Gödel's Incompleteness Theorem - Numberphile

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Recommends AlgorithmsX · 2017-06 · Mathematics Stack Exchange · on “Gödel's incompleteness theorem: if the statement is unprovable then how did we prove its true?”

@CarlMummert has a good explanation for what's wrong with this problem. Related Numberphile video that explains exactly why this happens: [this video]

Recommends BENG · 2019-03 · Mathematics Stack Exchange · on “Model Theoretical Interpretation of the Incompleteness of Number Theory”

This question was sparked by this Numberphile video: [this video] . Near the end, (12:05), he speaks about the Riemann Hypothesis. He describes that if Riemann is shown to be an unprovable statement, then this proves that Riemann is true. For if it were false then it would be provably false (demonstrable by the existence of a non-trivial zero not at $\frac{1}{2}$ ) and thus decidable, contradiction. My question is about the model theoretic interpretation of this. From my understanding, undecidable sentences are not valid and their negations are not valid. This is from the completeness of the logic i.e. $\{\vdash{A}\}$ if and only if $\{\vDash{A}\}$ . Thus, if $A$ is undeceivable then for some mod

Recommends Minnella · 2020-04 · Mathematics Stack Exchange · on “Gödel's Incompleteness Theorem - follow up question”

I recently rewatched the great video that Numberphile created about Gödel's Incompleteness Theorem ( [this video] ) and I had a follow up question of similar nature. It essentially addresses the objective truth of a statement, outside of whatever system we impose to try and prove it. So, is it true that "every true statement can be proven true in SOME system"? Or on the other hand, do there exist statements that can never be proven true no matter what system is used, aside from paradoxes. I'm a junior studying mathematics and physics so feel free to be somewhat technical in responses :) Thanks! Edit: let's assume that the statement isn't an axiom and can't just be made one

Recommends Muber17 · 2022-09 · Mathematics Stack Exchange · on “How do we find mathematical equation of any statement which is not about mathematics with Gödel numbering?”

I have 3 questions and they are closely related so I asked them in same post. With Gödel numbering we can encode statements like "0 = 0" or maybe "a then b". And this is basically just transformation of symbols or group of symbols to a unique number. My question is how do we encode This statement is cannot be proved from the axioms (which bring us to Gödel's Incompleteness Theorem) or any other statement that is not written with mathematical symbols but with words in any language? Other than that, what are the limits of Gödel numbering? I mean is it possible to encode "I like to play violin"? Does that mean if we can write any statement with Gödel number, it can be also written with mathemat

Cites Sriotchilism O'Zaic · 2017-06 · Mathematics Stack Exchange · on “Decidability of the Riemann Hypothesis vs. the Goldbach Conjecture”

In the most recent numberphile video [this video] , Marcus du Sautoy claims that a proof for the Riemann hypothesis must exist (starts at the 12 minute mark). His reasoning goes as follows: If the hypothesis is undecidable, there is no proof it is false. If we find a non-trivial zero, that is a proof that it is false. Thus if it is undecidable there are no non-trivial zeros. This constitutes a proof the hypothesis is true, thus it is decidable. This makes perfect sense, however only seconds earlier he states that the Goldbach conjecture might be undecidable. It seems to me that the exact same reasoning we applied to the Riemann hypothesis could be applied to the Goldbach conjecture. Just switch o

Cites Matthew Ciaramitaro · 2017-06 · Mathematics Stack Exchange · on “If the Collatz Conjecture is unsolvable is it true?”

Recently Numberphile uploaded a video [this video] on Godel's Incompleteness Theorem, and the Professor in the video made the conclusion that if you can prove a statement cannot be proven true or false by the axioms and if that statement happened to be false by some counterexample, there would be a contradiction, and so the statement would be true. Then if someone could prove the Collatz Conjecture was unsolvable would they have proven that it is true? Or is that overreaching from the conclusion in the video. Also, has anyone attempted a rigorous proof that the Collatz Conjecture is unsolvable?

Cites Jerry Guern · 2017-06 · Mathematics Stack Exchange · on “Can you prove a theorem by proving it’s undecidable? (Numberphile question)”

Numberphile posted a vid on 5-31-17 about Godel’s Incompleteness Theorem. Near the end of the vid, the professor argues that proving that the Riemann Hypothesis is undecidable would PROVE the Riemann Hypothesis. His logic, as I understand it, that a RH counter-example would render the problem decidable, therefore if the RH is undecidable no counter-example can exist. [this video] This an Oxford math prof, so I'm sure he knows what he’s doing, but deciding something by proving it can’t be decided seems like a wee contradiction to me. Can anyone clear this up?

Cites Michael Novak · 2018-07 · Mathematics Stack Exchange · on “Why does, in theory, you can prove Riemann hypothesis with Godel Incompleteness Theorem, but can't prove the consistency of mathematics?”

Some time ago, I watched this fascinating episode of Numberphile about Godel Incompleteness Theorem: [this video] In the video, Professor du Sautoy explains the theorem (with big simplifications) and he also mentions that if we could prove that the Riemann hypothesis is one of those propositions that can't be proved, then it means that the theorem is true, since if it was false we would have been able to find a contradiction. So we would be able to disprove it. In the first Extra footage video [LINK] Professor du Sautoy mention that one of the conclusions from this theorem is that the proposition mathematics \ arithmetic is consistent is one of those propositions that we can't prove. I don't unde

Cites user645636 · 2019-05 · Mathematics Stack Exchange · on “Plurality of arithmetics? Or absoluteness of arithmetical truths? (i. e. Are all mathematical truths actually conditional?)”

All of math comes with conditions. Here's a list of different and related arithmetics: Presburger arithmetic [LINK] Peano arithmetic [LINK] Robinson arithmetic [LINK] Second-order arithmetic [LINK] Modular arithmetic [LINK] ( $2+2\equiv 1 \bmod 3$ ) This list is far from complete. All either generalize other arithmetic, or have different conditions. Some not included ( probably subsets of more areas of math), would be p-adic arithmetic, matrix arithmetic, vector arithmetic, polynomial arithmetic, etc. All arithmetic rules you might learn in school, need not apply generally in anything less than a specific Field [LINK] for example. All we prove with proofs, is consistency with a given framewo

Cites Yash Jain · 2019-11 · Mathematics Stack Exchange · on “Assumption of Gödel's Incompleteness Theorem”

I am not a mathematician and have not read Godel's Proof. My knowledge about Godel's theorem is limited to this [this video] . So my question is this (please correct me if I have misinterpreted or misunderstood the meaning): According to Godel's theorem, every statement is either true or false. And there are statements which cannot be proven true or false with the axioms of that mathematical system. Godel's Theorem assumes that all statements can only be true or false. Now as obvious as this statement might be but does this not bound Godel's theorem to mathematical systems which have statements which are either true or false. I mean there might be a system of mathematics where statements might ei