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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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 ...
75
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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 ...
120
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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 ...
6
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4answers
641 views

What is the “correct” reading of $\bot$?

I have some doubts about the "natural" interpretation of $\bot$ in Natural Deduction and sequent calculus. In Prawitz (1965) $\bot$ (falsehood or absurdity) is called a sentential constant [page 14] ...
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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 ...
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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 ...
5
votes
3answers
422 views

Aftermath of the incompletness theorem proof

This is somewhat of a minor point about the incompletness theorem, but I'm always a little unsure: So one proves that there is a formula which is unprovable in the theory of consideration. Okay, at ...
4
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1answer
461 views

Can it be shown that ZFC has statements which cannot be proven to be independent, but are?

I am familiar with the concept that a statement can be proven indepent such as in the case of the continuum hypothesis where both ZFC+CH and ZFC+(CH is false) are both proven consistent, but I would ...
35
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4answers
2k views

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 ...
13
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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 ...
19
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5answers
3k views

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?
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" ...
6
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4answers
436 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 ...
10
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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 ...
6
votes
4answers
2k views

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: ...
5
votes
3answers
756 views

Is the negation of the Gödel sentence always unprovable too?

The incompleteness theorem says that certain theories+deduction system contain at least one sentence (the Gödel sentence "$G$"), which can't be proven (in the system in which it holds). (i) Is ...
7
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1answer
464 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; ...
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6answers
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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. ...
40
votes
12answers
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Are proofs by contradiction really logical? [closed]

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 ...
34
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11answers
2k views

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 ...
35
votes
2answers
426 views

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 ...
14
votes
7answers
2k views

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 ...
13
votes
2answers
1k 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 ...
21
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7answers
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Can you use both sides of an equation to prove equality?

For example: $\color{red}{\text{Show that}}$$$\color{red}{\frac{4\cos(2x)}{1+\cos(2x)}=4-2\sec^2(x)}$$ In high school my maths teacher told me To prove ...
12
votes
4answers
473 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 ...
11
votes
2answers
513 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 ...
18
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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 ...
13
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3answers
467 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 ...
8
votes
1answer
206 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
votes
1answer
270 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 ...
5
votes
1answer
247 views

What are various proofs good for?

There are plenty of questions around here, which are proven to be right or wrong in various ways. I wonder, what one can learn from these differing ways of how to prove something, despite the fact ...
6
votes
2answers
282 views

Ideas about Proofs

If there are two different proofs for one theorem, at some level are the two proofs the same, or can they be fundamentally different? In other words, if you have two proofs of a theorem, can one show ...
5
votes
5answers
186 views

what is a valid mathematical proof?

from what i have seen in my experience with math we can say that a valid proof is one that uses some form of logic (usually predicate logic) and uses logical rules of deduction and axioms or ...
4
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2answers
358 views

Model-theory and Proof-theory in Propositional Logic

I'm trying to link results of model theory and proof-theory in propositional language. Here i will use $\models$ to denote logical consequence, in the model-theory sense. Being $x,y$ two formulas of ...
4
votes
3answers
656 views

What do we mean by an “Elegant Proof”? [closed]

What do we mean when we say that a mathematical proof is elegant? Of course one can say that the proof is beautiful, but what do we precisely mean when we say that a proof is beautiful ? Is there a ...
3
votes
2answers
239 views

Double Negation is sequent calculus systems LK and LJ

In sequent calculus LK (see Gaisi Takeuti, Proof Theory (2nd ed - 1987)) we have a "standard" derivation of Double Negation in the form $\rightarrow \lnot \lnot A \supset A$. We have to start from an ...
3
votes
3answers
415 views

Impossibility theorems

I've been wondering how you go about proving an impossibility e.g. when I looked up Abel's impossibility theorem it says nothing about the proof and only restates the theorem when I'd like to know how ...
1
vote
2answers
208 views

Simple proof theory - Propositional Logic

When addressing the questions, which are featured below, I use the following definition and two lemmas. Definition: $\phi$ is a tautology if $[[\phi]]_{v}=1$ for all valuations $v$. Moreover, ...
1
vote
2answers
646 views

Axiom Systems and Formal Systems

I'm a really beginner in Mathematical Logic.I'm currently reading Shoenfield Mathematical's Logic and i'm having a hard time trying to relate the concept of Formal Systems with the concept of Axiom ...
1
vote
0answers
84 views

Examining every mathematical result in purely formal, ZFC language.

My main interest is physics. However, being self-taught in mathematics for the most part, my proofs tend to be more intuitive than it is acceptable. Yet, I recognize my inaptitude in rigor, and I ...
1
vote
1answer
258 views

Complexity of verifying proofs

My question can be read on many levels and so I welcome answers to any reading. The general question is: What is the computational complexity of verifying a proof? One way of looking at a ...
7
votes
1answer
171 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 ...
5
votes
1answer
215 views

What is the proof-theoretic ordinal of the first-order theory of real closed fields?

I recently asked a question on MathOverflow, concerning a predicative second-order theory of real numbers. Now the standard way of developing predicativity in the case of second-order arithmetic is ...
3
votes
2answers
398 views

Proof that SAT is NPC

I am not really sure I understand the idea behind Cook theorem (it says that SAT is a NP-complete problem). I read the proof with all its parts corresponding to the Turing machine TM solving it (TM ...
3
votes
2answers
593 views

How to prove consistency of Natural Deduction systems

In Dag Prawitz, Natural Deduction A Proof-Theoretical Study (1965), we have the system I of intuitionistic (first-order) logic based on eleven introduction- and elimination-rules : the 3 couples for ...
3
votes
1answer
152 views

Non-self-referential undecidable sentences in arithmetic

Are there any known undecidable sentences for PA are neither "self-referential" (like a sentence equivalent to its own nonprovability) nor imply consistency of PA (like in the Paris Harrington ...
2
votes
1answer
131 views

Proof negation in Gentzen system

I am provided with the L¬ and R¬ Gentzen rules for negation (besides “Cut” rule and some rules for ⋀ and →): $${\Gamma\vdash\Delta,\varphi\over \Gamma,\lnot\varphi\vdash \Delta}\ L\lnot \\[4ex] ...
2
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1answer
93 views

Quasi-interactive proof on real numbers

[This is a cleaner and simpler restatement of a question I asked earlier on Theoretical CS forum. Please re-tag as appropriate.] Suppose you have two oracles (black boxes) that represent real ...
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2answers
156 views

Does generalization of axioms apply also to theorems?

In Enderton's book "A Mathematical Introduction to Logic" (second edition), he includes six axiom groups, and allows also for a generalization of those axioms such that if $\Psi$ is an axiom then ...
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1answer
317 views

Löb's theorem and provability

I learned Löb's theorem. As I understanding, if a statement is formed like "I am provable", the statement should be provable. I want to ask further about Löb's theorem. There is two sentences, P and ...