This is also true of the current one, which is why he made a second one right after that, but that cannot cut it either, so he'll just keep uploading infinitely many videos without ever being able to adequately explain the topic. Sad.
I came out of it with more questions lol For example, why does a statement, that can't be proven/disproven by the axioms, become an axiom? That's not very logical, or I didn't understand him right
G Knucklez An unprovable statement doesn't automatically become an axiom, but you can add it (or its contradiction) to the system as a new axiom if you want to. This is because neither the unprovable statement nor its contradiction can create a new contradiction in the system; if they could, you could use that in a proof by contradiction. So an unprovable statement lets you create multiple new systems which assume different truth values for the unprovable statement. The classic example of this is the parallel postulate in geometry. People tried for millennia to prove it from Euclid's other postulates, but they failed because it's an unprovable statement. Instead, you can assume either that the parallel postulate is true, in which case you get plane geometry, or that it's not true, which (depending on how it's not true) give you either elliptical or hyperbolic geometry.
It would've been more self-referential if it was believed much later that someone actually was trying to poison him. A true statement that can't be proved
Dying by the fear of dying is pretty self-referential, negative feedback loop, perhaps, but how much more self-referential can it get? I can even imagine the fear of getting poisoned being stronger than the fear of starving, up to the point where you're too weak to eat, even if the starvation fear would have become stronger. Being poisoned would make it a paranoia becoming reality, I wouldn't call that self-referential at all.
@@z-beeblebrox It's still death by mental illness. Even if you were certain that someone was trying to poison you, you could still find a way to eat. Just go to a grocery store and buy some food. Problem solved. Go to a different grocery store every time. Throw in a few restaurants as well.
@@bmoney1860 I don't know why you're trying to refute a comment I wrote 3 years ago, but I'm pretty confident none of what you said had anything to do with my observational joke about self-referencing
Speaking of infinity and Gödelization, it is also noteworthy that every mathematical statement will map to a natural number and therefore the entirety of mathematics is countable. So whenever one encounters an uncountable set, mathematics can't describe every individual member, only the set itself.
I don't know. It strikes me that it should be possible to diagonalize (a la Cantor) and show that the number of mathematical statements is uncountable, and that as a result, the assertion of this mapping has some hidden flaw. Part of the problem, I think, might come from the two-value (true/false) logic systems, or just from a lack of rigor in the creation of language (or both).
Well, the statement that all of mathematics is countable is tied to the Church-Turing thesis, which, informally, says that everything a human can compute, a computer can also compute. Since every computer program is just a very long number in that computer's memory, the number of possible computer programs is definitely countable. OTOH, the number of thought a human brain might ever produce is at its very least bound by the number of configurations in the volume of that brain, which is finite. And even if you'd allow for an infinite brain (as idealized computer memory is also infinite, so that's fair enough), you'd still have a countable number of states AFAICS. Neither computers nor brains are bound by binary logic, although I'm not sure what you mean by "lack of rigor in the creation of language."
Well, saying that all mathematics is countable is just the result of it being a language, since what is meant by mathematics is the valid sentences made in the math we are doing. There are only finitely many symbols able to be used and every sentence has finitely many symbols. One can also say that the English (or any other language) is countable. I do think that the countableness only refers to the formal symbols and stuff, i.e. on the logic/axiom level, since it is clear that all of math is not countable since there are things with uncountably many elements and they are understandable. (I.e. the Cantor set) It just turns out that all the things that we can describe turn out to be countable, because describing them uses finitely many symbols. E.g. there are uncountably many real numbers, but of the real numbers that we can describe in a mathematical sentence, there are countably many because in describing them we are using a language with finitely many letters and sentences of finite length. E.g. 1 is not equal to 2. 1 is not equal to pi. 1 is not equal to e. ... is a way of talking about all the numbers that we can talk about, so in no way can we talk about all real numbers because of the limitation of language. When you invoke an axiom that brings infinity into the picture, like the power-set axiom applied to the axiom of infinity, then you can get uncountably many things and you can say things like: For all x in the real numbers, x^2 >= 0. There are uncountably many real numbers, but we are not counting real numbers, we are counting the sentences.
I haven't done any formal logic but I think part of the issue with trying to diagonalise is that statements must necessarily be made of up a finite number of symbols - if you tried a diagonalisable argument, you'd create a statement with a symbol for every natural number. It's the same reason why you can't use Cantor's diagonal argument on the natural numbers (like you do on the real numbers, but right-to-left)
And admittedly, with "words" (at least in the sense we normally think of them), yes, you could even give every letter an ASCII value, just concatenate them all, and come up with a single, unique numeric value. Mind you, statements about mathematics *do* include all irrationals, so that would mean that you have strings of *infinite* length. For example: "The ratio of a circle's circumference to its diameter is 3.1415926535...." would go on forever. Computers only have finite precision and memory; statements do not. We do know that all possible strings of infinite length are, in fact, uncountable.
Kind of surprised the halting problem wasn't mentioned when he talked about going to other fields to see if they had acknowledged "limitations on what they could possibly know." That's essentially what the halting problem amounts to in terms of computation theory(I don't want to stretch too far and say CS) and is something any intro to CS course would at least mention, I think, and probably the easiest example of it in another field as an example of a fundamentally unanswerable question.
The question about what happens if your theorem is undecidable, or how will you know has already been covered to some extent. Euclid's 5th Postulate and the Continuum Hypothesis are both formally undecidable within the mathematical system. It has been proved in each case that they are independent of the remainder of the axiom system. These undecidable propositions then give us options in terms of how we progress (as alluded to in the video). In the case of the 5th Postulate we proceed in one of Euclidean geometry, spherical geometry or hyperbolic geometry. I'm not aware of any work having been done based on different options related to the continuum hypothesis, but there are surely choices that can be made and there must be consequences of those choices.
I really like that last statement about Gödel's death. Reminds me of Nietzsche who ended up completely psychotic, John Nash, etc. There's definitely a fine line between madness and genius.
"...pull ourselves outside a system". This right here is an absolutely crucial part of understanding emergent behaviour. As soon as we are cognizant of a structure, we go about figuring out the shape of that system. As soon as we understand the shape of a system, we can envisage the exterior of that shape/structure.
This is all very interesting. One idea I do want to present is the fact that because no set of propositions can prove itself consistent due to incompleteness, it follows that whichever set of propositions Gödel used to deduce and conclude the Gödel Incompleteness Theorem, such set is by his own Theorem unprovable, implying that the Theorem itself is unprovable. Therefore, the very truthiness of the Theorem renders the Theorem as not decidably true since it cannot be proven, and this creates another infinite loop/contradiction.
I have not read Gödel's work and probably I am not in position to do so, but while viewing this video, a question came to my mind: Could Gödle's coding actually be introducing incompletness? I mean, could the outcome of his work is exactly the result of some characteristics of this coding? On the other hand, if this is the case, then the inability of mathematics to describe itself may be a proof of its incompleteness in the first place...
Panagiotis Rentzepopoulos It is conceivable that the terms in which Gödel couched his proof necessitated its truth. That is always a potential problem with a proof - it could be circular. I think it's fair to say that this proof has been scrutinised carefully. No one (who has mastered the subject) doubts its truth. It is ironic that this subject was triggered by an attempt to solve that very problem. The hope was that we could reduce all proofs to a series of simple manipulations of the axioms that would guarantee correctness if rigorously followed. Gödel proved that this could not be done.
PompeyDB, how very presumptuous of you. If you had done any digging regarding the information of the time of when this video came out in correspondence to the implicit time given by the comment, you would have realized that the person is, in fact, not American. The American East Coast would have had this video available during the middle of the night - around 4 AM by their standards. I am doubtful that anyone would call that the morning, rather than the middle of the night. No, judging by the comment, I would say the person is located in Europe.
There's a wonderful book that elaborates on this, The Eternal Golden Braid, by Douglas Hofstadter. It touches upon music, computer science, the visual arts, and formal mathematics. Highly recommended, if you find this video interesting
But if we prove that mathematical consistency is unprovable, doesn't that by the same logic imply that there are no inconsistencies? Or at least that you cannot come across them? Because if you come across an inconsistency it would disprove mathematical consistency, but that is impossible as we proved.
The problem is that you would never be able to prove that your true statement leads to an inconsistency because you have to prove it for it to be true, and a proof which contains an inconsistency or which causes one in another part of mathematics is not a proof. The statement would have to remain unproved which means you do not know if it is true or not.
As soon as there is just one inconsistency in a theory, you can prove ANY statement, so also it's consistency (if you're able to formulate it). Gödel proved (from the outside, not within the theory) that if maths (or arithmetic to be more precise) is consistent, it cannot prove that within.
"if we prove that mathematical consistency is unprovable, doesn't that by the same logic imply that there are no inconsistencies?" If we found an inconsistency, we would have proven that mathematical consistency is unprovable, so no, proving that doesn't imply that there are no inconsistencies.
I don't see that circular time / the grandfather paradox is really a problem: • firstly, it all rather depends on your radius of curvature - it the loop's sufficiently long then there will never be any consequence for a locally-experienced universe • secondly, the range of things that can possibly happen in a curcular time universe may be constrained to resemble something like a collection of fixed-points, but if the complexity on view is sufficiently huge then you would never be able to notice the constraints anyway. (There's also possibly something about stability in there too...)
12:00 Incorrect. The grandfather paradox does not imply time travel is inconsistent. It just implies that specific sets of events must be consistent within the universe. Therefore, for us to even time travel in the first place, we may have to lie to the travelers that they can accidentally screw up the past - which will allow them to be able to time travel in the first place.
@@matthewstuckenbruck5834 The Halting Problem shows that there are decision problems (yes/no questions) for which there are no algorithms that can reliably provide the answer. In other words, there are questions that no algorithm can solve.
@@willmcpherson2 Ah but can you prove how much effort you ought give to try to find out before you give up on something you don't know wether is solvable or not?
@@cyberneticbutterfly8506 In general, no, because creating an algorithm that knows exactly when to give up is equivalent to creating an algorithm that knows whether an algorithm halts (making it equivalent to the Halting Problem). Although with both problems, you can use approximations. “Give up after 1000 steps” for example.
Best thing IMHO about Continuum Hypothesis is that both parts of derived mathematics eg. infinities galore or organised have shown to be usable by scientists. So it's still usefull mathematics.
1:00 - A corollary of Turing’s solution to the Entscheidungsproblem (literally: decision problem) says that this question is undecideable, which I suppose is just another way of the universe giving mathematicians the middle finger.
The limitations of any given formal system are intentional and purposeful, designed to focus on specific domains. While no single system can prove every truth, any true statement can be proven within a stronger, consistent framework. Even the unprovable statements within a system arise because that system is deliberately not designed to handle its own self-analysis, a limitation that can be addressed by stepping outside it.
The translation of: "Wir müssen werden, wir werden werden.", at 10:03 would be: "We must become, we will [or shall] become." The correct german translation for: "Wir must know, we will know.", would be: "Wir müssen wissen, wir werden wissen."
Godel's Theorem is more about the Incompleteness that results when using the Axiomatic Method: he proved there MUST be Theorems which are true but that they cannot be proven to be true using a given set of axioms. Godel did not believe that this was the final word -- humans may agree to loosen the rules of inference and accept the result as a reasonable proof (here he was drawing on Plato's approach to Mathematics). Goodstein's Theorem illustrates what Godel was talking about. The statement of the Theorem only involves the Integers with Addition and Multiplication. Peano's Axioms gives us this part of mathematics. Goodstein's Theorem was proven about 1944; however it was subsequently shown that it could not be proven using only the Peano Axioms (which does seem to be surprising). To prove Goodstein's Theorem one needs to use Transfinite Numbers (which need additional Axioms to the Peano Axioms). Goodstein's Theorem does have some practical uses: it can show that certain Computer Programs will come to a conclusion, rather than continue for ever. Turing had a theorem concerning Computers: that a Computer itself cannot judge whether a reasonably complex program will come to a conclusion or not.
but if you take the factorization approach instead of the additive approach you can show using mathematics that there will come a point where all theorems will rely on theorems past a certain point. any composite above n^2 has to have a divisor greater than n for example.
In the previous video he mentioned it is possible to prove a theorem is undecideable, and therefore true, since no false statements are undecideable. Is it possible we can prove that every Gödel problem can be solved in this way? That would sort of sodestep the entire issue
Let me see: Say "+" is assigned the number '2' . Then the number "2" has to be assigned to some 'x1'. But then "x1" would have to be assigned to 'x2' and then "x2" ... so "+" builds a ladder through the coding. So for every theorem, I could find a step in the ladder having a value greater than that?
One of Gödel's contemporaries also met a tragic end: In contrast to Gödel being paranoid about other people trying to poison him, Alan Turing poisoned himself.
I am not an expert, so I apologize for the question, but the following question came to my mind. If given a certain number of axioms, there is a theorem that cannot be proved, but if we add other axioms, the theorem can eventually be proved, Is there a theorem that cannot be proved even by adding infinitely many axioms?
I love how eloquently he speaks. Btw, does anybody knows the DENSITY of statements that are true but not provable in the usual set of axioms of arithmetic? They are infinite, but how infinite in relation to the set of all statements?
I am very much not qualified to give a definitive answer but I expect they are extremely dense in the same way that irrational numbers fit between every rational number on the number line, but so many of them don't have a constructive use or are simply down some logic path that people haven't bothered to look at yet. We as people use math and numbers all the time, but the numbers we use are very specific ones, the ones that constructively come off of the other numbers (the 'axiomatic numbers' if you will :) ), like the whole numbers extends off of the naturals, then the rationals and the integers extend off of the whole numbers and then the constructables off of them and then the irrationals as the mathematical logic increases in complexity and power, then the imaginaries and the complex numbers in one direction and the transcendentals in another direction... I suggest looking at the Numberphile video "All the numbers" featuring Matt Parker to get a feel for what I mean by the numbers 'building' off of each other and there being so many more than the numbers humans use under normal circumstances. I certainly don't buy Chaitin's Constant (the halting probability 'set' of numbers as the number itself changes depending on the program being checked if it halts or not) numbers of pairs of shoes at the store, because that doesn't make particularly much sense. It's also possible mathematical logic can't lead to inconsistency because not every number may be 'reachable', or something similar.
I'm not sure if you're asking about topology on the collection of sentences within a formal language, or if you're one of those people who dislikes referring to cardinality as "size" and chooses to call it "density" instead (my personal opinion is that size is a significantly better word than density when it comes to cardinality, but there are enough people out there in RU-vid comments declaring it should be called density instead of size, so what can you do?). I can't speak to actual density in a topological sense (which is the sense in which the word "density" actually makes sense), however I can talk about cardinality. In a formal language, one always starts with a countably infinite number of symbols. Every sentence consists of _finitely_ many symbols. Hence, there are a countably infinite number of possible sentences. This, in and of itself, gives us only two options: there are finitely many undecidable statements or there are countably infinitely many undecidable statements. As you correctly noted, there are certainly infinitely many, so this gives us countably many undecidable sentences. It's the same size as the number of sentences to begin with as well as the number of decidable sentences.
Regarding consciousness, the human brain is operating on a higher dimension of thinking than the 1 or 2 dimensions of simple mathematical thinking, which is why we can do math, because we're thinking more complexly than simple logic. A normal computer can only do one calculation at a time, while a human adult brain (functioning well) can do multiple calculations at the same time. So we can sort of triangulate answers to complex problems, whereas a normal computer can't. This is why the idea of artificial intelligence is so confusing to many people. A simple computer algorithm won't function the way a human brain can, since it's still looking at only one problem at a time. From what I can tell, the human brain can model up to four logic problems (which I define as the difference between a starting state and a goal state) at a time, at least in a mature (wise) adult brain (over the age of about 40 when the prefrontal cortex starts being able to operate at it's full ability, according to current neuroscience).
Seems to me that Dr. du Sautoy's explanation boils down to saying that no completely consistent system* of mathematics can be generated from a finite set of Postulates**. *Leaving aside the question as to what, exactly, is the definition of "a system of mathematics." For instance, does "algebra" in the broadest sense overlap with "topology," in the broadest sense? Naïvely, it seems to me that all the branches overlap here and there, in which case there could be just one "system of mathematics." ??? **I know that most (not all) mathematicians disagree with me on this, but I think that great confusion is caused by the use of the word "axiom" when what is really meant are POSTULATES (á la H.S. Plane Geometry, and Birkhoff's and Maclean's nomenclature of the "Peano POSTULATES" when they explain these in their classic text /*A Survey of Modern Algebra"). I'm using "Axioms" and "Postulates" according to this distinction: Properly speaking, Axioms (here, in abstract systems of thought) are always the same. For instance, A=A, or to more-or-less quote Aristotle, "A thing cannot both be and not be in the same sense and at the same time": The Law of Non-contradiction, which is axiomatic in all logical thought (even in consideration of real-world phenomena or issues). In this sense, Axioms are simply the rules of logic that we humans MUST assume if our words or thoughts are to have any meaning, regardless of what we're talking or thinking about. But POSTULATES are assumptions which are made without proof that they are true ("true" within the context of the logical system in question). They, along with definitions, FORMALLY are, along with definitions, the foundation of every logical system (including philosophical systems of metaphysics). I don't mean that the system is generated over time, actively, by thinkers, from these; usually it's not. Usually, the system is developed by humans and as they go along they also figure out postulates that are required if the system is to be consistent (non-contradictory) and fruitful. (If two such postulates turn out to be contradictory, then "something is rotten in Denmark.")
Heisenberg's uncertainty seems like the physics version of this theorem. The presenter talked about whether similar "unknowabilities" exist in other fields in science, and this seems like a good example.
Fantastic interview, but I was really hoping he would talk about nonstandard models of arithmetic, i.e. what you get when you take as an axiom that Godel's statement is false.
I will say again what others have said, slightly differently. Gödel's theorem is not an axiom of arithmetic. If you were to assume the Theorem is false, you would need to label one of its logical steps false (even though it is thought to be true) and go from there. Alternatively, you could start by assuming that one of the axioms Gödel assumes is false. Now that approach has been used. For example in Geometry. The results are worthwhile. But Gödel's proof was about all such axiomatic systems. Drop or alter one axiom and you get a new system. It's still an axiomatic system and it is precisely such a system that Gödel's proof has as its subject
NETWORKED MINDS, being used already in polymath think-tanks. Yet, this is where the next growth will come from (before Quantum Computing AI takes hold altogether) as with James Maynard & Terry Tao s work together. Their groups each got so far, then their amalgam took them STEPS farther.
since when does labelling an unprovable statement as false mean the original statement is actually right only because we correctly labelled it? The new statement might be right but the new statement only says that the old one is WRONG. they are mutually exclusive.
There are several answers depending how I parse this question. Goedel's are called Incompleteness theorems a little misleadingly, since they don't establish that there are truths that cannot be expressed in terms of consistent logic, only that they cannot all be proven in those terms. If you are asking 'How we know the system that Goedel proves incomplete is consistent' you seem to beg your own answer since Goedel proved that a consistent system cannot be proved consistent within the system, and proved that by stepping outside the system by coding it. However, if you ask 'How we know the [coding] system is consistent in which Godel proved incompleteness [of the lower, coded system]' it is a fun question, because if this higher (coding) system is consistent we can't know it by proof. However, not being certain of the consistency of this system does not negate any proofs that are possible in the system, any more than attainable proofs in the first system are undermined by not knowing it is irrefutably consistent. It is always a Goedel sentence that undermines any effort to prove that the system in which it is expressed is consistent, because it has an undecidable truth. What Goedel proved was that any consistent system is limited (incomplete) in the sense that it will contain unprovable truths: i.e. will have postulates with lines of proof and disproof that cannot be shown to be false in that system. Goedel sentences (truths) in that consistent system will appear to be as likely true as false. In the system's terms, the conjecture (eg. Goldbach's conjecture) and its contrary will for ever appear to be plausibly true and plausibly false. This means the conjecture that (eg. Goldbach) is "true and false" has a certain permanence - even here where consistency entails the law of non-contradiction. It is the self-consistency of a logical system that forces it to be unable to prove every well-formed-formula (expression) in that system as plausible, or likely true. Like the uncertainty principle, there's a great 'cost' to having the 'certainty' of consistency. This leads to the inescapable conclusion that to have confidence at being able to assert the truth of everything conceivably true requires not consistent logic, but some form of modal logic which allows intermediate truth values "yes and no" (reminiscent of entangled states in physics) -- such as we see in human language all the time. For man to have come to such an uber basis of reason is already proof against natural selection (a system presumed to operate on consistent survival rules), and it's the reason that Goedel was able to posit his theorems in the first place. However, it's not the reason he was able to prove them once he'd divined them, as all he had to do was adopt the economy of stepping out of the arithmetical axiomatic scope to its meta-level of coding arithmetical axioms: a level that is itself also pursued using consistent logic. That economy was wise, since the only way to convince [non-contradiction]-bound mathematicians of its truth was to do so in a consistent system where Goedel theorems do not happen to be Goedel sentences (i.e. unprovable there). That system of coding will have other truths that are not provable in that system. Now, a most interesting sentence would be one that is a Goedel-sentence at every meta-level of 'coding' or 'representation.' Possibly Anselm's declaration of "that than which nothing greater can be conceived" is just that. Turing found that within a modal multivalued logic of his devising, this sentence was sound (therefore true, if the 'atomic truths' or axioms are true). It convinced him to be a theist (well, at least a deist).
@@garyknight8966 I do not see any reason for such a long explanation. Things are so much more simple. Let's see arithmetics for instance: a) There are sentence that are not true and also are not false. b) there are so, three kinds of sentence - true; false; not true and not false. Abou consistence, we don't know - if some contradiction appears we do not use this sistem any more. Definition: a sentence is true if it can be proved. Definition: a sentence is false when its negation is true. Definition: we always assume that the system is consistent and if a negation leads to a contradiction so the sentence is true by definition. Also, there is no such thing as conjecture in mathematics, because you can not ask: is this affirmation true? In tha question you suppose that or it is true or it is false. But we agreed that there is a third option, that is, it may be not true and not false. This third option is fantastic because it permits to construct two mathematics from that duality. This happened with the fifth axiom in plane geometry and always happens with the axiom of choice. (You see that there is no sense to say that a sentence is true but can not be proved.) I think that Godel's theorems are super estimated.
@@garyknight8966 Also, it is obvious that a sentence is true if and only if the axioms are true. Indeed, we are not allowed to ask if an axiom is true or not, since an axiom is BY DEFINITION) true. All rational thoughts are based in that situation, veracities are something subjunctive, that is, they depend on veracity os the axioms, or the creeds.
I'd like to ask some questions (sorry for my bad english): Gödel assigns a number to every mathematical sentence. This number is a product of prime numbers which are assigned to the axioms used to demonstrate the sentence. Does Gödel consider the chance that a true mathematical sentence has more than a proof? In that case, shouldn't a sentence have more than a number as a "surname"? We could use the product of those numbers as the new surname, but are we sure that it doesn't create some homonymy cases? We could chose to use both surnames with a comma between them (like 14 , 15 which is a sentence demonstrable thanks to the "axioms" 2 and 7 or thanks to the "axioms" 3 and 5) but this logic takes me to ask an another question. Is possible that two axioms could be used to demonstrare two different sentences? In that case, what should we do? p.s. while writing I realised that we could just put a digit at the end of the number that are assigned to sentences that have the same demonstration(s) of other ones: using the example I used above of [14 , 15], if it's possible that two sentences are both demonstrable with 2 and 7 or with 3 and 5, we could call them [14, 15 - 1] and [14 , 15 - 2]. I decided to post this comment anyway for everyone that has the same doubts I had while watching the original video. If anybody wants to correct me or wants to add something else, he's welcome. Have a nice day
Based on my expierence following statement is expected to be true: "If equations that refer to lows of phycisc can be solved means those solutions aplly to the real world in a certain sense".
OK, if the value assigned to a correct, provable statement is the product of those related to all the proven statements used to achieve that one particular proof, what if I can find multiple deduction pathways to the said statement? Does that mean that particular statement can take multiple values or do we arbitrarily select the smallest number to encode that statement? Also, if some statements are used more than once in that particular proof, do we multiply the code number of those statements twice or do we just count them once? I know I have also asked (quite impertinently I regret) a list of questions in the other video, but please do excuse my burning desire to get to the bottom of this! (Really this theorem is so mind-blowing I can't even speak right now!)
He said in the beginning: the theorem implies that there will be an infinite number of undecidable sentences. If this is true, then Godel would have the right to prove that halting problem is undecidable, right? I do not understand what is useful of Turing's result if we have undecidable sentences by Godel. Could someone explain this to me.
The explanation of the active role of a verb in a sentence was "connecting word", and the mathematical equivalent was an equal sign, so if the subject of an identity is true then the objective realization is the same equally true. It's normally a reversible action in mathematics, so if either side of the equal action is false, both sides are false or the statement is simply a lie. That there are many systems of lies dressed up as "trueisms" (truthiness!) is inarguable. So researching ways to identity veracity is still the core business of mathematics. yay
2 and 3 as the only single gap prime in our number system has always been interesting for me. There's a strange tension point between two and three -- they are the smallest even and odd integers in the natural numbers, if we start counting after one -- that can interestingly exploit things. It's why I think the Collatz conjecture can never be proven false ... but I can't prove that my observation is true, nor can I disprove some number is out there which will never reduce to 1.
Hmm, If it isn't provable that Mathematics has inconsistencies then according to your previous statement: 1: It is True. 2: It is an axiom. But I am probably wrong. Do tell where though, so I can increase my understanding.
Gödel's theorem is a statement of the impossibility of solving a recursive object in the form of a direct solution. In arithmetic where there is no recursion, such as arithmetic without multiplication, all propositions are provable. From here, one more small step and you can prove Fermat's theorem in the way that he kept silent about in his diary: most likely Fermat understood the meaning of recursion - this is a function whose domain of definition is the product of the set of values of the function itself by the set of the domain of definition of its predecessors. And after a couple of manipulations, you can come to Fermat's conclusion. But the boundaries of RU-vid comments are too narrow for me to provide this proof here 😜 p.s. No, I'm not a crazy Fermatist
@Sirius GG “All right, Mr. Burns, you win. But beware, we Germans aren't all smiles und sunshine.” from the "Burns Verkaufen der Kraftwerk" episode of the Simpsons ...
I suddenly understand the problem with proof by contradiction. Proof by contradiction requires that mathematics is consistent (i.e. that p AND not p cannot be true for any statement p). But the consistency of mathematics is not provable, therefore proof by contradiction is not a proof. I guess this is fixed by taking consistency to be an axiom...
Actually proof by contradiction works off of the principal of the law of excluded middle. That is to say that every valid truth assignment to sentences of mathematics is in a sense a total function (every sentence has exactly one truth assignment that exists in a Boolean Algebra). When the truth values are binary, you can very easily derive P OR ~P. Granted, everything will fall apart if the theory is inconsistent, but an inconsistent system is absolutely useless since that implies a False sentence is a theorem, and since false implies true, then EVERY sentence and its negation is also true. Additionally, adding in "this system is consistent" (if that is even expressible within the language) will NOT fix an inconsistent system since the sentence "this system is inconsistent" can be derived by whatever contradiction already exists from the other axioms. In effect, mathematicians have to have a little faith in the consistency of ZFC (the standard axiomatization), because without it, studying mathematics is useless, however I think this faith is much less than the faith required to study modern physics, of which observational error and inductive inference cause much higher concern for doubt, at least fundamentally.
I feel like having faith in ZFC's consistency is equivalent to taking its consistency as axiomatic. I agree though that you don't gain anything really except a formal declaration that "I hold this to be true", because no system is safe from Godel. Also it seems to me that the law of the excluded middle and consistency are equivalent. If and only if a system is consistent does it uphold the law of the excluded middle.
Actually a thought occurs to me... If you take consistency as an axiom (call it A), you can prove that it is true with the trivial proof: A is true, therefore A is true. But Godel's second theorem tells us that any system that can prove its consistency must be inconsistent. Therefore our system containing A is inconsistent... Okay my head hurts a little.
LydianLights if A equals A is true, it has proved its consistency and therefore now inconsistent because it is both true and false because if "A=A is true" and if "A=A is false", it means the statement "A=A is true" is false and "A=A is false" is true which means it is both true and false. Therefore it is an inconsistent system. ...whatdidievensay
Perhaps it is more an operational decision. In fact, if we let in inconsistencies, we can prove anything. A system in which everything is true is useless, and of course, completely uninteresting.
To summarize orochimarujes's comment, you can add amendments that make it easier to add more amendments, eventually leading to a complete breakdown of the system of checks and balances and democracy.
Yeah. They didn't account for a foreign country using the internet to interfere with the election by hacking email servers, using an army of bots, spreading fake news, and using complex algorithms to psychologically profile and influence potential voters.
No one actually knows, since Godel died before he published it. Guesses are that either it's the fact that it can be amended in possible way (including to remove amendments), or that judges have complete freedom to interpret them (including other judge's interpretations).
How can we know if the coding that Gödel came up with is correct? I mean, in the original footage we saw that the number 3 was assigned to the Then statement, but how do we know that it should be 3 and not any other number? If it was any other number, then the outcome of any number assigned to an axiom would be different. Would the system still work if every number assigned to axioms was different? I have so many questions
I believe the inconsistency he referred to is the fact that the Federal Government is the ultimate arbiter of its own powers, which is a positive feedback loop. If you take the microphone of a public address set up and put the microphone against the speaker (i.e. input is fed by output which in turn becomes input), you will blow the amp. We're actually pretty close to blow up at this point in the history of this republic, wouldn't you agree?
I wouldn't support the thinking that a computer(or something else) can never achieve our conciousness, since a pseudo-random process like evolution could create it, therefore we can at least match it as well. Can we improve it though? Thats up for debate. But this is likely since otherwise we have to assume that the pseudo-random evolution created a perfect mind.
_"How can you assign an integer to every mathematical idea after exausting them because you need to list all of the real numbers first?"_ Ideas aren't assigned numbers, but mathematical expressions are. That is, things that are written down, which are necessarily finite.
I mean, technically every sentence, book, website, program, convolutional neural network, network protocol, etc... can be represented by a number. This is all just data at the end of the day. That would be an interesting conjecture, I mean outside of mathematics, is data inherently imperfect? For instance, this comment is saved onto a database somewhere and assuming the data doesn't degrade, you should be able to just copy it from one source to another. You can even include more data to ensure the user knows that this comment should be formatted using UTF-8, maybe a checksum, a string of trailing zeros to ensure the end of the data gets interpreted even include some kind of mathematical statement or function to communicate that the data should be meaningful however, you still need a way to interpret the data. Basically, what I'm trying to say is any statement can't exist on its own, it requires to be in some kind of system or require some kind of context, it cannot exist on its own which also means that everything can't be reduced into data, there has to be an extra component to interpret that data. I don't know though, probably just confusing myself.... haha
One question: Have anyone concidered that the ”axiom of mathematics” is flawed in itself? I might have missed it in the video but for me that would be an important part of debugging a problem like this.
If all mathematical statements were coded with prime numbers and to be provable it has to be divisible by the axioms, then that's saying that there are no provable statements in mathematics.
the set of all sets does not contain itself if you state that any set "contains itself" then... ...the set of all sets with even members may have odd members...
Yup, that is what gödel said, that any system capable of arithmetic (including math) can't be complete and conistent at the same time. Mathematicians rather choose incompleteness over inconistency. That means that if maths is consistent there are certainly statements out there that can't be dervied (proven) from the axioms.
I still don't get it how Godel made it possible for Math to refer to itself by encoding sentences. After you encode a sentence you would get a number - sure a big number, but still one exact number (and unique at that), so lets say for simplicity that such number for some encoded statement is 7. How do you even ask in Math if 7 is true or false? 7 is 7 and that's it.
"We cannot prove that mathematics does not have contradictions" If you can't prove it with math itself what about this: *Mathematics that is also overlapping with physical reality i.e. maths found in physics must be true.* Now: *Can matematics be proved not to have contradictions if the previous statement is taken as true?* For instance using the things that overlap to prove the rest?
About the inconsistency of the US Constitution, does anyone know if Godel referred to the US Constitution in its original form, or in its present state at the time Godel was granted citizenship. This is a very interesting topic.
So wait, to determine that G is true we can not used the mathematical system otherwise maths is inconsistent. We must come outside of the system and use a level of human reasoning/logic to recognise that G is itself true because there is not proof with the system? In turn this points to two conclusions? Either we hold Godel's theorem to be true and then conclude that human reasoning outside the mathematical system is not mathematically based (which would imply a refutation of determinism since we can describe the base units of chemical interactions in our brains mathematically) or we hold that all human thinking can be deterministically described and thus mathematically described and conclude that whilst human reason can be described by our system it is fundamentally flawed, and it in stepping outside the system and saying "but we can see that G is true" we have made an error that we can not understand.
I take it that an unprovable statement is said to be true, but what prevents us from reasoning that the converse that is true because it is unprovable?
That would certainly work. The problem is that we CAN'T show that a statement is unprovable (what if it's just really tough and we aren't smart enough to figure it out? We can't tell the difference). In fact, because you are correct that just proving a statement unprovable would prove it's true, that by itself is enough to prevent that from ever happening. Because you just proved that it's true. Which means the statement wasn't unprovable, since you just proved it. :)
I'm not familiar with orders of logic, but from what I understood in other comments, it would seem that if proven unprovable in first order of logic, then the proof happens in second order. From what I understood, you don't prove that it's always unprovable, just unprovable in first order of logic.
timelike loops occur in black hole physics...? circular light paths in time like "spatial" dimension inside the schwartzchild radius, EH appear to exist. Where Tlike D and Slike D swap coordinates
So the axioms of logic are outside of all axiomatics? Because if you add the proof by contradiction to the axions then the sentence results in a contradiction, a neither true nor false statement?
So what Gödel proved was that any finite number of axioms is insufficient. On the other hand, his coding of formal statements shows that you only have countably many statements that would need a proof. Can you define a formal system with a countably infinite (transfinite) number of axioms, that is in some sense "the simplest" or "smallest" from which you can prove all true statements? In analogy with something like the countably infinite set of prime numbers that describe all natural numbers (sans 1) through multiplication? In the case of primes you can also prove that a finite number of primes is insufficient. But from that you just conclude that the primes must be an infinite set, BUT they are still a well defined minimal set that you need to reach all integers through multiplication. I'm thinking of something like this for mathematical axioms, like "prime axioms", the minimal set of axioms from which all true statements can be proved. Has something like this been attempted?
"So what Gödel proved was that any finite number of axioms is insufficient." It's actually a little bit worse than that. Gödel's theorem still applies to certain axiomatic systems which have infinitely many axioms. It applies to any axiomatization that has a recursive axiom set. A recursive axiom set means that there exists an algorithm which is capable of taking in any sentence in your language and will determine whether or not that sentence is an axiom. So even if you have infinitely many axioms, if you are still able to tell apart axioms from non-axioms, Gödel's theorem will tell you that your axiomatization is either incomplete or inconsistent. So your thought of trying to construct a minimal set of axioms from which all true statement can be proven would not be helpful, because in order for it to have the property you desire, we would lose the ability to tell axioms apart from other statements. Of course, Gödel's theorem is about first-order logic. I know that second-order logic is a bit different, but I have never studied it.
Very interesting stuff! I didn't realize it was that bad. I feel that this should be part of the video extras at least, since it expands the (practical) implications of the theorem immensely. Thanks for the answer! :) P.S. You also write that second-order logic could be different. Could there potentially be a way around the theorem using second-order logic then, if I understood the implication correctly? ;)
I haven't studied second-order logic at all, but I know that Gödel's Completeness Theorem (which was his first landmark theorem - and by the way, the "completeness" in his completeness theorem is a different notion from the "completeness" in his incompleteness theorems, so no contradictions there) only holds for first-order logic. It is false for second-order logic. After a precursory search on mathstackexchange and mathoverflow, it seems that his incompleteness theorems still apply to second-order logic, but I was a bit confused by what I read, so I'm not willing to make any declarations.
A system with a large number of axioms is pretty bad. If you had an infinite number of axioms, then you would not need to prove anything. Any statement you can make (and its negation) is already an axiom. Mathematics would then simply be an infinite list. Very uninteresting. Suppose we made a very short list. All the solutions to the engineering problem of building a particular bridge, where the bridge does not collapse for certain loads. Well, that would be interesting, but impractical with today’s technology. The neat thing about mathematics is it allows us to calculate a small number of solutions by using limited axioms.
If I got it down correctly, Godel used logic to fabricate these codes, and we think about the axioms as logical statements. What if logic itself is inconsistent?
Márton Kardos Then you can prove everything, and its negation. Demonstrate this and fame and fortune will be yours...just be warned that your bank account will and will not be secure.
Is it really true that we have a gap between truth and proof? Is it not just that we can't have a definite proof by being self-referential? All the Theorem is saying is that we can't prove a true statement from the same system we derived this truth from but we sure can do it from a superior one. In fact Goedel created a higher level system precisely to proof that about mathematics. This would in fact make criteria for truth more consistent. Would this be a correct interpretation?
Science is based upon mathematics, so if mathematics has statements that can't be proven, then so does science. A hypothesis in physics should be a mathematical statement, such as F = ma. The only difference between physics and mathematics is that physics is concerned with applying mathematical statements to material objects.
