0:30 What is a math system? 0:47 Axioms 1:09 Theorems 1:42 Sufficiently Expressive 2:10 Completeness 2:32 True and its opposite are incompatible. 3:04 Inconsistent System 3:31 Unicorns exist 4:16 Principal of Explosion 4:36 Godel's 1st Incompleteness Theorm 5:36 Godel's 2nd Incompleteness Theorm 6:31 Beauty of Incompleteness
I believe that many people fail to understand how deep the halting problem and Gödel's incompleteness theorems is. They together with Church's Lambda calculus and Kolmogorov complexity are all models of computation and can be converted between each other. And they say something about the limits of our knowledge, about the definitions and expressiveness of languages, what randomness is or isn't. They are also related to the "there's no free lunch" theorem as in we can never create an algorithm which finds the optimal algorithm for a given problem, as well setting limits to what we know about optimization. In my opinion, the models' usefulness in mathematics and computer science pales in comparison to their implications for philosophy. All four models define limitations on knowledge and language, and yet it makes you wonder.
I think I might understand what you are saying. There Implications might spread to all fields of study. Maybe any ethical system we come up with can not proof itself.
He proved the truth is larger than proof and that you never actually proof the truth. What is even more amazing is that the was able to prove this using logical symbols. Proof is always using indirect methods - proof is a sub-domain of truth.
It doesn't say anything about our limit of knowledge and understanding, it says it about all sets that can be imagined. You could be an alien and still have the same problem.
Understanding it is the biggest mistake in mathematics is also a conversion experience. Unfortunately, only a small number of people on Earth understand this. Read Gödel's Mistake. Modern mathematics is a castle in the sky. It will be entirely replaced by new methods I have developed that will be able to solve problems that most mathematicians believe are "impossible" to solve.
@@JohnSmith-ut5th you provide no explanation for why you think the incompleteness theorem is wrong; it makes perfect sense and I fell like you are just trying to hold on to your Maths (not to be too ad hominem)
"The mathematic, then, is an art. As such it has its styles and style periods. It is not, as the layman and the philosopher (who is in this matter a layman too) imagine, substantially unalterable, but subject like every art to unnoticed changes form epoch to epoch. The development of the great arts ought never to be treated without an (assuredly not unprofitable) side-glance at contemporary mathematics." - Oswald Spengler
We never will know anything.for sure. That is a guarantee. At some level we have to be pragmatic about it and deal or accept the uncertainty of everything...
5:00 You have forgotten one extra condition: it must be algorithmically decidable whether or not a particular statement is an axiom. Because we can create a theory whose axioms are all statements which are true in the standard model of Peano arithmetic; such a theory is consistent (if Peano arithmetic is) and complete (in the sense that every statement expressible in the language of the theory can be proven true or false). But such a theory isn't particularly useful when we can't even know what the axioms of the theory are.
I think he meant decidability of the axioms to be built into the idea of a “math system”. But yeah, it would have been better if he’d made that explicit.
Well done intro to Godel. I just happen to row ( sculling ) with a world class mathematician and he keeps boggling my mind with examples like this. He handed me a book called "Everything and More"... a wonderful romp on the outer fringe... and one more book called "Naive Set-Theory" which left me hanging in mid air. It was "naive" in that the proof assumed you understood the axioms and theorems. It was anything but "naive" in the layman's sense. Once again, a very well done intro to Godel. Onward!!! Where are the follow up videos to this one? He asked naively.....
this was an amazing video, as others have already said. I just wanted to express my gratitude for you posting this very clear video. thank you so much.
Very good video, this time I think you managed to explain things very clearly. I'm looking forward for the next one! By the way, as a technical suggestion: consider getting one of this pieces of foam for your microphone (or maybe using some kind of post-processing) to suppress the "s" sound, it's kind of annoying when using headphones.
Apologies for the hard s's! I do have a pop filter, but I may need to adjust it around a little. There were also a ton of s's in this video (sufficiently expressive, consistent math system).
A very nice and brief introduction to the Incompleteness Theorems. And an easy to undertand example of how an inconsistent system vis a vis Principal of Explosion can both prove and disprove the existence of Unicorns. Like others who have posted, the deeper, philosophical implications have not been explored sufficiently.
Just because it cannot be proven doesn't mean it cannot be known. There are knowledge systems beyond proofs (experience, intuition, meditation), eventhough they've not been as successful in producing truths as rigorous proofs.
Gödel's Incompleteness Theorem has several implications for the future of Artificial Intelligence, specifically in the areas of completeness and consistency. Completeness: The theorem states that any formal system that is powerful enough to express basic arithmetic will always contain statements that are true, but cannot be proven within the system. This means that there will always be some truths or knowledge that an AI system, no matter how advanced, will not be able to discover or prove. For example, Tesla's "Full Self-Driving" system uses advanced AI algorithms to navigate the car, but the system will never be able to prove that it can handle all possible driving situations. Consistency: The theorem also states that any consistent formal system powerful enough to express basic arithmetic cannot be both consistent and complete. This means that if an AI system is consistent, meaning it does not produce contradictions, it will not be able to prove all truths. For example, ChatGPT is a powerful AI language model, but it is not able to prove all the statements it generates are true or consistent. Limitations: Gödel's Incompleteness Theorem highlights the limitations of any AI system, regardless of how advanced the technology becomes. It suggests that there will always be some areas of knowledge or tasks that an AI system will not be able to understand or accomplish perfectly. Human oversight: The theorem implies that AI systems will always need human oversight to ensure they are working as intended and to address any limitations or gaps in knowledge.
First of all, this is a great video. In order to understand way better these topics, I would suggest to read the book Gödel's Proof by Ernst Nagel & James Newman. Likewise, the books of the philosopher and mathematician Ludwig Wittgenstein - who was Russell's alumni- "Tractatus Logico-Philosophicus" and "Philosophical Investigations". If anyone is interested, then read those works.
1st theorem reminds of Heisenberg uncertainty principle. While uncertainty principle is for material sciences, godel theorems are for information systems.
Your proof of the Principle of Explosion is using the Disjunctive Syllogism and a Contradiction to prove any proposition *q*. HOWEVER, assuming that there is a contradiction renders the Disjunctive Syllogism as an invalid rule of inference. Therefore, the proof is not actually valid. I will try to explain this below: The disjunctive syllogism says this: *Premise_1: p OR q* (we call "p" and "q" disjunctives) *Premise_2: ~p* (where "~" indicates "not") *Therefore, we conclude q.* This Rule of Inference is Valid assuming that NO CONTRADICTIONs EXIST. The reason that it is valid is the following: The truth table of "OR" requires at least one of the disjunctives (p,q) to be true in order for the disjunction (i.e. premise_1) to be true. Since the premise_2 (~p) is true...then, GIVEN NO CONTRADICTIONS, we have that p is false...then since premise_1 requires at least one disjunctive to be true and the disjunctive p is false.... IT FOLLOWS NECESSARILY that q must be the true disjunctive making premise 1 true. Again....one of them must have been true...and it wasn't p. NOTICE how the assumption "NO CONTRADICTIONs EXIST" is required for its validity. Lets try abandoning this assumption. Given the existence of a Contradiction (p & ~p), notice how premise_2 stating that ~p true DOES NOT entail that p is false....because the contradiction says that p IS ALSO true. Therefore we can have premise_2 (~p) true....and ALSO have premise_1 true because the disjunctive p is true. Therefore, it is NOT NECESSARY for q to be true....p already took care of premise_1. So the validity of the disjunctive syllogism is lost in the presence of a true contradiction....we can have both (p OR q) true and (~p) true, and q being false.....the premises would still hold...without requiring q to be true. In summary, Disjunctive Syllogism in the presence of contradiction becomes invalid, and cannot be use to proof explosion...it doesnt follow.
Am I wrong but is math stuck in a classical world.? The cat can be alive and dead at the same time in the real world. So a statement can be false and true at the same time as long as you don't look at it. Here is another question. If I make a sentence as follows: "this sentence is a lie" So whatever the sentence says is not true, so then the sentence is true and then it is not true ....... This sentence is fluctuating in time between two meanings and it is an example of a Turing machine that won't stop ??? Also I really think the problem with Godel is induction. How do you count to infinity ? Since you can't physically do it , you bring in the concept of induction. So addition works for small numbers. Numbers small enough to be counted. How do we know induction will work on the big numbers. Where is the proof induction works? If you start counting and use the whole universe and its particles to count on you need a machine with a memory, once this machine gets to a certain size it will collapse into a black hole or some other physical limitation will arise. So you can only count so high and there is a number, the largest number , that is not infinity.
Yes Math is Created but Its also discovered. Math is created in a sense that all mathmatical symbols have no relevance in tangible Reality other than being specific of specific abstract communication medium which is Latin but are useful to laydown the cognitive ability of counting bestowed upon human beings. However Geometry do exist in our perception and we can see qualities and quanitities as well as analyze it and projecr abstract communication medium onto it to approximize it and solve its problems snd we have to build ontop of contradictions and paradoxes, we can never find reach Absolute Truth via Reasoning and Rational faculties, all we gonna do is put lego blocks of knowledge ontop of one another. Then You will be talking about Metaphysics which current science foudations it has no ability to do Metaphysics, metaphysics requires first person inquiry about Truth and Truth can not be conveyed in first person, it requires consensus and consensus requires abstract communication medium which itself is limited and does not convey a real time experience except a distorted pictures
A theory is incomplete, if there is some statement expressible in the language of the theory which it can't prove true or false. A theory is inconsistent, if there exists a statement which can be proven both true and false; it can be seen that an equivalent condition is that it can prove every statement. (Both definitions can't be simultaneously true.)
For anyone confused but want a firm grasp, try grabbing something. Good, grab some more stuff. Very good. Let’s assume you can grab anything. Very very good. Is there anything you can’t grab assuming your hand is infinite in size or strength? That’s right, the hand itself. The hand itself will always be out of grabbing reach. That’s incompleteness.
@@pedrovillanueva6767 the analogy is also old as shit lol. That analogy is an ancient Hindu idea. Going back thousands of years. The advaita master Mooji uses it a lot for psychological purposes. You can check out his videos on RU-vid
Gödel’s incompleteness theorem is actually The Ultimate Murphy's law. Standard Murphy is about "everything going wrong", but Gödel hammers you with "and you will never even know why" (you may think you know something, but you cannot be sure as your thinking system is incomplete or even inconsistent in the first place).
So, if math system A is consistent, we can't use A to prove that A is consistent due to Gödel's second incompleteness theorem. But could we build math systems B and C such as we can prove C to be consistent using B and prove B to be consistent using C? Or would that fall apart too?
Really good question! The short answer is that this doesn't work (en.wikipedia.org/wiki/Gödel's_incompleteness_theorems#Implications_for_consistency_proofs). If B could prove the consistency of C and vice versa, we'd be able to formalize a proof in C of its own consistency, which we know by the second incompleteness theorem is impossible.
Okay, what implications would it have if there was a situation where in which system A could prove the consistency of system B, but system B couldn't prove the consistency of system A? Would that not work as well?
+Noah Mendez That situation is pretty similar to the one Teddi P is describing. What if system A is inconsistent? If system A is inconsistent, its proof that system B is consistent isn't useful, but how do we verify system A is consistent? We've pushed the incompleteness theorem back one layer, but we still have the same fundamental issue.
All languages are descriptive. If a language is inconsistent, it isn't compatible with being descriptive of reality where inconsistencies never occur (that's called causality).
Math is perfect, in the usual sense of the word. The axioms that math is based on allow different models, or interpretations. It is a bit like starting from a description of a piano. Two different people build different pianos. Each does everything a piano is supposed to do, and yet there are variations in which extra things each can that are not proscribed by the description. Far from being a bug, this is a feature of axiomatic systems. The existence of non-standard models of mathematics is very useful in practice. It enable mathematicians to have their cake and eat it to. For example infinitesimally small numbers do not exist, and yet there is a model in which they do exist, but they are not accessible from within the axiomatic system. There is some subtlety in understanding what this means exactly. In summary mathematics is way more amazing than people realize, but it is hard to convey the details.
What does it mean to say a theorem is true? Godel Incompleteness makes a distinction between truth and provability. A provable theorem simply means that starting with one or more of the axioms and the rules of inference a theorem can be arrived at by simple string manipulation. It's like a system of dominoes, and a "domino" theorem says that a particular domino will fall if we give a push to the first few dominos (the axioms). A true but unprovable theorem means that no amount of string manipulation can take you, step by step, from the axioms to the theorem, yet the theorem is nonetheless true. OK, but what does it mean to say the theorem is true? There isn't some set of truths which transcends all formal systems and whose members are the true theorems of all possible formal systems. So a true theorem only makes sense within the context of a formal system of axioms and string manipulation rules. It appears that the truth of a theorem means that the theorem is a consequence of the axioms and string manipulation rules, but that for unprovable true theorems there is no amount of string manipulation steps that will get you from the axioms to the theorem. In the domino system, the analogy would be that a particular domino falls over at some point after we give a push to the first few dominos but that there is no step by step process that you can calculate ahead of time (without actually starting the dominos toppling chain reaction) that would show you (prove to you) that the target domino will fall. This sounds absurd. However, the domino system would need to be nonlinear (contain self-reference/recursion) for this to happen. Any system of standing dominos that I've ever witnessed didn't contain recursion. So this analogy breaks down, but nonetheless, it illustrates what is meant by truth and provability. The statement that Incompleteness only happens if the system is sufficiently expressive simply means that the system must exhibit self-reference. My novel Shards Of Divinities examines these ideas in more detail. See @t
Indeed, proof and truth are separate concepts in mathematical logic, and this is represented by two subbranches of mathematical logic: proof theory and model theory. Proof theory deals with, well, proof. And model theory deals with truth. Mathematical logic begins with a formal language. You have infinitely many symbols at your disposal which fall into certain classes of symbols: logical symbols (like and, not, or, etc.), constant symbols, variable symbols, function symbols, and relation symbols. Besides the meaning bestowed upon a symbol by what class it belongs to, these symbols do not have any inherent meaning. One can produce an interpretation of the language. This gives meaning to all of the symbols. First, one sets a domain of discourse by taking a set of objects. Then, the interpretation assigns each constant symbol an element in the domain of discourse. And it assigns each function symbol a function from the domain of discourse to itself. And it assigns each relation symbol ordered tuples in the domain of discourse. A relation on certain constants is said to be true if the tuple is associated to that relation within the interpretation. Proof theory does not concern itself with interpretations. It takes a set of sentences in the formal language, calls them axioms, and then studies what can be proven from those axioms. Model theory concerns itself with interpretations. Given a set of axioms, a model of that set of axioms is an interpretation under which all of the axioms are _true._ In first-order logic, there are two main theorems which connect proof and truth. The Soundness Theorem: If a sentence is provable from a collection of axioms, then that sentence is true in all models of those axioms. Gödel's Completeness Theorem (not a typo): If a sentence is true in all models of a set of axioms, then there exists a proof of that sentence from the axioms themselves. Now, it is possible that an axiom set can have multiple fundamentally different models. For example, the Peano axioms were developed as an attempt to pin down the natural numbers. But it turns out that there are other structures beside _just_ the natural numbers which follow all of the Peano axioms. The actual natural numbers then are referred to as the standard model of the Peano axioms, and there are also nonstandard models of the Peano axioms. This naturally leads to the idea of statements which are neither provable nor disprovable from a set of axioms. If you can find a sentence that is true in one model but false in another, it absolutely will not be provable from that axiom set. Now, Gödel's First Incompleteness Theorem found a method of concocting a sentence in the language of arithmetic which is true in the standard model of the Peano axioms but false in a nonstandard model. (It does a lot more than that, but when people say the whole "true but unprovable" thing, this is the setting they're talking about.) The original goal of the Peano axioms was to describe (the standard model of) the natural numbers, so we really only care about truth within the standard model (unless maybe you're a logician!) when dealing with the Peano axioms. Nonetheless, the fact that the sentence is false in a nonstandard model while being true in the standard model shows that it is not provable from the Peano axioms. If his theorem only applied in this one context, it wouldn't be remembered though. Gödel showed something more powerful. Any first-order axiom set which has a model in which all of the Peano axioms are true will have more than one model. In other words, there will be some statement which is true in one model of that axiom set while simultaneously being false in another model of that axiom set. In some sense, this shows that no usable axiom set in first-order logic can _ever_ uniquely pin down the set of natural numbers. If you take any useable axiom set which has a chance of describing the natural numbers (meaning that _the_ natural numbers is a model), then it will have another, fundamentally different model as well. Don't know if this really answers your question, but I thought I would share in case it's helpful.
You're correct; with an inconsistent system, you can prove anything, including blatantly contradictory conclusions (in fact, by definition of an inconsistent system it already has two contradictory statements). A system in which you can prove everything is not useful to us, which is why it's extremely important that our system is consistent.
@@UndefinedBehavior well no - you are not 'proving' anything except in the particular parameters of a logical system - eg. a sentential logic system. Unicorns do not exist in any logic system - people have to learn that Godel says nothing about the external world
"Inconsystent system can prove anything." - What about paraconsistent logic? It is enough to get rid of the disjunction introduction or disjunctive syllogism (one of these) and contradictions don't explode anymore - they become isollated, you won't prove everything.
I think that the fact that are numbers are a constants in a universe of infinitys. U can not use numbers that aren't constantly changing to understand. A constantly changing universe
I don't get the use of the 'puzzle' to describe it, and the thumbnail used, you don't really believe 'puzzles' are a description for reality do you? The white image behind the puzzle means what? Empty space is nothing, there wouldn't be a puzzle for empty space.
We won't be able to prove the consistency of maths using math itself. As math is the fundamental and even it cannot be used to prove it. Only way is to take it to be true. On the other hand it can still be incomplete but we can still be sane.
"Any sufficiently expressive math system must be either incomplete or inconsistent." Suppose that, using a given "sufficiently expressive math system", you could prove that you can't prove whether a given statement is true or false, within that same system. That would mean that you had proved that the system is incomplete, and therefore not inconsistent, e.g. that it is consistent. However, "A consistent math system cannot prove its own consistency". Would that mean that the system is inconsistent, because it is able to prove its own consistency? But then again we would then be basing ourselves on Gödel's Theorem(s) as part of the deduction, so maybe you could argue that the system couldn't prove it's own consistency because it also had to rely on Gödel's Theorem(s)? Maybe you could prove Gödel's Theorem(s) (or the one(s) you base yourself off of) within the system, though, so that is also a part of the system? Confusing.
Another funny thought: Imagine if we could prove that our math system is inconsistent, but were able to find a way to figure out which areas are inconsistent, and avoid them, while the rest of the system works properly.
You are right on the first paragraph. It is incomplete because you can prove that you can't prove whether all statement are true or false. But you could said that with certainty because it(math) is consistent always except that it can't prove its own consistency but anything else it can prove with certainty. It can prove or disprove( not all) but certain things with certainty because it is consistent but you can't use it to prove its (math's) own consistency. You have to take it as it is. You have to take it to be true, there is no other choice. As a matter of fact the only system that can prove its own consistency will be an inconsistent system. So, in short, we will never know/have the proof for the truth to be truth as truth is higher and beyond proof itself. That's something about the ultimate truth itself or the fundamental. You can't go beyond it. Because that's the fundamental, that's why. But we have the proof that we will not know the truth about everything else. So we will not know everything. That's to sum up. That's it.
It cannot prove its own consistency. It cannot prove everything (that is proven). But if it proves something then that is certain or consistent as it is always consistent except when it comes to prove its own consistency. Simply.
people see this as unfortunate, but I see this as beautiful. This, along with the various other uncertainty principles, are the main reason for why I am agnostic. As someone who loves understanding the world, there is some strange comfort in knowing that it is impossible to have complete knowledge.
It's cute that you use Yoda and Vader to liven up the discussion, but perhaps it would be better to somehow represent "not Yoda" to stay more conceptually consistent.
could anyone critique my notions here? Would really appreciate it....................As a follow up to my recent posts on (Goedel’s incompleteness theorem) the architecture of materiality and that of the realm of abstraction, the two structurally linked, which prohibits for formulation of conceptual contradictions, I present the following for critique. After watching several video presentations of Geodel’s incompleteness theorems 1 and 2, as presented in each I have been able to find, it was made clear that he admired Quine’s liar’s paradox to a measure which inspired him to formulate a means of translating mathematical statements into a system reflective of the structure of formal semantics, essentially a language by which he could intentionally introduce self-referencing (for some unfathomable reason). Given that it is claimed that this introduces paradoxical conditions into the foundations of mathematics, his theorems can only be considered as suspect, a corruption of mathematic’s logical structure. The self-reference is born of a conceptual contradiction, that which I have previously shown to be impossible within the bounds of material reality and the system of logic reflective of it. To demonstrate again, below is a previous critique of Quine’s liars paradox. Quine’s liar’s paradox is in the form of the statement, “this statement is false”. Apparently, he was so impacted by this that he claimed it to be a crisis of thought. It is a crisis of nothing, but perhaps only of the diminishment of his reputation. “This statement is false” is a fraud for several reasons. The first is that the term “statement” as employed, which is the subject, a noun, is merely a place holder, an empty vessel, a term without meaning, perhaps a definition of a set of which there are no members. It refers to no previous utterance for were that the case, there would be no paradox. No information was conveyed which could be judged as true or false. It can be neither. The statement commands that its consideration be as such, if true, it is false, but if false, it is true, but again, if true, it is false, etc. The object of the statement, its falsity, cannot at once be both true and false which the consideration of the paradox demands, nor can it at once be the cause and the effect of the paradoxical function. This then breaks the law of logic, that of non-contradiction. Neither the structure of materiality, the means of the “process of existence”, nor that of the realm of abstraction which is its direct reflection, permits such corruption of language or thought. One cannot claim that he can formulate a position by the appeal to truths, that denies truth, i.e., the employment of terms and concepts in a statement which in its very expression, they are denied. It is like saying “I think I am not thinking” and expecting that it could ever be true. How is it that such piffle could be offered as a proof of that possible by such a man as Quine, purportedly of such genius? How could it then be embraced by another such as Goedel to be employed in the foundational structure of his discipline, corrupting the assumptions and discoveries of the previous centuries? Something is very wrong. If I am I would appreciate being shown how and where. All such paradoxes are easily shown to be sophistry, their resolutions obvious in most cases. What then are we left to conclude? To deliberately introduce the self-reference into mathematics to demonstrate by its inclusion that somehow reality will permit such conceptual contradictions is a grave indictment of Goedel. Consider; As mentioned above, that he might introduce the self-reference into mathematics, he generated a kind of formal semantics, as shown in most lectures and videos, which ultimately translated numbers and mathematical symbols into language, producing the statement, “this statement cannot be proved”, it being paradoxical in that in mathematics, all statements which are true have a proof and a false statement has none. Thus if true, that it is cannot be proved, then it has a proof, but if false, there can be no proof, but if true it cannot be proved, etc., thus the paradox. If then this language could be created by the method of Goedel numbers (no need to go into this here), it logically and by definition could be “reverse engineered” back to the mathematical formulae from which it was derived. Thus, if logic can be shown to have been defied in this means of the introduction of the self-reference into mathematics via this “language” then should not these original mathematical formulae retain the effect of the contradiction of this self-reference? It is claimed that this is not the case, for the structure of mathematics does not permit such which was the impetus for its development and employment in the first place. I would venture then that the entire exercise has absolutely no purpose, no meaning and no effect. It is stated in all the lectures I have seen that these (original) mathematical formulae had to be translated into a semantic structure that the self-reference could be introduced at all. If then it could not be expressed in mathematical terms alone and if it is found when translated into semantic structures to be false, does that not make clear the deception? If Quine’s liar’s paradox can so easily be shown to be sophistry, how is Goedel’s scheme not equally so? If the conceptual contradiction created by Goedel’s statement “this statement has no proof” is so exposed, no less a defiance of logic than Quine’s liar’s paradox then how can all that rests upon it not be considered suspect, i.e., completeness, consistency, decidability, etc.? I realize that I am no equal to Goedel, who himself was admired by Einstein, an intellect greater than that of anyone in the last couple of centuries. However, unless someone can refute my critique and show how Quine’s liar’s paradox and by extension, Goedel’s are actually valid, it’s only logical that the work which rests upon their acceptance be considered as invalid.
Gödel's "Theory" only holds true until P=!NP is solved showing NP is false all problems are P. When that's solved you will get any and every answer instantly, provided the correct way of asking is invented. Which is where the Pig vs Unicorn breaks down, It should be understood that is not a math question about expressible geometries upon the existence in which we find ourselves. It's speculative conjecture built off historical fantasies. I don't understand why people don't understand what math is. Math is a mental abstraction of geometrical realities of force phenomena that govern the ebb and flow of reality. Math is just a language another way to say triangle spheres and currents.
logic is the way we defined the language to work or to have meaning i cant say he is tall and he is not tall then language has no meaning so the truth value of logic is no thing but is the way we defined the language to work some tried to prove that mathematics is extension to logic by reducing numbers to logic they failed because they use numbers while doing such a proof this is the incompleteness theorem there will be at least one fact about numbers un provable
I don't get the definition of consistent, Like what do you mean by "proving at most one of a statement and its opposite is knows as consistency". Like give an example, Maybe I don't know much english too understand that XD
Dunno if this makes sense but here is what I have to say. Inconsistent makes sense as well we built the whole system so flaws will always be there, there is no such thing as perfect the thing is how great of a degree is it though. It makes sense for it to be both in this case there is no way to stand outside of all the bubbles to see them all and a system was created by imperfect humans thus it is imperfect and will always be. The definition of perfect is something us humans made so nothing will ever be perfect thus everything is going to be to an extent chaotic and we will never know truly how chaotic because we form the world around the chaos we created thus we will always see it as something consistent because we force it into that state. Even though a lot of what we create math rules etc does a decent job of cookie cutting everything but as gravity already proves we are inconsistent. You supposedly weigh x on earth due to earth being y size etc but in fact, throughout different places, you lose and or gain weight thus we just round it thus us accepting the chaos as something that is not special thus us ignoring both issues above.
Gödel especially shown that classical mathematics is doomed, only computable mathematics makes sense and has no Gödel issues. Constructive mathematics ftw.
Seems to me a pretty self-evident theorem. There's a reason philosophers talk about first principles, that is, principles that can't be proved but only assumed. But again, so what? Philosophically, we recognize that contingencies ultimately presume necessities . . . so this ought to be regarded as (2500 year) old news.
Chris, yes, the difference between the necessary and contingent is a) important, b) widely misunderstood, but that’s not really what Gödel and pals were dealing with. The core of Gödel’s results seems to point to something deeply limited in the structure of our interaction with the rest of reality. So even if we started with a set of axioms which were necessarily true, such that we don’t need to assume them, still we cannot proceed from them to reach all well-formed true statements (assuming that the system as a whole fulfills the usual criteria in terms of expressive power.) That limit is _not_ a function of us having to start from some kind of first principles about whose truth we cannot be sure. It is a function of ... well I dunno, and I’m not sure anyone else does either. I speculate that it is something inherent in the structure of thinking and language, but I speculate too - a la Wittgenstein - that this very speculation is itself subject to the very limits about which we wonder. It does feel very much like we are in the vicinity of that about which we cannot speak and so about which we must remain silent. 🙂
@@bakedutah8411 Rewatching the video, I think I might see my misunderstanding and I wonder if you can comment further. Per the explanation here, an explanatory system must be incomplete, which means that it must contain some statements that cannot be derived from axioms or statements derived from axioms. I presume from this that the theorem requires that no number of axioms can ever get you to a complete system -- perhaps theoretically (I'm completely guessing here), every axiom designed to plug a hole, so to speak, is by necessity going to open another hole, again so to speak, somewhere else. So our necessary and therefore assumed statements don't get at Godel's point, because those necessary and therefore assumed statements are simply the axioms of our necessarily incomplete system (if, again, we want our system to be explanatory). Is that correct? If so, do you know of any example outside of mathematical systems to illustrate this problem in the real world? Or are all the examples relegated to paradoxical statements along the lines of the liar's paradox?
Chris, well the devil is in the very technical details with this stuff, but basically, yes, that’s correct. Suppose you have a formal system based on set of axioms A1, for which there is an unreachable theorem T1. Well of course we could simply add T1 to A1 to get a new set of axioms, A2 and associated new formal system. That new system can now reach T1 (obviously, because we loaded it in up front as an axiom), but Gödel shows that there will then be a new theorem, T2, that is unreachable. And we can repeat the process ad infinitum. As to a non-mathematical system displaying the problem, I’m not sure that would be practically possible simply because such “real” systems aren’t typically defined with the precision needed to apply this kind of proof. However, there are non-trivial, “real” mathematical problems that at very least show that Gödel’s results are not just playthings of the pure theorist. One example is Paris-Harrington (en.wikipedia.org/wiki/Paris-Harrington_theorem )
@@karihotakainen5210 in order to prove a system with finite number of axioms is consistent you need it to be complete (self contained and without any outside dependencies), but the first theorem already states that no system with finite number of axioms is both complete and consistent. listen to 5:55 what are you hearing there? can you paraphrase?
@@real_one Gödel's Second Incompleteness Theorem is, indeed, a quick corollary of the first, but its implications to mathematical logic are so important that it deserves the title of "theorem".
Thomas Fisherson Wittgenstein has this sort of expressive limitation thing going on in the Tractatus. A language cannot express its expressiveness, that is, relate its relation to the objects it designates, that is, be its own meta-language.
metaphysics is a bitch... or maybe not really... unless we don't really exist, in which case we wouldn't have thought about levels of understanding... i'm humbled by your intelligence sir
This sounds like a question of who has the burden of proof. Aren’t the proofs for Gödel’s theorems subject to Gödel’s incompleteness theorems and thus self-refuting? So the whole problem of incompleteness is due to the inability to prove consistency? Lol, that shouldn’t be a problem if you just go with the way things seem and let the burden of proof fall to the person claiming the inconsistency.
"Burden of proof" is the stuff of science and law, where one relies on evidence to prove things. In the realm of logic and mathematics things are proved by reasoning rather than evidence. One does not require evidence to know that 1+1=2 or e^(i pi)=-1. Mathematical facts follow from what we already know. the proof of Gödel’s incompleteness theorems are not subject to Gödel’s incompleteness theorems because Gödel’s incompleteness theorems are about systems of logic and proofs are not themselves systems of logic.
