Questions tagged [proof-theory]

Proof theory is an area of logic that studies proof as formal mathematical objects. If you'd like advice on the presentation of a proof you have in draft, use proof-writing instead. If you'd like feedback on its validity, use proof-verification. If none of the above apply, you do not need a proof-* tag.

This tag is NOT intended for questions asking how to write proofs or for checking a proof posted in the question. If you'd like advice on the presentation of a proof you have in draft, use instead. If you'd like feedback on its validity, use . If none of the above apply, you do not need a proof-* tag.

Proof theory is an area of logic that studies proofs and deductive systems as formal mathematical objects. Formally, a proof is a sequence of symbols that obeys the rules of the deductive system.

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Can every proof by contradiction also be shown without contradiction?

Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantages/disadvantages of proving by contradiction? As an aside, how is proving by…
sonicboom
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How do we prove that something is unprovable?

I have read somewhere there are some theorems that are shown to be "unprovable". It was a while ago and I don't remember the details, and I suspect that this question might be the result of a total misunderstanding. By the way, I assume that…
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Do we know if there exist true mathematical statements that can not be proven?

Given the set of standard axioms (I'm not asking for proof of those), do we know for sure that a proof exists for all unproven theorems? For example, I believe the Goldbach Conjecture is not proven even though we "consider" it true. Phrased another…
Jeremy
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Are the "proofs by contradiction" weaker than other proofs?

I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the reason? Do logicians think that proofs by…
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Are proofs by contradiction really logical?

Let's say that I prove statement $A$ by showing that the negation of $A$ leads to a contradiction. My question is this: How does one go from "so there's a contradiction if we don't have $A$" to concluding that "we have $A$"? That, to me, seems the…
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Proving the existence of a proof without actually giving a proof

In some areas of mathematics it is everyday practice to prove the existence of things by entirely non-constructive arguments that say nothing about the object in question other than it exists, e.g. the celebrated probabilistic method and many things…
Damian Reding
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What is exactly the difference between a definition and an axiom?

I am wondering what the difference between a definition and an axiom. Isn't an axiom something what we define to be true? For example, one of the axioms of Peano Arithmetic states that $\forall n:0\neq S(n)$, or in English, that zero isn't the…
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Is it possible that "A counter-example exists but it cannot be found"

Then otherwise the sentence "It is not possible for someone to find a counter-example" would be a proof. I mean, are there some hypotheses that are false but the counter-example is somewhere we cannot find even if we have super computers. Sorry,…
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Is it possible to prove a mathematical statement by proving that a proof exists?

I'm sure there are easy ways of proving things using, well... any other method besides this! But still, I'm curious to know whether it would be acceptable/if it has been done before?
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When are two proofs "the same"?

Often, we find different proofs for certain theorems that, on the surface, seem to be very different but actually use the same fundamental ideas. For example, the topological proof of the infinitude of primes is quite different than the standard…
William Stagner
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If all sets were finite, how could the real numbers be defined?

An extreme form of constructivism is called finitisim. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who supported such a system. I can see that the natural…
AgCl
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Can proof by contradiction 'fail'?

I am familiar with the mechanism of proof by contradiction: we want to prove $P$, so we assume $¬P$ and prove that this is false; hence $P$ must be true. I have the following devil's advocate question, which might seem to be more philosophy than…
MGA
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Under the Curry-Howard correspondence or loosely "proofs-as-programs", do we also have "programs-as-proofs" and what would some arb. program prove?

Curry-Howard Correspondence Now, pick any 5-30 line algorithm in some programming language of choice. What is the program proving? Or, do we not also have "programs-as-proofs"? Take the GCD algorithm written in pseudo-code: function gcd(a, b) …
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What is the difference between ⊢ and ⊨?

I want to know the difference between ⊢ and ⊨. http://en.wikipedia.org/wiki/List_of_logic_symbols ⊢ means ”provable” But ⊨ is used exactly the same: A → B ⊢ ¬B → ¬A A → B ⊨ ¬B → ¬A Can you present a good example where they are different? Is it…
Niklas Rosencrantz
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Why an inconsistent formal system can prove everything?

I am reading a Set Theory book by Kunen. He presents first-order logic and claims that if a set of sentences in inconsistent, then it proves every possible sentence. Since he does not explicitly specify the inference rules, I became curious as to…
Gadi A
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