentence corresponds to a completely concrete statement to the effect that there is no number whose decimal representation meets a certain list of concrete criteria. You might want to read a rigorous general-audience book on Godel's theorem like Torkel Franzén's 'Gödel's Theorem: An Incomplete Guide To Its Use And Abuse.'
Gotta love the fact there is no definitive answer to this set of concepts. Turing’s halting problem say it all as one cannot predict the future. Great video to be sure. In indirect fashion, this relates to complementarity as you must connect seemingly opposite notions to see the complete picture. Take Zoe and infinity or light and dark or better yet, the Yang and Yin symbol of the Tao. Keep posting, you have something significant to say on all things regarding reality. :)
I am ( and as I look back on my life have always been) a DEVOUT skeptic. Your videos talk about many of the issues I always have (and still do BTW) struggled with. Thank you for your efforts. When I have consumed all of your inputs, I will have to process all of them, as a whole, before I can even think (at more than the surface level) about their implications. Thank you for your service.
This commonly misunderstood paradox exposes the catastrophic problem of self-reference which forms the sand-like foundation upon which the entire apparent universe rests. That is, the inherent presupposition (or assumption) of truth. Take any self-referential statement and it is bound to have an unfounded assumption of truth baked into its premise (“this sentence is…”) which depending on what is ultimately asserted (“this sentence is false”) creates a paradox. Self-reference assumes the truth because it has to, there is no other option, and so it is unable to judge its own reliability without first presupposing it. A prime example of this is the incomplete system of mathematics which hides its fatal (self-referential) flaw behind smokescreens of technical jargon it uses in order to “proof” itself true by itself which from the get go is assumed to be true (ie the “self-evident truths” or axioms of math). No amount of math however will change the fact that it is down-right impossible to prove the validity of 2 without first making the unreasoned assumption that 2 exists. Rather than dismissing the notion of truth altogether, the incoherence of this paradox appears to place truth outside the reference of “self”. In other words, truth is not (nor can be) self-evident. What exactly does this mean? Firstly, it means that so-called objective knowledge (in and of itself) is an enigma - analogous to subjectivity (because some self said so). While objective knowledge is assumed to have a one-to-one correspondence with reality, the truth of it can only be judged from a standpoint outside of itself - that is, independent of the mediating mind. Is that even possible? Yes, because you are NOT your “self”. There is a self reading these words that “I” call “you” and “you” call “me”. It is a caused fact existing in three dimensional space and passing through time, manifested as perception and conception. Its purpose is to generate the world-for-me (a massive collection of apparently isolated objects it calls “things”) from the “thing-in-itself” or that which representations are of. It is bound in experience to self-reference, forced to rely on tools (sense, language, thought) to describe, understand and manage the apparent world of “things”. The truth of what anything is, however, is ultimately a complete mystery, with one exception. Beyond the self-generated world (the insatiable, thinking, wanting, not wanting self) exists the one thing-in-itself that I have direct inward access to, that I can be, that I am - consciousness - the ultimately ineffable experience in which exists no separate facts, no space, no time and, ultimately, no difference between me and the rest of the universe - the state of being ‘I’ call ‘I’. In being conscious, I experience truth independent and free of self-reference.
The Liar's Paradox can be easily fixed: The *Liar's* "Paradox" Logic deals with arguments. "This statement is false." ....isn't a valid argument. It's a statement. If we were to look at it and ask if it is a valid argument or not, we would find it's not valid. It's a conclusion without any premise claiming to support the conclusion, at best. So, it would be given a truth-value of "unknown", instead of true or false. Meaning: the video is spot on saying that the two statements are ostensibly the same. This points to something much deeper....
The difference, however, is that the strengthed liar is a valid conclusion when used with a premise or set of premises that support a claim. The "paradox" is giving a truth value to the argument being presented (the theorems mentioned). It's a metaphysical means of finding physical evidence for truths. In other words: Support I come up with a theorem that sounds ridiculous and, scientifically speaking, is absurd. If I claim that my theorem is purely conjecture, it will likely be ignored. However, (while unlikely with me being the author) the same theorem may contain some new way of viewing things that could lead to a deeper revelation about reality by others. .....Then again: the truth of everything I've posted isn't provable. 🤷♂️
"True but not provable" is misleading. The statements are undecidable rather than true, and one may construct a consistent system as easily by declaring them "false" as by declaring them "true" even if the "true" interpretation seems more intuitive. For example, "a number x satisfies x*x=--1" is undecidable in conventional or real arithmetic, and we may intuitively declare it false, but we may also declare it true and develop complex arithmetic, and complex arithmetic has practical advantages like the fact that all polynomials of order n have n complex zeros. In this case, the undecidable proposition is not like "this statement is false", but if complex arithmetic is consistent, then the "existence" of the imaginary number i must be undecidable in real arithmetic because complex arithmetic only extends real arithmetic by adding the existence of i axiomatically. Hofstadter's "Godel, Escher, Bach" doesn't purport to explain consciousness or anything similarly metaphysical in terms of incompleteness that I recall, and it denies that Godel's undecidable proposition is "true" in any meaningful sense by positing a system of "supernatural numbers" in which the statement is false. Hofstadter is being playful rather than mystical by calling the numbers "supernatural". Complex arithmetic similarly extends the "real numbers" with "imaginary numbers" to construct "complex numbers". All numbers are artificial abstractions, so all are in some sense imaginary. In terms of consciousness and the like, the book is more concerned with self-reference than with incompleteness per se. Hofstadter does not make the Penrose argument that I recall.
Godel developed a 3-valued logic for this purpose. Nowadays we know RM3. What the theorem REALLY says is Incomplete OR Inconsistent. It turns out you can have Completeness! No worries. You need to learn to live with inconsistency but you already do. You use three-valued logic every time you go through a stoplight. Turns out that intermediate, inconsistent, both true and false (different from neither true nor false!), third truth value is quite useful, not only for unraveling the Liar, but also Paradoxes of Relevance. (They are often called "informal" fallacies, because "classical" logic can't see them. You need three-valued logic to solve them.)
If you have any inconsistency in a logical language, you can trivially deduce any conclusion you want to arrive at. You may as well throw the whole system out the window.
@@Elrog3 No the entire system doesn't need to be thrown out the window if shown to be inconsistent. There are systems that are wholly inconsistent with each other and yet are useful for certain applications like modular arithmetic would be wholly inconsistent with normal arithmetic without either making a distinction in your operations or developing another system that can underly the mechanics of both.
@@christopheriman4921 Two separate logical languages being inconsistent with each other isn't the same as a logical language being inconsistent with itself, which is what I was talking about.
@@Elrog3 I know but if you look at any inconsistency in a mathematical field as another potential useful mathematical construct I can see how math in general could be wholly contradictory with itself and still be useful.
@@Elrog3 Sorry for the lateness of my reply. Your definition of logical language is the problem. You are assuming binary logic. I said RM3, it's not binary. There is no triviality.
To me, I see to problem with these statements being trivial. That is their purpose, to trivially show an inconsistency with the axioms of a given system. Their purpose is explicitly not to prove can be undefined, but to disprove that something can't be undefined. Perhaps one day we will come up with a structured system that does not have this problem, but to me these statements serve their purpose correctly. After all, you only need one example of something existing to prove it exists. These models are black swans, if you will. Obvious, even trivial contradictions to the idea that all behavior in the system they are created within could be defined or evaluated.
Thanks. I lack the mathematical / logical understanding to make much of this sort of thing one way or another. I do enjoy watching / reading about it, though. However, I can say that it teaches me something important. It teaches me about the limits of any definable system, which has application to life in general. In designing systems, one simply cannot make them perfect; there will always be possible events that lie entirely outside the system's parameters. I think the best we can do is to design simple, flexible systems, and get good people to use them; people who are aware of the system's limitations, as well as their own. Unfortunately, humans - and the systems they've put in place - hate this sort of thing. Bureaucracy, hierarchy, pecking order, rule of law... People don't like others who step 'out of bounds', or overstep their authority. Humans have very strict, pyramidal ideas about such things, which go back to tribal, feudal, military, city-state notions that are now thoroughly outmoded. But explain that to a prospective employer, & you won't get the job! ;) tavi.
I find the presentation intriguing. But I wonder what is being presented here. Are these Dialect's own thoughts? Or is this a condensed presentation of peer-reviewed work from multiple experts in the field? After all, while I do have some familiarity with the discussed concepts, I am by no means on a level where can I confidently asses the correctness of what is presented here. For that reason, I will put significant value into the opinions of experts on this subject matter and I'd appreciate it if Dialect could share any references to such work.
These are our own thoughts, presented after a great deal of study of the topics. However, this work should be considered "investigative" not "conclusive" as we are still looking into the theorems and their meaning. The problem you will find with the "opinion of experts" is that they offer little consensus about what the theorems actually mean. The theorem itself is so esoteric almost no one understands it, and those who claim they do argue with one another about differences in meaning. To us, the wide divergence in professional interpretations is a big red flag that something is at issue with the theorem. As to resources, in this video we consulted the works of Gödel and Tarksi directly. The English translation of Gödel's Incompleteness Theorem we sourced from www.jamesrmeyer.com, and Tarksi's "On the Semantic Conception of Truth" can be found in multiple places via a quick googling.
@@dialectphilosophywho are you behind Dialect? The endings of each of your videos seems very mysterious, hopefully your not some type of esoteric philosophical cult or string theorist grifters 😂. The way you tease a higher understanding to these deep questions of the universe is intriguing. When is the other shoe going to drop?
Try the example of Russell's Paradox. Set A = {the set of all sets that do not contain themselves} Suppose we define two other sets. Set A1 = Every set that is in Set A but not Set A itself. Set A2 = Every set that is in Set A and also Set A itself. Both of these are perfectly good sets that you can do whatever you want with, with no paradox or problem. (I guess, I haven't proven that.) So the problem is that any definition you can make for a set is supposed to be workable. And the general case of Russell's Paradox is: Set B contains element x and also does not contain element x. Why is that supposed to work? Are we supposed to have a language which does not allow us to create self-contradictory statements?
Coincidentally, I just started on Davidson's _Truth and Predication._ Are you considering covering Tarski's semantic conception of truth, since it does take the self-reference out of the picture? I'm usually with most philosophers in barring The Liar from being a truth-bearer, and just treating it as indexical.
You need to up your game, buddy. The purpose of lying is to deceive. Any competent liar would have the ability to lie with only truth statement. In more mature environment, where competent liars breeds competent truth detectors, liars morph into … something else.
@@chamorvenigo3128 @Patrick Chuan To clarify: I'm using "The Liar" as shorthand for The Liar's Paradox, following the convention of the video - nothing to do with deception. Truth-bearers are things which can have a truth-value (TRUE or FALSE), and an indexical statement is one whose truth-value relies on when, where and/or who uttered it.
@Eta Agreed. Liar is a syntactic paradox that manifests strictly across lines of a proof. No one who uses a language would use it for anything. I get why Dialect is targeting this, if only because of all the hype around Incompleteness. But really it's just a result that a very niche community used as a kind of self-corrective of foundationalist excess. WGAF? That said, i think T-sentences would be interpreted by Dialect as the "strengthened" Liar. I"m totally with following Davidson in terms of philosophy of language, which makes this all a tempest in a teapot. Particularly the last paragraph of "Nice Derangement of Epitaphs."😂
I'm sorry, but the argumentation in this video is, unfortunately, very problematic. While the liar's paradox uses a sentence that directly refers to TRUTH, the statement that is proved by Gödel very clearly does not do the same. Instead, it refers to provability - a concept that can be formalized, in contrast to truth, which cannot. The sentence "This statement is unprovable" is a very rough paraphrasing of the actual statement that was proved by Gödel. A much better rendition would be something like: "There exists no number P that is the encoding of a proof of the statement with number S." where S so happens to be the number encoding of that very statement. The important part here is the ENCODING, which is something that is specific to a particular logic (a particular set of inference rules plus a particular set of axioms). So the statement says that in this particular logic there is no derivation starting with only axioms, only following the rules, and ending with our statement. If we ever find such a derivation, then we would have uncovered an inconsistency within the logic. But if the logic is consistent, then a derivation (i.e., a proof) will not exist. Notice that, therefore, the statement is specific to the logic and conditional on the logic's consistency, the latter being something that can also not be proved from "within" that logic. In other words, the truth of the statement is very much a meta consideration that happens outside the logic in question. The logic itself, and the proof, only talk about the existence of (encodings of) proofs. Proofs are concrete mathematical objects with a clear definition (which makes their encoding possible in the first place). There is nothing wrong with Gödel's proof, and similarly, nothing wrong with Turing's on computability nor with Church's earlier analogous result about the lambda calculus. In this video the entire argument seems to hinge on a deliberate conflation of truth with provability. The whole point of the Incompleteness result is that these two things are very clearly not the same. Finally, a question: If the strengthened liar's statement "This statement is not true." is undefined/indeterminate/whatever BUT NOT TRUE, then isn't it clearly true because it "correctly" states that it is not true? Isn't it then still a contradiction anyway? I don't think you can wiggle out of that one so easily.
I do think self-reference paradoxes are telling us something profound, and that is that any set of abstract axioms will always contain a contradiction. That is why we cant simply deduce everything from pure thought, we will always have to run experiments and observe the real world to confirm whatever we think.
I would assert that math is not the God that we thought it was. Faith is back in the drivers seat folks. I'm encouraged buy this channel. Finally some folks are applying rigor to the rigor assumed to be in math and science and coming up with sensible explanations.
> Formalism treats math like a religion because everything derives from some small set of axioms what? > The incompleteness theorems prove (for Godel) that math is just another kind of language game Yeah right, for Godel the Platonist. If anything, the whole program of Hilbert was nominalist in nature. Hilbert was the one who believed math wasn't grasping any real truths. Why not at least mention the obvious possibility that the point was just to debunk any kind of formalist or logicist program. Ie. to debunk the idea that math can be done in an entirely formal way of axioms and their deductions. > statements which have no empirical content are intuitively taken to be (essentially) meaningless Now you're saying that everyone's born a logical positivist? > This is only because those statements which are true but not provable are in fact only statements about the meaninglessness of other statements Preposterous. The whole example all of this is working with is self-referential, it's explicitly not about other statements. Anyway, I also don't think the footnote makes much sense. Yes, the statement being not-true is consistent with it being indeterminate, obviously. But the point is that this agreement of the content of the claim (that it is not-true) with its indeterminacy, itself makes it true, which is the opposite category. Just how the agreement of the claim of the basic statement (that it is false) is in agreement with its falsity, thereby making it true. My personal opinion is that the details of Godel encoding matter more than they're given credit for. If these claims are legitimately constructed out of axioms capable of basic arithmetic, then that's that. What doubt is there? And, if there's no other problems or objections, then that's that. There are true unprovable statements in certain formal systems and that's that. At most you can peddle this empiricist rubbish about "where is muh sensible content" but that's it. What reason even is there to believe this? In this essentially Humean idea that absolutely everything and anything we think about is reducible to sense impressions, and if it isn't, well, we just don't think of it! Can't people see how ridiculous this is? To be frank I don't really see how truth is essentially replaceable with provability. They're clearly pretty different. But even when it comes to the liar's paradox I don't see the issue with just saying that it is a true contradiction (the statement is both true and false). The insistence on the opposite just stems from a highly dogmatic commitment to the law of non-contradiction. One which, ironically, isn't a claim of any empirical content either.
Godel also explain that the undecidable statement "This statement has no proof" is a true statement. The issue is that it cannot be proven within the system if was stated in, and in fact all such statements become decidable in a type higher than them is stated by Godel an (infamous) footnote 48a at the end of his article. No "strengthened liar" is needed, so the theorems of Godel need not drop the assumption of the law of the excluded middle. A more interesting idea that appears in Godel's article is "w-consistentcy", that there are consistent systems where you can make a proof for the validity of a statement of a single number, but the universally quantified "This statement is true for all natural numbers" is false (Think about the current situation with Golbach's Conjecture to get an idea of how this could possibly be- although we do NOT know that golbach is an example of w-inconstencey). In such a system, Godel's theorems may not hold even though they the systems are consistent.
Yes, we read that same footnote. The "higher system" which "proves" that the undecidable statement is true is exactly the strengthened liar argument, and Gödel states quite clearly in his introduction. Without applying the reasoning of the strengthened liar, nothing proves "this statement has no proof" is a true statement, and so you'd still be stuck at the paradox of the liar. The theorem does require dropping the law of the excluded middle, as Gödel clearly acknowledged.
"The meaning of a word doesn't change if you say it in English and then translate it to French." In the colloquial sense sure, but rigorously they not only change but it would be extremely unlikely that two words carry identical semantic meaning across languages. Or to put it another way using the example of text embeddings if we we're to convert the semantic meaning of words into a high dimensional vector space that relies on the relation it has to other words in that space all words wouldn't just have a unique relation to other words but the semantic space as a whole would be constantly changing and morphing into something else like natural languages have always done. But I only bring that up to say that there is a fundamental difference in stating something colloquially and stating it using a formal system of logic. To use the example of the text embedding again what we usually do is take a huge corpus of text and DESCRIBE the relationship of all words in that text to obtain the semantic relationships. However in a system of formal logic the intention is to PRESCRIBE those relationship by definition. They wouldn't change like natural language changes and it wouldn't matter if it were a person who speaks English or Chinese or whatever other language that is interpreting that formal system. I'm not saying that is actually the case but that is the intention. Or at least the intention is to do it prescriptively using the best methods we currently have and if anyone were to come up with better way we would then convert to those methods. So on the surface, the sentences "this statement is untrue" and "this statement is unprovable" might appear similar because they are both self-referential. However, their deeper logical and philosophical significance diverges drastically. "This statement is untrue" forms the basis of the classic Liar's Paradox. If the statement is true, then it must be false, leading to a contradiction. This paradox challenges the coherence of truth values in a self-referential manner. "This statement is unprovable" (often associated with Gödel's first incompleteness theorem) is fundamentally different. If the statement is true, it means that it is true but cannot be proven within a given formal system. This isn't a paradox in the same sense rather, it points to the limits of formal systems to encapsulate all truths derivable from their own defined axioms. This goes for any formal system. Of course we can ask whether any given statement isn't just a nonsensical collection of words but then it becomes a matter of asking what makes a collection of words sensical. USING THAT EXACT SAME LOGIC AND MEANS of describing what makes a statement sensical or not is also the same system that ultimately shows the incompleteness theorem. You can't have it both ways in that sense.
I'm a little confused by your analysis of the strengthened Liar, particularly when you claim that a contradiction can be avoided by saying that "this sentence is not true" is true because it is indeterminate. If it is indeterminate then it is not true, but it is true by stipulation, hence it is both true and not true, and that is a contradiction. The formal definition of a contradiction is any statement of the form P and ~P. To see that we can derive a formal contradiction with your proposed solution, let T be the truth predicate and l be the strengthened liar sentence, ~T(l). Then if we assert T(l) we get that l which is just ~T(l). From this we can infer T(l) and ~T(l), and a formal contradiction is reached.
At 11:19, false does not imply not true. You cannot say “This statement is not true” is false implies “This statement is not true” is not truw, because that implies that the statement could be either false or indeterminate.
Yeah, once you move to a three-valued system things get a little messier. We used "false" in that scene in a meta-sense. That is, we have two levels of language going on here. You have the liar's statement, 'This statement is not true' and then you have the meta-statement, " 'This statement is not true' is {true/false}". It's assumed that although the liar's statement adheres to a 3-value logic, the meta-statement does NOT. If you attempt to treat the meta-statement as also adhering to a three-valued system, then the strengthened liar argument has a weird gap where you can say " "This statement is not true" is false" without contradiction. We aren't sure what this would imply for the strengthened liar and Gödel's reasoning, but certainly you could still build the meta-statement "This statement is not true" is true" likewise without contradiction.
The strenghtened liar paradoxe is based on the fact that you accept the law of excluded middle which is not the case in every logic but is indeed used in the classic logic. In the end logic is just a set of rules, these rules even as trivial as they seem can be rejected.
@@dialectphilosophy @dialectphilosophy you said accelaration is not the solution of twin paradox then what's the real solution of twin paradox?you stopped making video on it and never told the real solution of twin paradox?
@@dialectphilosophy @dialectphilosophy you said accelaration is not the solution of twin paradox then what's the real solution of twin paradox?you stopped making video on it and never told the real solution of twin paradox?
You don't understand, for example in the halting problem, saying that the halting problem contradiction program is a "trivial program that does not have any significance about the vast set of all computer programs", is not accurate. I am going to loosely use the term "subgraph minor" in a very generalized sense, in that, in the context I will be using it, it will mean that a certain program A is embedded inside another program B, and part of its computation involves A. I am not using the phrase "subgraph minor" in its usual sense, I abstractly generalized it and hopefully you can derive its generalized meaning. Call the hypothetical program that is supposed to decide of another program halts, halts(). Call the halting problem program H, which says "H = if halts(me) is true, dont halt, otherwise, halt". I can embed H as a subgraph minor in any other computer program, in fact, I can show that it can **emerge** as a subgraph minor of any other computer program unpredictably. You don't know if H emerges somewhere in the computation of the program, you don't know. A computer can't tell. It can be anywhere. Because H can be explicitly, or emerge, as a subgraph minor of an infinite variety of computer programs, then H affects an infinite family of computer programs, and thus prevents computation on an infinite variety of tasks. Far, far, far, from trivial. This H can be embedded into infinite computer programs and you won't be able to tell it is there -- especially because it can appear through emergence. Call the set S the set of all computer programs that are either H, or have H as a subgraph minor (either emergenty or explicitly). The real kicker is that you can show that H, if existed, would allow computers to output more things not previously producible from the set of programs in the complement of S. Meaning that computers cannot output certain sequences of 1s and 0s. These sequences of 1s and 0s that a computer cannot output are *dense/equidistributed* in the set of all possible strings of 1s and 0s, meaning, these sequences are everywhere unpredictably. There are holes EVERYWHERE in the set of all things that a computer can compute. A computer can't know where they are and because it is dense/equidistributed in the set, it will affect computers uniformly over the vast distribution of all things that are may or may not be computable. Don't believe me? One way to see that a theorem is in correspondence with reality is if the direct relevant consequences of that theorem are also implied by other previously known theorems. The relevant direct consequence of the halting problem in question is, "computers cannot output certain sequences of 1s and 0s" But first, let's backtrack: The whole task of the halting problem was to disprove the idea of a kind of Generalized Strong Turing Completeness. Turing Completeness means that any machine in question is isomorphic to a Turing Machine. But what does a Turing Machine do? We hope it means it can output anything, and so we can define Strong Turing Completeness: A machine possesses Strong Turing Completeness if it can output any infinite sequence of 1s and 0s. We know that a Turing Machine can output infinite sequences of 1s and 0s, take the digits of pi or e for example. But the question is if it can output all infinite sequences of 1s and 0s? The answer to this question is false, via uncountability. The set of all programs of finite length is computationally enumerable and is a countable infinity. You cannot provide a bijection between the set of all computer programs and all sequences of ones and zeros. Thus we have a failure of Strong Turing Completeness for any machine. Here we have shown that the consequences of the halting problem theorem, namely that computers cannot output certain sequences of 1s and 0s, is also directly implied by yet another theorem about uncountability. What's more, the statement "computers cannot output certain sequences of 1s and 0s" is logically equivalent to the theorem of the halting problem, they both imply each other. Proof left as exercise to the reader (think Busy Beaver function). Thus, uncountability also proves the halting problem theorem. Even though they are different results. I can go further and continue proving the statement "computers cannot output certain sequences of 1s and 0s" with other ways too, even besides uncountability and the halting problem, there's the Compression Theorem, which also implies it, and the Compression Theorem is a *combinatorial result* which doesn't rely on "philosophical handwavy ideas", combinatorics is the least hand-wavy thing in mathematics itself. Or even the uncomputability of Kolmogorov Complexity proves it too. The list goes on... ---------------------------------------------- As for the Incompleteness Theorem, one way to know it is true is if we have *seen its affects already upon us*, and we have: We already have empirical evidence that it is correct, take Goodstein's Theorem, Goodstein's Theorem is a true result in mathematics that is unprovable in Peano arithmetic. The Incompleteness Theorem already predicted them, and that is one. Here is another Paris-Harrington principle, or the Kruskal's tree theorem. These are all theorems that were unprovable in some axiomatic systems, but we managed to find other stronger axiomatic systems to prove them eventually. This was all predicted by Godel's Incompleteness Theorem. Another commenter said: >One explicitly constructable unprovable statement that might be essentially unique or might be just one amongst an infinitely vast and unconstrained set of statements and that's all the info we have. I actually have the exact answer to your question. I have thought about this for YEARS. Take P -> G is unprovable G -> P is true. Two separate statements, not a relationship between a thing and itself, those are two separate statements. You might say, "but but this 'trivial' statement has no bearing on and no consequence on any other statements other than itself, thus its is an isoated incidence" But this is wrong. This relationship of P and G, can be embedded into other logical-relationships as a subgraph minor, and can emerge OUT of other logical-relationships as a subgraph minor in an emergent and non-explicit sense. I am using the term "subgraph minor" in a very generalized sense as a metaphor. It is only an educational metaphor to explain the nature of the concept I am describing. If you look up what a graph minor is, you will be able to understand my argument: This unprovability relationship can literally be embedded into certain other statements and you can't even tell by looking it it. Thus it will affect an infinite variety of relationships, nontrivially. And it will be unpredictable when this is unprovability relationship is a subgraph minor of some logical relationships because of emergence. Ie, something as simple as "every even number is the sum of 2 primes" can have this relationship embedded inside it deep within its emergent consequences, you can't tell. It is far, far, from an isolated incidence. Statements and relationships such as these form a dense and equidistributed set amongst the distribution of all possible statements. A computer won't know where they are, a computer can't know where they are, and a computer will never know where they are. So take that for "trivial". :P Also, to computer scientists and mathematicians, if you want to use the statements which I have said in this RU-vid comment in any papers/research, please cite me and ask before hand :) - Morphism.
That was spectacular. I have been working on these things for two decades and the author and I seem to be at about the exact same point of this. The author seemed about 100-fold more articulate than myself. He seems to have an identical view to mine on Gödel 1931 Incompleteness and Tarski Undefinability. I have additional technical details that support our identical views.
Feel free to share your work with us via our contact email! We've suspended our ambition to do a walk-through of Gödel's theorem for the time being, as we've made our way through it several times and still cannot quite fully penetrate the obscure symbolism of it.
@@dialectphilosophy I contacted you by email and also posted the key essence of the technical details that support our shared view immediately above. My system seems to be anchored in proof theory rather than model theory. Montague Semantics allows this to be extended to natural language reasoning.
This is all just IMHO: Those self-referencing statements can be read in more than one way. It is possible to read (this statement is false) as ((this statement ist false) is false), but it's not necessary to create that recursion. Instead, (this statement) can be an external reference, so that the sentence is out of context. Ok, even if we deliberately go down the route of recursion, the solution is quite simple: It does not exist since the recursion never ends. One cannot read the terms in a recursion as valid states of a system, like intermediate results from a calculation are not the solution of it. It's a bit like finding out the color of a chamelion by putting it on top of another chamelion. Thanks for interesting content like this. Love it :)
Correct. It is indeed not even all axiomatizations of arithmetics: Peano Arithmetic and Robinson Arithmetic are affected by Gödel incompleteness, while Presburger Arithmetic is not. Boolean Algebra, for example, is both complete and consistent (and also decidable).
@@Elrog3 It is probably the most widespread misinformation about Gödel incompleteness theorems (and YT video titles and/or thumbnails like "The Man Who Broke Math" really don't help). But it is easily dissolved by noting that all theorems have hypothesis, which must be true for the results to follow. Thus, Gödel's incompleteness applies only to a subset of all formal systems, and this was expected. The "surprising" part, historically, was realizing that this subset was much _larger_ than initially anticipated.
Binary logic is a useful but very "limited" construct. Analog also has its (human) limitations but does appear to more closely mimic the universe. Everything in the universe exist somewhere in the balance between either -∞ and +∞ or 0 and +∞ to infinite degrees of precision. Maybe the best we can do is create an approximation with fuzzy logic and remind ourselves that there is no "ultimate" true or false.
"This program does not halt" is not an adequate summary of the church Turing thesis, and the proof of the church turing thesis does not rely on the liar's paradox. Turing machines can be implemented in the real world, in hardware, so you can't claim the contradiction at the heart of the proof of the Church Turing thesis is "purely semantic", when this "purely semantic" phenomena is isomorphic with physical phenomena!
There is no proof of the church-during thesis; it is an assumption that allows one to cross from the theoretical implications of lambda calculus to real-world computation machines. Nor is "this program does not halt" a summary of the Church-Turing thesis, it is a summary of Turing's The Halting Problem, which relies on more than a few assumptions and methods to demonstrate its self-referential point. It is important to be careful with these distinctions and not confuse such subjects in the study of logic.
@@dialectphilosophy You're correct in that I used the wrong term. The Church Turing thesis is not the halting problem. But aside from the misplace vocabulary, the argument still stands. All Turing machines definitely either terminate, or do not terminate. And Turing Machines cannot decide whether they do. This isn't a semantic thing. termination is very well defined. It doesn't make any sense to claim that the halting problem is "purely semantic".
Is this going to be linked to the relativity series or is it just a separate topic you are exploring ? I am only interested in the relativity stuff and not anything that won’t link to that.
I definitely remember from my time as a philosophy student that Brouwer and his school of intuitionism considered Gödel theorems utterly trivial and I tended to agree 'intuitively' (hehe) but could you address that in a future video, ty in advance!
Sometimes, in mathematics and formal logic, intuition acts as a shortcut, and gets there first. The "hard part" (and "hard" may be an understatement here) is making it rigorous.
I just think to myself: How would one go about defining something that is imbedded in the language that is being used to define it? That is not possible. To me, this is what the larger scale truth that Gödel is describing, hints at. In my opinion, using language to define language is always going to be circular. Trying to define 3 dimension with 3D math will always end with incompleteness.
That's an insightful remark, and likely at the base of Tarski's ambition to construct a "hierarchy of languages". Indeed, you've struck on what we think the deeper meaning of these incompleteness theorems truly is, which is that every language is going to be stuck ultimately with undefined or circularly-defined terms, and is going to have to source meaning for those terms from elsewhere.
@@dialectphilosophy Thank you for the reply. I have really been appreciating your perspectives on topics that seem to need fresh points of view. This truth, (whatever that is...🤔😉) does not inherently mean that, with properly constructed conventions and rigorous definition use case, they are useless to construct...in my opinion, obviously. Chemistry, and classical physics being prime examples on our macro scale. I think quantum mechanics has a harder time because it is going down the rabbit hole as opposed to climbing a ladder, like astronomy. That is where things get trickier. Not only do you have to already have a rigorous language model to interpret observations with, you also have to invent new conventions before describing what observations are telling us. That is, in itself a type of conundrum... I spent some time pondering the usefulness of phenomenology, only to conclude that meta cognition still requires the material institution of language to be shared, effectively neutralizing it's broader usefulness, except for learning how to learn how to teach... (What a mouthful nested ideas become). My final conclusion comes to this: even if solipsism, or Boltzmann brains, are really the truth, language still needs to be shared. Even if a person creates a technology with ideas that are not themselves shared, that technology will only counter our need for inventing language and therefore work backwards against the institution of language necessary for maintaining an advanced society... 😌 Phew... I'm tired now.
@@dialectphilosophy Other minds: and the allegory of the cave also means that because the onus for learning ultimately falls on the individual, it is up to the elder wise people to broadcast teach at all times... Which is how I imagine a competent University should operate... Then again, it's all arbitrary, and on a relative scale.
@@dialectphilosophy I feel that a good antidote to all this over-hype (of the paradoxes), in my reading experience anyway, is the critique of the notion of a language which comes at the end of Donald Davidson's article "A Nice Derangement of Epitaphs." I can't justify this intuition in a YT note, but perhaps you'll run across that one at some point and find it relevant. I'm really only mentioning it here because there seems to be some slippage in the discussion between the "language" that humans use/create and the formal "languages" that these theorems are in and about. It seems important not to conflate those two.
I’m very fond of your relativity videos, but I’m not sure i agree with you on this subject. We do know of undecidable problems which are easy to understand in natural language. For instance, the tiling problem: the statement “you can tile the plane with a repeating pattern made up from this given configuration” is in most cases unprovable. It seems to me to be a clear case that shows that the undecidability and unprovability theorems are not about trivialities at all. If you Google “tiling problem and undecidability” you can find a lot of information about it which I’m sure will interest you very much. Anyway, thank you for your dedicated work!
Hey thanks for watching! To address your objection, it's important to note that questioning the validity of the incompleteness theorem (which, the more we study it, the more lacking of such validity we feel it shows) does not mean incompleteness or undecidability is not an actual thing -- it would simply mean that dressing up the liar's paradox in mathematical clothing isn't a proof of such incompleteness. In general however, "undecidability" proofs do make us suspicious, since a mathematical proof can contain no more information than the assumptions put into it. Additionally such proofs are often extraordinarily abstruse, highly-technical and only claimed to be understood by a few individuals. Thus, for instance, the fact that experts can't agree on what the meaning of Gödel's incompleteness theorem actually is, is a giant red flag for us.
Does the "indeterminate" option not still contradict the "true" option? Like, indeterminate means we cannot say whether it's true or false. If we /determine/ its truth, then we cannot assign the truth value "indeterminate". So the statement "this statement is not true " would still /truly/ and /determinately/ say that it is not true, if it is indeterminate. Thus, it would self-contradict. Or did I miss something?
Check out the footnote in that section. Some people do argue that "indeterminate" and "true" truth values contradict each other (which is why the paradox is called "strengthened") but Gödel's and others thinking is that these truth-values are not mutually exclusive.
@@dialectphilosophy That's interesting, thank you. I still find it difficult to wrap my head around one statement being both true and indeterminate at once, but I might try to look it up in Gödel's own argument (if it's not too formalised for me :D).
@@dialectphilosophy If the statement says it is indeterminate, and it is proven to be indeterminate, doesn't that make it true? But if the statement says it is indeterminate, but it's proven true, doesn't that make it false? But if it's proven true and false, doesn't that make it self-contradictory? But if it says it's indeterminate and you can't prove it's indeterminate, then maybe it's indeterminate. If you can't prove it's true, and you can't prove it's false, and you can't prove it's indeterminate, and you can't prove it contradicts itself, then you can't prove there's any problem with it.
What about sentences though like "this statement is neither true nor false"? Or even "this statement has no meaning"? Or "this statement is indeterminate"?
I generalize these concepts in the statement that all systems are limited. You could perhaps imagine an infinite system, but to verify everything within that system would require infinite resources, therefore would be practically impossible. It might be possible in theory, granted infinite resources, but that amounts to it being potentially true but unprovable. We can generally always _imagine_ things which we can't actually do.
Gödel's proof is profound because it reveals "the impredicativity of self-referentiality". The implications in understanding epistemological failures of all sorts of modern insanities (most of which have to do with politics, or with legitimate subjects that have been politicized, and thus become immune to reason) are absolutely non-trivial. For context, ready Robert Rosen's _Essays On Life Itself._ My impression of what you are trying to accomplish with this channel is to make subjects that seem counterintuitive more understandable, bypassing the esoteric syntax of formalism and reconnecting the underlying concepts to *_relatable_* semantic content. What Gödel did was to show that pure syntax is ultimately of no practical use. You can't do much with it. He showed that the entire enterprise proposed by Russell was trivial. The philosophical and epistemological implications would be trivial only if the human species understood them so well that we consistently avoided the logical and practical pitfalls they warn against. But we don't. I see it everywhere, every day. Very seldom does such a work of pure logic have such profound implications for what should constitute rational behavior, if only those implications were understood rather than ignored. Stated alternatively, people's inability to eschew the absurdity of trivialities like the liar's paradox is the source of most of the insanity that poses any real existential threats to human survival.
This has been said before, probably in the comment thread, however "This statement is a lie" and "This statement is unprovable" are somewhat different in that one is in the vernacular and the other is a mathematical statement. The formalism of mathematics is not exactly the same as regular speech. "This statement is unprovable" is likely true, in fact, since if it can be proven, it is an invalid assertion.
All this not as mysterious as Dialect appears to make it. If decisions (about truth) are restricted to formal reasoning and representational semantics, antinomies and contradictions occur either as undecidable (over infinite time) or temporally contradictory (alternating truth values in time). If judgments are rendered in real time, they must be passed for reasons relating to human, social requirements. And still, they can be revised or even contravened in time. So anyone hearing, "this statement is a lie" can just stop there and conclude "the speaker is lying." Or go one more step and decide, "yes, that's true." No step of thinking it through produces a contradiction, which we nominally understand as a statement asserting a thing and its opposite at the same time. The whole idea of undecidability, "paradox," halting, or incompleteness relates to a kind of perfection that doesn't transcend the boundaries of mathematical logic as an atemporal infinitude. As far as incompleteness goes, language changes, so no problem there. That there are 'paradoxes' in this realm only show that the way we use natural language cannot be reduced to formal logic. What is maybe more toward Dialect's project is that the mathematical idealization of reality (and perhaps Foundationalism in general) is being critiqued.
This also gave me some pause, but maybe it can be resolved by saying that the first "true" refers to the "true" in true/false/indeterminate 3-valued logic (let's call it true₃), while the second "TRUE" refers to the "true" of usual true/false 2-valued logic (let's call it true₂). And so, the statement "This statement is not true₃" is indeterminate₃, which is definitely different from true₃, and so it is correct in its assertion, i.e. it is true₂.
@@AdenoidHynkelThe2nd You're correct! We regret were we unable to make that fact explicit in the video, but it just took the presentation too far off-track, as it requires a lengthy discussion of language vs. meta-language, content vs. form, etc.
More over Rice's theorem generalizes Turing's theorem to a whole broad range of properties and is pretty important when you are dealing with any type of meta program, whole architectures have been rewritten as a result of Rice theorems. It is pretty important in cybersec... Hop this helps 👍
You can find numerous academic papers arguing that quantum indeterminacy is related to formalistic incompleteness (we showed one in the video). It's certainly an intriguing idea, but we'd recommend caution and critical thinking before jumping to any conclusions.
I agree that the perception of Godels incompleteness theorems suffer from popular mysticism, and very much require more contextualization. I especially appreciate how you compare the incompleteness theorems to the liar paradox, because it is, as you say, at the heart of the issue. I think you could have gone much further and explicitly though, and hope you will in subsequent videos, that the incompleteness and undefineability theorems aren't really at all about the arithmetic systems they are purported to be, but the logical systems in which their arithmetic systems are embedded in. And in all cases what's really going on is that they are able to construct a liar paradox within their system, and use that to prove by contradiction that the attempt to do something in full generality, such as define all the true or provable statements within this system, is impossible. A bit of a slight of hand, surely, because you'd ordinarily conceive of truth or provability as excluding those kinds of statements, and their method of constructing things like 'all true statements' only run into the problem because they start from the set of all possible formula(even malformed ones). I don't agree with your interpretation of the strengthened liar paradox, though. There is a further paradox, and the reason that the strengthened liar paradox is in fact strengthened is that it actually frustrates the attempt to weasel out of the paradox with a third logical category of 'indeterminate': If it is indeterminate, then it is true, which is a paradox. The contexts you're trying to assert which differentiate those statements are the statement being viewed as a whole from the outside versus the self-reference from the inside. The fact that those references seem to take different values illustrates perfectly that the source of paradoxes, or rather the discontent with the existence of paradoxes, is trying to view logical statements as static when the real meaning of them is in their evaluation: a dynamic, computational process. What the strengthened liar paradox demonstrates is that may not be a panacea to extend your logical system with a third truth-value of indeterminate, that it doesn't completely resolve paradoxes. And that should be right, because what indeterminacy really means is that the process of evaluating a statement does not terminate(the word terminate is even in the word indeterminate, terminating evaluation on the value of indeterminate is an abuse of language if not a paradox in itself); it's a statement about the process of evaluating a statement, not the value of the statement. And determining, in full generality, which statements' evaluations will definitely terminate or not is, supposedly, impossible: that is the halting problem. I was comfortable with the halting problem, but you're making me second guess my assumptions(thanks!). What the proof seems to be actually saying is that any program made to determine whether a program halts can be made to return the erroneous(post factually) value on a program which invokes it. That seems like a dramatically different conclusion than the standard interpretation, and depending on what you mean by 'universal' halting, may be perfectly good enough to allow for what you'd call a solution of the problem. It also seems to be more a property of the invoking function, that it can flip whether it halts or not, than the halting algorithm itself(It reminds me of the idea that if you could predict the future, or were given a prediction of the future, you could change it with your free will). On the other hand, you could view it as the halting algorithm which is given the decision of whether to halt or loop the programs which would invoke it to try to thwart it(it could even log, separately, that "I told this function it would die/live forever to trick it into living/killing itself"). And given a real description of the halting algorithm, you would know whether it would return true/false/loop forever in this instance. A situation which is a far cry from 'a halting problem is unsolvable'. Theoretical computer science is a bit divorced from practical application in this regard: as computer programmers, we determine whether our algorithms halt all the time. I have never seen an algorithm that I can't decide will halt or not; the algorithms which invoke halting algorithms seem to be a special class, and the problem should be re-framed as how halting algorithms fail rather than their flat impossibility, because both solving the halting problem practically and the special cases where it fails are interesting. All that being said, I think it's ultimately naive to characterize all these theorems as essentially meaningless self-justifying exercises, to categorize them as gibberish is arguing with bad faith and missing anything actually interesting about them. The reason that the liar's paradox is significant, despite being a patently stupid statement, is that it is the simplest example of a statement whose evaluation results in a indeterminate loop, which is a class of logical behavior well worth understanding. What sets of statements/systems demonstrate similar behavior in a more complex way? Is it always a bug, or can it be a feature? And as to the question of whether or not those systems have any meaning in reality, I would point out that reality, apparently, never seems to terminate... and also references itself ubiquitously. There are isomorphisms(and automorphisms) between indeterminate logical systems(paired with an interpreter) and physical(time evolving) systems. Does that mean anything? Maybe. Seems like it does. It's true at least, and I'd like to see the details more fleshed out. ("from whence'" is redundant: 'whence' itself means 'from where'(and 'whither' means 'to where'. 'Whence derives Incompleteness...' is succinct and beautifully archaic. Keep it up I love your videos)
Thank you for watching and for your very excellent analysis! We're certainly open to the idea that the liar's paradox is significant in some way that is more subtle than what it first appears to be, and if one wishes to argue this, then arguing for the significance of incompleteness theorems certainly gets a whole lot easier. Possibly our issue is more along the lines of the fact that, if more people understood what level of similarity these incompleteness proofs bear to the liars paradox, they'd be less inclined to assign them such wide-ranging mystical interpretations in the first place. And we have by no means made up our minds on this topic, as these mostly are first impressions coming from outside the subject. That said, your analysis of the halting problem and computability strikes us as being dead-on. We have made our way through Gödel's proof a few times already, and want to begin a video series on it, but we still are having a great deal of trouble understanding it and interpreting it. As such, it may be some time before we can come forward with any real significant insights, but nonetheless stay tuned!
"This statement is untrue." The statement is half true since it is a true/false statement it can only have two answers. By taking it out of an expression that only gives two answers ie. half true, both options can be fact. It is true that the statement is untrue hence half true.
Don't all of these systems have a circular nature that have no complete meaning? The real problem here is to try and understand what exactly axioms are and how they provide a structured system while at the same time not being fully and exactly clear on what it is that these axiomatic statements precisely are in the first place. It looks like we accept them on the basis of seeming self-evidence, but it is the 'seeming' that provides the initial solidity of these systems until we look more closely and see its circular nature. What I mean to say here is that meaning isn't exact and I have a hard time thinking that you could find a system that isn't contingent on some type of meaning based structure. In my opinion, this shows how meaning itself is not, and can never be, exact. Strangely enough, we don't seem to require perfect definitions to use these systems which I find interesting. There seems to be objective consistency despite never being able to have perfect exactness in any system. I think this is potentially the more fundamental mystery; the ability for us to make connections and perceive meaning in a consistent and inter-subjective way where we didn't perceive any before.
Am I missing something here? Supposedly... _"This statement is not true"_ is TRUE ...because it's indeterminate, hence not true (as it says of itself). And since it's not FALSE, in which case it would be both true and false, "there's no paradox"... (?!?!) Welp, there _is_ a contradiction: the statement is true and not-true after all. Btw, contrary to what is shown in the video, a contradiction is for two propositions P and not-P to be true - not for a proposition Q to be both true and false (although that will imply a proper contradiction right away, unless we are doing some dialetheistic stunt). Also, an indeterminate or meaningless statement _can't be true to begin with,_ for if it's true, it's surely determinate (as true) and meaningful. I'm afraid the core argument in this video doesn't fly at all.
We addressed this point in the footnote and recommend you go back and pause the video and re-read it. Your mistake is in assuming that "not-true" and "true" are mutually exclusive categories. In three-value logic, they are not defined this way and there is no contradiction with a statement being both true and not-true.
@@dialectphilosophy Thx for the reply. Hope you don't take my disagreement as hostility, as I love what this channel is doing: shamelessly thinking freely about classical questions, without dogmatic respect for what is deemed "settled" received wisdom. Of course, such a project is bound to step into all sorts of mistakes, and the obtuse dogmatic _clerics_ of "standard knowledge" will use that to disqualify and even shut-down free inquiry. Hope you just keep going despite that. But speaking of mistakes, you just said, guiltlessly, that _"there is no contradiction with a statement being both true and not-true."_ But surely you can see why I'm baffled: a contradiction is just a pair of statements P and not-P and the case 'P is true' and 'P is not true' is just a clear example of that. Now to your point, 'true' and 'not-true' might very well not be mutually exclusive categories in (some) three-valued logic - but that just means that such contradiction _can be true_ in that system, and not that there's no contradiction. Btw, I would be actually fine with that, as Graham Priest convinced me that some contradictions are indeed true! The problem here is deeper though as, contrary to what you said above, 'true' and 'not-true' _are_ mutually exclusive categories in your presentation (the Venn diagram used makes that quite clear). I will try to show that while presenting my take on the footnote. Another viewer (LeapDaniel) already found issue with it, in a comment that quotes it, and I agreed with him; maybe you wanna reply him. The footnote: _"The traditional presentation of the Strengthened Liar takes the two mutual assertions listed above to be proof of a further paradox, arguing that a given Liar statement L cannot have the truth values "true" and "indeterminate" at the same time without contradiction. However, the statements "L is true" and "L is indeterminate" have somewhat different contexts; the former asserts the fact that L does not belong [to] the true category, while the latter asserts the fact that L does belong to the indeterminate category. These truth values not being mutually exclusive, there is no contradiction..."_ You are correct that _"the statement 'L is true' asserts the fact that L does NOT belong to the true category"_ (for L itself says it doesn't) and that _"the statement 'L is indeterminate' asserts the fact that L DOES belong to the indeterminate category."_ So both assertions _in that respect_ are consistent with each other: L indeed doesn't belong to the *true* category, obviously, for it instead belongs to the *indeterminate* category. So far, so good. The problem, of course, is that L does belong to the true category _as well,_ for _it is also true._ But the true category and the not-true category (false + indeterminate) are intended to be mutually exclusive in your presentation, just like the true and false categories were in the bivalent example. The category 'indeterminate' is a subset of the 'not-true' category (hence we can have L being both indeterminate and not-true), but it's not a subset of the 'true' category. You can put L inside the 'indeterminate' space in your Venn diagram, and that puts it inside of the 'not-true' space as well; but you can't put ONE L inside both the 'indeterminate' and the 'true' space. That, in a Venn diagram, has to mean mutual exclusion, no overlap. I take it that the thrust of your presentation is that L is at bottom _just indeterminate,_ being inside the 'indeterminate' space in the Venn diagram, so it's not true, just like it says of itself. Of course it can be both not-true and indeterminate, no paradox. But what about it being true as well? It seems the fact that L is true is just being _brushed aside_ as L "being what it _says_ it is, that is, NOT true" which, as we just saw, is no problem! And that much _is right_ as far as it goes. But it doesn't _erase_ the additional fact that L is _also true_ and, therefore, is paradoxically inside the 'true' space as well. That can't be avoided. L can't just be 'timidly true' in the innocuous way of "just being what it says of itself, that is, NOT TRUE" without being fully and problematically TRUE for real. And then paradox ensues. I tried hard to be clear. Did I move you at all? Thx for the attention anyway.
You cannot just doubt the Liar's Paradox because it sounds strange! If you did, you would be able to doubt everything else as well! 2023-09 Edit: The Strengthened Liar is a magnificent way of showing why calling the Liar "meaningless" is incorrect!
@@BelegaerTheGreat If you doubt, that is encouragement to go find out. If you are sure that all food is bad for you, then you won't eat until you are ready to eat food that's bad for you. If you are sure that only white food should be eaten, you will eat only white food until you feel sick. If you don't know what's good for you, you might try various things and see which of them leave you feeling better after you eat them. This is not ideal since doubting that something is a deadly poison can leave you dead if you try it to find out. The first guy who ate a tomato after everyone around him said it was poison, was maybe a fool and I'm grateful to him.
Please humor my offtopic question regarding time dilation and length contraction... the thought experiments we use to support these are about observing moving clocks and objects.. but isn't the act of observation itself reception of light from those clocks and objects, and would hinder the correct result? Also, these concepts arise out of the equivalence principle and the unchanging speed of light, as I understand. The first, I am completely ok with. But the second irks me to no end, although unfairly. I just cannot fathom how everything will contort and delay/hasten to make sure light stays at the same speed, but put that light in water and voila it has changed? I know it is not a logical criticism, but it makes me feel like there is something we are missing there.. I hope you take a moment to satisfy this fool's curiosity ... great videos thank you!
the liar is something else than goedels sentence in my opinion. the liar says something about his own truthness value and when you try to assign true or false you encounter an endless loop. for goedels sentence there is no paradox. he constructed a true sentence in a non-contradictory formal axiom system which says: this statement is unprovable. a true, unprovable statement means your system is incomplete. ps: how do we know it is true? if "this statement is unprovable." was wrong, we hence had a false, provable statement. that would mean our system is contradictory. but the assumption was a non contradictory system.
Anyway!, I like your post' on RU-vid. It's vitamines for the brain. So a big thank for your work. It is done very professionel and i feel it is done with passion and love ❤️
It seems to me that math is a language, albeit a quantitative language rather than a qualitative one. Insofar as there are real and relevant differences between quantitative and qualitative ways of describing the world we can treat them as different; however, insofar as both are languages in the broad sense, both must be subject to the limitations of language, i.e: being vulnerable to semantic meaningless when constructed self-referentially. Thank you for this video helping me tease out this distinction.
Hello Dialect, can you answer me this question? As everyone knows travelling at the speed of light C means not travelling along the Time axis of a 4D space-time diagram. When a body travels at less than C such as a particle with Mass then it moves more along the Time Axis and less along the Space axis. This can be represented with a line that is of fixed length C that moves through the 4D diagram at an angle related to speed. Going along with this math principle then a body in an inertial frame is moving at C. This is difficult for many to accept and understand however as you know this is mathmatically correct. The thing is if we are moving a C in Time then does it not follow that rotating the 4D spacetime we can be observed from another prespective of an observer in Time that we and all that are in that inertial frame, are now seen as massless particles with differing energies. In other words if we are moving at C along the Time axis.. do we become like a photon? and consequently all Massless particles must therefore aquire mass? I have always had the idea that the large scale universe moves away in a progressively increasing noninertial frame and is length contracted back into the small scale Universe.. kind of like a 4D mobius strip, and as such there is a parallel mirror Universe the same as ours hidden in the Time dimension. Indeed is has been said that Time behaves more space like and Space more Time like in Blackholes. So again what do think it mean to you that we move at C in Time?
Disclaimer: I like your videos about physics, but with this video about logic I cannot agree on many points. The Strengthened Liar is not a solution to a paradox. It is an attack to a presumable solution with third logic value. And indeed, it doesn't solve anything: (1) Let Liar always speak FALSE. (2) They then say: (3) "This statement is not TRUE". With the set of logic values of {TRUE, INDETERMINATE, FALSE}, does this lead to inability of assigning some logic value to the statement? First, it is provable, that (4) "(3)" is FALSE. The consequence of (1) and (2). Then, it is provable, that (5) "(3)" is TRUE. The consequence of (1) and (4). Therefore, we reached to the conclusion, that given (1), (2) and (3) we inevitably come to the paradox. And yes, we need to specify "INDETERMINATE" to the same category as other truth values, otherwise it is strange logic with other rules. Then, it was mentioned that we can repeat this proof for "This statement is not provable". But this is not correct, because there is no distinction between "UNPROVABLE" and "INDETERMINATE" from perspective of formal logic. There is a distinction between "There is proof that THIS" and "There is proof that NOT THIS". And Godel's Incompleteness Theorem is about non-provability (a) "There is no such proof which has this statement as the object of proving", because if there would be such a proof, it would be counted as numeral, and it would lead, by Consis, to "NOT (a)". 12:25 "This statement is TRUE, because it is INDETERMINATE" No, it is wrong. For it to be TRUE, we need an assumption that Formal Arithmetics is consistent. We don't have this assumption. Therefore, there could be a proof for it. Therefore, it could be FALSE. That's why it is INDETERMINATE. We don't have proof for TRUE or FALSE. The true meaning of this theorem is actually that a strong theory is either consistent or incomplete (not both), and in fact consistency we cannot even prove. 12:38 "In other words, we just recreated, ..." Yes, it was recreated, but using rhetoric rather than logic. 12:53 "Only statements about meaninglessness of other statements" Is there actually a proof that only these statements are affected? But I could believe in this (because there are such constructs as "Primitive recursion"). Other thing is, actually we would have wanted for this to be not this way. Because we would want some machines which would analyze other machines (for bug finding purposes, for example). There is whole area of activity named Verification in which people seriously try to use various methods to verify correctness of software products. This is an argument about usefullness of Incompleteness Theorem. Unlike (maybe) natural language, in programming we use self-reference on every step (recursive functions, while-loops). So it is not only gibberish that is the object of these theorems.
"There is whole area of activity named Verification in which people seriously try to use various methods to verify correctness of software products." Yes. In theory, some routines can be proven reliable for some range of inputs, and others cannot. In theory, some verifiers can be proven reliable for some range of inputs.... If they sometimes reveal problems, they are useful whether or not they find all the problems.
Why do we take it so much for granted that there is no external and inclusive idea to our own propositions that constitutes a universe where our undeterminate propositions are actually fully determinate? Is our mind all inclusive, so that such loopholes are comprehensible? Do our propositions reflect a total reallity?
Not bad. I always knew there was something fishy about these kind of results mathematical logic. It never set well that certain mathematical statements such as twin prime conjecture, Riemann Hypothesis, P vs NP, etc could lie in this "true but unproveable" category. It seems like the only kind of mathematical statements one could that are "true but unprovable" are really silly self referential statements that we aren't interested in making because it does not progress mathematics in any meaningful way.
This is not true. The Halting Problem, for example, tells you that your compiler can't remove all unused code and your tests can't be proven to test every possible bug. Dialect dismisses this stuff, but it's very important for computer science.
Notice that the actual statement that is being proven unprovable is completely non-self-referential. It is a statement about natural numbers, namely that some particular integer with a particular arithmetic property does not exist. Self-referentiality comes in via the encoding: If the statement were false, i.e., if there were a counterexample - which would just be a particular natural number, then one could decode that number and unravel it into a proof of the very statement that it contradicts. That is the connection - if a counterexample exists then we have a false statement with a proof - which would mean that the logic itself would be inconsistent. So under the assumption of consistency, the statement must not have a proof.
@@MatthiasBlume There are a big variety of mathematical systems. Suppose you had a finite collection of axioms that are true for the natural numbers, and are also true for various algebras that might not be the natural numbers. And suppose you had a statement that cannot be proven with those axioms. And suppose it turns out that this statement is true not only in all of these algebras, but it is true for every algebra that those axioms are true for. It's impossible to find a counterexample in any algebra that fits the axioms. But there's no possible way to prove it. Wouldn't that be interesting! Is there a known example like that? Or could it be that for every statement about the natural numbers that can't be proven from some given set of axioms that fits the natural numbers, there has to be an algebra that fits the axioms where the statement is false?
@@jethomas5 There is no example such as that. In fact, there cannot be. The statement that cannot be proved in the system can always be added as another axiom - and so can be its negation - resulting in two different new logics, with their own undecidable statements therein. Philosophically this is quite interesting, because in the negation case we are adding a statement that by the above reasoning should be considered false. Hofstatter's Goedel, Escher, Bach discusses this in some length, and I recommend it as further reading. (Hint: The "true" but unprovable statement has the form "There does not exist an N such that ...". Adding the negation of this statement simply means stating that "there is an N such that ...". To make this true we just have to make up such an N. It can't be a natural number as previously conceived. In particular, it cannot be zero, or obtained by applying successor a finite number of times to zero. And, thus, it would also not decode into a finite proof of the original non-negated statement. Admittedly, this is all rather mind-bending, and I don't blame anyone for being at least mildly confused...)
I might be missing something here, but if Hilbert's program was to derive all of mathematics from the axioms, couldn't we just add all the unprovable statements to the axioms and carry on?
The short answer is "No", and this is one of the key points of Gödel's first incompleteness theorem. The slightly longer answer is that, if you start with a formal system which satisfies the hypothesis of Gödel's first incompleteness theorem, even after adding unprovable propositions as axioms, the resulting formal system will still have (new) unprovable propositions; thus, it is "intrinsically" incomplete. The more you add axioms, the more unprovable propositions will come up. However, what is typically missed is another key aspect. Gödel's incompleteness theorems do not apply to _all_ formal systems: like all theorems, they have _hypothesis_ and, if those hypothesis aren't satisfied, then the theorems results do not apply. Thus, there are actually formal systems which are both complete and consistent, such as: Boolean Algebra, Euclidean Geometry, Presburger Arithmetic. In fact, the whole point of Gödel was to rigorously show that not all axiomatizations of arithmetic are equivalent: some of them (like Peano) are more "powerful" ("expressive" is probably a better term), but they "pay the price" for such "power" by being either incomplete or inconsistent, while others (like Presburger) are "weaker" (less expressive), but they are both complete and consistent.
@@QuantumMechanicYT There's a subtle point here that I hadn't gotten before. Say you have an axiom system with 15 axioms, and there's a collection Y of unprovable propositions with those axioms. You add a 16th axiom. Are there now additional unprovable propositions that were provable before? Or is it that there were an infinite number of them before, and there are still an in finite number, but some that were unprovable before can now be proved? Perhaps with a new axiom Goedel's theorem gives you a way to find additional unprovable propositions that were unprovable before but that the theorem did not provide a method to find before?
@@jethomas5 I am afraid that I do not know the answer to your question, which I find very interesting. My best guess is that the collection Y is infinite right from the start, but adding one (or more) axiom(s) still adds new elements (possibly infinitely more) to Y.
@@QuantumMechanicYT Thank you! I believe that Goedel's method provides a way to find an infinite number of them. I expect that if a proposition is unprovable with 18 axioms, it will still be unprovable if you remove one of them and have only 17 axioms. Because if you could prove it with 17 axioms, adding another will not make the proof stop working. So each new axiom removes some items from Y leaving only an infinite number remaining. I read into your statement above a claim that it isn't true, and wondered whether my intuition was known to be wrong.
