# Tagged Questions

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-* ...

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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 advantage/disadvantages of proving ...
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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 ...
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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 ...
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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 ...
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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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### 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. ...
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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 ...
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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 ...
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### Proof-theoretic characterization of the primitive recursive functions?

The total recursive functions are exactly those number-theoretic functions that can be represented by a $\Sigma_1$ formula of first-order arithmetic. Is there a similar characterization of the ...
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### Why are $\Delta_1$ sentences of arithmetic called recursive?

The arithmetic hierarchy defines the $\Pi_1$ formulae of arithmetic to be formulae that are provably equivalent to a formula in prenex normal form that only has universal quantifiers, and $\Sigma_1$ ...
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### How is the standard model of number theory specified, and why can't we use that specification to prove any number theoretical sentence of interest?

According to Gödel's incompleteness theorem, there exists a sentence $G$ in the vocabulary of number theory ($N$) which is not provable from any (recursively enumerable) consistent set of axioms $T$, ...