Questions tagged [provability]

For questions on provability, the capability of being demonstrated or logically proved.

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Is there any conjecture that has been proved to be solvable/provable but whose direct solution/proof is not yet known?

In mathematics, is there any conjecture about the existence of an object that was proven to exist but that has not been explicitly constructed to this day? Here object could be any mathematical object, such as a number, function, algorithm, or even…
lone student
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What is a simple example of an unprovable statement?

Most of the systems mathematicians are interested in are consistent, which means, by Gödel's incompleteness theorems, that there must be unprovable statements. I've seen a simple natural language statement here and elsewhere that's supposed to…
Michael Harris
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Is there any conjecture that we know is provable/disprovable but we haven't found a proof of yet?

I know that there are a lot of unsolved conjectures, but it could possible for them to be independent of ZFC (see Could it be that Goldbach conjecture is undecidable? for example). I was wondering if there is some conjecture for which we have proved…
Abc
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I don't understand Gödel's incompleteness theorem anymore

Here's the picture I have in my head of Model Theory: a theory is an axiomatic system, so it allows proving some statements that apply to all models consistent with the theory a model is a particular -- consistent! -- function that assigns every…
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How long can proofs be?

Let $$f(n) = \max\{\text{length of shortest proof of }\varphi \mid \varphi \text{ is a provable ZFC sentence of length } \leq n\}$$ How fast does $f$ grow? Is it polynomial, exponential, more than exponential, etc.?
user623070
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BIG LIST: Statements that look obviously false but cannot be disproved

I'm looking for statements that look obviously false but have no disproof (yet). For example The base-10 digits of $\pi$ eventually only include 0s and 1s. To make this question a little objective, I'm thinking about the "Vegas gambling odds" I…
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Statement provable for all parameters, but unprovable when quantified

I've been reading a book on Gödel's incompleteness theorems and it makes the following claim regarding provability of statements in Peano arithmetic (paraphrased): There exists a formula $A(x)$ such that the statements $A(0), A(1), A(2), \dots$ are…
Matěj G.
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Gödel's paradox: Why is "a proof that some universal statement is unprovable" not a valid proof that this statement is true?

Here is a paradox I have some difficulty resolving: As far as I understand, by one of Gödel's incompleteness theorems, in a first order logic theory with Peano arithmetic, one can find some non-trivial universal closed sentences (starting with a…
jam
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Can unprovability unprovable? Is there an $\omega$-fold unprovability?

I was just thinking about unprovability. I just wanted to know if it is possible to make a concrete boundary between provable problems and unprovable problems in a certain axiomatic system. We know that there is a statement that is true yet…
Henry
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Is it really impossible to lose the Hydra game?

In a number of blog posts I found the claim that the game described below cannot be lost, which is to say, every possible strategy is a winning strategy. In each case, a sketch proof is given that involves ordinal arithmetic, and the claim is made…
N. Virgo
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Semantics for minimal logic

Minimal logic is a fragment of intuitionistic logic that rejects not only the classical law of excluded middle (as intuitionistic logic does), but also the principle of explosion (ex falso quodlibet). Essentially, from proof-theoretical viewpoint,…
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Growth-rate vs totality

How can one prove the statement, "If a function grows fast enough, it cant be proven total in PA, unless PA is inconsistent"? How fast must it grow to be not provably total?
user77335
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Is there a decision procedure for intuitionistic propositional logic?

Is intutionistic propositional logic decidable? If so, what is a decision procedure for it, like tableaux for classical propositional logic? EDIT: In the first revision I mistook "predicate" for "propositional".
Pteromys
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What's the difference between "unprovable" and "undecidable"?

It seems to me that there is a difference between an unprovable sentence, and an undecidable sentence, but sometimes I have the impression that some authors use the terms interchangeably. In my understanding, if something is undecidable, then it is…
fonini
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Prove the inequality is true

Here is a question that I need to prove Prove that for $a, b \geq 0$ $$a^8+b^8\geq a^3b^5+a^5b^3$$ So far I have managed to simplify to $$(a^3-b^3)(a^5-b^5)\geq 0$$
Alex
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