Proof theory is an area of logic that studies proof as formal mathematical objects. For questions asking how to write proofs or for checking an informal proof, please use the proof-writing or proof-strategy tags instead.

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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 advantage/disadvantages of proving ...
104
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11answers
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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 ...
58
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8answers
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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 ...
49
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8answers
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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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13answers
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What is a proof?

I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra). Mathematics is a system of axioms which you choose yourself for a set of undefined entities, ...
43
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2answers
1k views

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?
30
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6answers
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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 ...
18
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5answers
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How to prove the mathematical induction is true?

I have no idea about the underlying theory from which the mathematical induction was derived. How to prove the mathematical induction is true?
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5answers
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If it takes infinite steps to prove a statement, is that a valid proof?

In Cantor's diagonal argument, it takes (countable) infinite steps to construct a number that is different from any numbers in a countable infinite sequence, so in fact the proof takes infinite steps ...
15
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16answers
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Beautiful, simple proofs worthy of writing on this beautiful glass door [closed]

What are some of the more beautiful proofs you know? I am measuring beauty in two dimensions -- first, how conceptually elegant is it and second, how aesthetically pleasing is it. Context: I work ...
15
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2answers
261 views

How much math do we need to prove all simple numeric identities?

Consider real numeric expressions build only from integers, operators $+,-,\times,/$ and taking a positive expression to a power (no variables involved), e.g. $$\frac{2}{7},\ 2^{1/2},\ ...
14
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7answers
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Why do statements which appear elementary have complicated proofs?

The motivation for this question is : Rationals of the form $\frac{p}{q}$ where $p,q$ are primes in $[a,b]$ and some other problems in Mathematics which looks as if they are elementary but their ...
14
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4answers
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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 ...
13
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7answers
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Is a counterexample considered a rigorous proof that a property is not true?

This is my follow-up question to my own query earlier: How can I algebraically prove that $2^n - 1$ is not always prime? Almost half of the answers said that I provided my own proof by giving ...
13
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7answers
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Difference between “Show” and “Prove”

In many mathematics problems you see the phrase "prove that..." or "show that..." something is. What's the difference between these two phrases? Is "showing" something different from "proving" ...
13
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5answers
220 views

Does “=” have to be interpreted as equality?

To put it briefly: In model theory, we are allowed to interpret any relation symbol in any way we like. So why do people seem to require that "$=$" is interpreted as the actual equality? Let me ...
12
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6answers
2k views

Aren't constructive math proofs more “sound”?

Since constructive mathematics allows us to avoid things like Russell's Paradox, then why don't they replace traditional proofs? How do we know the "regular" kind of mathematics are free of paradox ...
12
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4answers
443 views

Existence Proofs

This may be a stretch, but are there examples of proofs that prove that a proof exists for a theorem. For example, if A is a theorem, and it is too tedious to prove that, is it possible to show that ...
12
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1answer
168 views

Can all math results be formalized and checked by a computer?

Can all math results, that have been correctly proven so far, be formalized and checked by a computer? If so, what type of logic would need to be used there? I've heard that the first-order logic is ...
12
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3answers
406 views

On Pudlak's “Life in an Inconsistent World”

In his Logical Foundation of Mathematics and Computational Complexity (2013), Pavel Pudlak invites the readers to ponder about fictitious people whose natural numbers are nonstandard. His exposition ...
12
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2answers
258 views

Is it possible to formalize the relationship between different proofs of the same theorem?

Some theorems have many proofs. Examples include the Pythagorean Theorem and the Law of Quadratic Reciprocity. I was wondering if one could formalize the relationship between these proofs. Sure, they ...
12
votes
1answer
204 views

Unprovable unprovability

In general, mathematical conjectures are resolved by proof, disproof, or proof that they are neither provable nor disprovable. Is it possible that some open conjectures cannot be settled in any of ...
11
votes
2answers
921 views

How to find the shortest proof of a provable theorem?

Roughly speaking, there are some fundamental theorems in mathematics which have several proofs (e.g. Fundamental Theorem of Algebra), some short and some long. It is always an interesting question ...
11
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4answers
442 views

Is there a connection between length of sentence and length of proof?

My basic question is: "Do longer tautologies take longer to prove?" But obviously this is underdetermined. If you are allowed an inference rule "Tautological Implication" then any tautology has a 1 ...
11
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3answers
511 views

Gentzen Cut elimination: Why do we have to “go infinite”?

I found some slides here that say you can't do cut elimination on PA with axioms like $$\frac{P(Z)\;\;\;\;\;\forall n,\,P(n) \implies P(Sn)}{\forall n,\,P(n)}$$ (which denotes infinitely many axioms ...
11
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1answer
938 views

The Power of Lambda Calculi

A simple question here, which likely demands a somewhat complex answer... Or rather, a set of related questions. What are the advantages of typed lambda calculus over untyped lambda calculus in ...
9
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5answers
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List of Mathematical Impossibilities proved using special tools

It is always weird to see a proof that something is impossible, especially when the tools used in the proof have nothing to do(at a first sight) with the original statement of the problem. I know a ...
9
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3answers
150 views

The category of theorems and proofs

On a philosophy website, it said that you could have a category with theorems as objects and proofs as arrows. This sounds awesome, but I couldn't find anything on the web that has both "category" and ...
9
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1answer
201 views

Gödel's way of teaching non-standard models to Takeuti.

In Memoirs of a Proof Theorist, Gaisi Takeuti relates how Gödel taught him about nonstandard models in an "interesting" way: It went as follows. Let T be a theory with a nonstandard model. By ...
9
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2answers
301 views

Minimal difference between classical and intuitionistic sequent calculus

Consider propositional logic with primitive connectives $\{{\to},{\land},{\lor},{\bot}\}$. We view $\neg \varphi$ as an abbreviation of $\varphi\to\bot$ and $\varphi\leftrightarrow\psi$ as an ...
9
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3answers
367 views

Prove by Hilbert deduction: $\vdash _{HFOL} \forall x (\neg(A \to \neg B))\to \neg(\forall xA \to \neg(\forall xB))$

I'd really like your help proving: $\vdash_{HFOL} \forall x (\neg(A \to \neg B))\to \neg(\forall xA \to \neg(\forall xB))$ Where $HFOL$ is the proof system which contains the Hilbert relevant ...
9
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1answer
421 views

Packing boxes and proof of Riemann Hypothesis

From Scott Aaronson's blog: There’s a finite (and not unimaginably-large) set of boxes, such that if we knew how to pack those boxes into the trunk of your car, then we’d also know a proof ...
9
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2answers
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How can I learn about proofs for computer science?

I study computer science at a university. My school offers several courses where various proofs are expected, but there is no course that introduces the fundamental concepts of proofs and how to write ...
8
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1answer
154 views

Can all theorems of $\sf ZFC$ about the natural numbers be proven in $\sf ZF$?

I know a proof of Hindman's theorem that uses ultrafilters on the natural numbers, and ultimately, the axiom of choice. But the theorem itself is essentially a combinatorial property of the natural ...
8
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8answers
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Tricks for Constructing Hilbert-Style Proofs

Several times in my studies, I've come across Hilbert-style proof systems for various systems of logic, and when an author says, "Theorem: $\varphi$ is provable in system $\cal H$," or "Theorem: the ...
8
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2answers
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Definition of “non-constructive proof”

I was wondering if it is possible to define exactly what a non-constructive (nc) proof is. I have often seen the concept associated with the use of principles such as the axiom of choice or the law of ...
8
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4answers
184 views

How or why does intutionistic logic proof negations from within the theory, constructively?

I'm having a little of a cognitive dissonance why, in intuitionistic logic, it's possible to show stentences like $(\neg A \land \neg B) \implies \neg(A\lor B).$ In plain text: If 'A isn't true' as ...
8
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1answer
238 views

Reverse Mathematics of Well-Orderings

In Simpson's book, a well-ordered set $X$ is a linear ordering such that there are no functions $f : \mathbb{N} \rightarrow X$ which is decreasing. However, a familiar definition of well-ordering is ...
8
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1answer
238 views

On the existence of closed form solutions to finite combinatorial problems

Is it possible that a finite combinatorial problem may admit a closed form solution, and for it to be impossible in practice to prove the validity of this solution? I'm not sure if a rigorous ...
8
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1answer
115 views

Is there a way to tell how many different ways you can prove a theorem?

Consider the question. Given the nature of a sentence $S$, it there any way to tell how many different ways you can prove this sentence? Proofs are not distinct if we have a situation such as: $P ...
7
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3answers
447 views

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 $\Sigma_1$ ...
7
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2answers
210 views

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 $T$, ...
7
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2answers
183 views

Theorems with one-line proofs [closed]

Inspired by this very concise answer, which proves that $$\sin^2(\theta)+\cos^2(\theta) \equiv 1 $$ as follows: $f(\theta)=\cos^2\theta+\sin^2\theta \quad;$ then it's simple to see that ...
7
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2answers
262 views

Consistency of PA: why other proofs?

Completeness theorem affirms that a formal first order system is consistent iff it has a model. The FOL number theory(PA) or First Order Arithmetic has a model, which is the natural numbers structure. ...
7
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2answers
456 views

Who stole the axioms in Natural Deduction?

The study of Gentzen's sequent calculus give me the opportunity to propose some reflections about the concept of logical truth. I'll refer to the english edition of Gentzen's works : The collected ...
7
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1answer
398 views

Can Robinson's Q prove Presburger arithmetic consistent?

I made an assertion in What are some examples of theories stronger than Presburger Arithmetic but weaker than Peano Arithmetic? that Q has higher consistency strength than Pres, Presburger arithmetic; ...
7
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1answer
136 views

Learning how to prove that a function can't proved total?

In proof-theory one can prove that in, say, Peano Arithmetic one can't prove a function $f$ total. Often this seems to mean $f$ is growing too fast to be provably total. I have some background in ...
7
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1answer
199 views

Are coinductive proofs necessary?

I've been exploring corecursion in Coq (specifically, infinite streams of natural numbers) lately and so far any coinductive predicate I've constructed and its coinductive proof can be transformed ...
7
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1answer
128 views

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 ...
6
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4answers
399 views

Are there “essentially non-constructive” statements?

There exist constructive and non-constructive proofs. Sometimes, for a mathematical statement, we can have both non-constructive and a constructive proof. However, are there statements for which ...