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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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 ...
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What if a conjecture were provably unprovable?

Suppose we found a proof that "The Twin Prime Conjecture cannot be proven", without any conclusion as to the conjecture itself being true or false. Is it then possible for the conjecture to be true? ...
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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 ...
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Prove that the set $C = \{x \in\Bbb R : ax\le b\}$ is convex

Prove that if a and b are real numbers, then the set $C = \{x \in\Bbb R : ax\le b\}$ is a convex set. My solution so far: To show that a set $C$ is convex it needs to be shown that for for every ...
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Trying to disprove a statement - some partial working included

I am trying to find a counter example to show that the statement below is false, but I am having difficulty in trying to find a reasonable argument. Here is the statement: $n^2-12n + 35 \geq 0$ for ...
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Is it possible to prove that the encoding of existentials in System F is valid?

In Girard's Proofs and Types, under item 11.3.5, second-order existential quantification is encoded in System F using universal quantification as follows: $$ \Sigma X.V \equiv \Pi Y. (\Pi X.(V \to ...
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Do we know that if $\pi$ is normal then there is a proof of it?

We do not know whether $\pi$ is normal or it is not and many other weaker statements, e.g. (*) $\pi$ contains infinitely many $0$s. Inspired by the Godel's incompleteness theorem that there are some ...
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126 views

What if 'proof by contradiction' is not a valid method of proof?

I've just been reading this question about the existence (or lack thereof) of contradictions in maths. I've been wondering: What if 'proof by contradiction' is not a valid method to (dis)prove a ...
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Use a proposition to prove another proposition

I'd like to ask for help with an exercise from Solow - How to Read and Do Proofs(3.16). I've tried to get through it but I can't make the proper connection between the two properties. I figured that ...
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Deriving $A \rightarrow ( B \rightarrow C ) \rightarrow ( ( A \rightarrow B ) \rightarrow ( A \rightarrow C ) )$ in the sequent calculus

I need to prove the following theorem: $A\to (B\to C) \to ((A\to B) \to (A\to C))$ using the sequent calculus method. Using the rules: $$ G, A \Rightarrow B,D \over G \Rightarrow A \to B , D ...
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Proof exercise: finding hypothesis and conclusion in a statement

I am starting learn mathematical proofs and I was doing some exercise that needed to identify the hypothesis and the conclusion in a given statement. And I'm having trouble trying to figure it out in ...
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Mathematical statements that cannot be proved or disproved [closed]

I've recently been reading about the continuum hypothesis and am fascinated by the fact that it cannot be proved or disproved, despite the fact that the statement itself is either true or false. What ...
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70 views

Proof by contradiction using counterexample

Why can't we use one counter example as the contradiction to the contradicting statement? Example: Let a statement be A where a-->b. We can prove A is not true by finding a counter example. Now, in ...
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Can we always give a direct proof? [duplicate]

This is something I was wondering about for quite a while. Is it possible to construct a statement that can only be proven by using 'proof by contradicition' or contraposition? Or to put it ...
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Biconditional statements with an “easy” direction.

I was reading through Lax's Functional Analysis, when I came across the following statement: Theorem: X is a normed linear space over $\mathbb{R}$, $M$ a bounded subset of $X$. A point $z$ of $X$ ...
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Examples of revisited proofs after new theorems are discovered… [closed]

Are there any nice examples of "old" complicated proofs that become much simpler after new math is discovered years later? For instance, we know now that Pn+16< Pn+1 occurs infinitely often (where ...
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1answer
60 views

Existential introduction - required or not?

Consider a theory $T$ in first order logic, and a formula $C$. If there exist a proof of $C$, and that all formulas in $T$ and $C$, none of them contains $\exists$. The question is: does there always ...
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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 ...
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particular property and completeness?

I was puzzeling with the almost standard definition of completeness: In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula ...
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Defining a partial function in a formal theory

Assume we have a first-order theory $T$ of arithmetic (i.e., number theory). Suppose I wish to introduce a new function symbol $f$ in the theory, so that $f$ is a partial number function (namely, it ...
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$\neg(A\Rightarrow B) \iff A\land \neg B$

When considering the question: Rewrite the following using only the symbols $A, B, \lor, \land, \neg$ : $$\neg(A\Rightarrow B)$$ I do not understand how to interpret this and what method to ...
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A question about $KP + V = L$ and $KP$ set theory.

In reading Rathjen (Choice principles in constructive set theories) and Jager (On Feferman's OST) I've come across two facts that are taken as obvious/well known, and probably are, but for which I ...
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1answer
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$HA^{\omega}$ is a conservative extension of $HA$. But why?

This is definitely a silly question, but I've no one to ask... $HA^{\omega}$ is an extension of $HA$ in all finite types. One can formalize a model of $HA^{\omega}$ in $HA$ using indicies of partial ...
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Idea of a proof by contradicton

Is the idea of a contradiction to prove that the desired conclusion is both true and false or can it be any derived statement that is true and false (not necessarily relating to the conclusion)? Or ...
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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 ...
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135 views

Is there a specific name for a corollary of a conjecture?

How do you call a corollary of a conjecture? Is there a specific name for it? Can it be called simply 'corollary'? Can't it be called 'corollary'? I mean, does the label 'corollary' imply that the ...
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342 views

Prove: if x is odd, then sqrt(x) is odd.

If $x$ is odd, then $\sqrt{x}$ is odd, where $x$ is an integer. Any hints welcome and preferred. Thank you!
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1answer
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Proove that cos(x / 2) + cos(y / 2) - cos(z / 2) = 4 * sin((pi - x) / 4) * sin((pi - y) / 4) * sin ((pi + z) / 4

Help me proove that cos(x / 2) + cos(y / 2) - cos(z / 2) = 4 * sin((pi - x) / 4) * sin((pi - y) / 4) * sin ((pi + z) / 4 where x + y + z = pi I've reached 2 * sin((x + z) / 4) * (cos((x + z) / 4) - ...
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Is proof by algorithm credible?

I have found this question How to prove that the inverse of a matrix is unique? And while the accepted answer is fine I was wondering if it's possible to proof the uniqueness by algorithm. There is ...
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Proving $A+2B+3C+4D < 2.5$ with given conditions

I want to prove follow inequality. Conditions: $$A+B+C+D=1$$ $$A>B>C>D>0$$ Prove: $$A+2B+3C+4D < 2.5$$ Thanks in advance.
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double implication proof

How would I go about said proof: I know how to do it with just a single logical equivalence, but how would I prove a double implication?
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Difference between bound and free variable

What is the difference between $\forall x (P(x)\implies Q(x))$ and $P(x)\implies Q(x)$ I know in the first one the variable x is bound but in the second one the variable is free. What are the ...
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1answer
139 views

Formal proofs and Deduction Theorem

In this question i will explain one idea i had about basic formal proofs and the use of Deduction Theorem. I'm considering a formula γ to be a logical consequence of a set A of formulas if and only ...
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Law of excluded middle. Do we need it in proofs?

Quite often when I am making a natural deduction proof, and I have no fixed idea on how to continue. I find myself thinking: "lets start with some form of the law of the excluded middle (LEM) and ...
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Graph that represents logical reasoning

The proof of a statement $X$ in terms of assumptions $A$, $B$ and $C$ can sometimes be represented using a directed graph: $$ \begin{matrix} & & X\\ & \nearrow & & ...
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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 ...
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General view of Theorems

I'm trying to see almost all theorems ( at least the non-existential ones ) as affirming that some formula ( mostly of first-order logic language ) is a logical consequence of other formulas. So, ...
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Hilbert calculus: Proof that every provable formula has a proof

For my indroduction to logic course I have to proof, that every provable formula has a proof. It sounds first very funny, second also very logic, still I don't get to make of formally work.. The ...
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Proving that operations give equal results given equal inputs

I was reading about the 9 or 12 basic properties of 'fields' (if that's what they're called) in a book called Spivak's Calculus, 3rd Edition, and got quite befuddled by dealing with as basic stuff as ...
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$LK-\Phi$ proof of $\exists y Pby$

I am having difficulty with the concept of $LK-\Phi$ proofs, here is a question I have been working on: Let $\Phi = \{Pafa\}$, where $P$ is a binary predicate symbol and $f$ is a unary function ...
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1answer
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Are there statments which do not have a constructive proof?

I understand that a lot of statements are just non-nonconstructive in nature (like negative statements), and I understand that a lot of statements are not provable without the axiom of choice. ...
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Struggling with proof, by contrapositive?

I am having trouble solving this proof. I tried to do a proof by contrapositive. Q = $(u+z)/(v+w) < z/w$ P = $(u/v < x/y \land x/y < z/w)$ Assuming $\lnot Q$ got me: $u/v \ge z/w$ ...
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ZFC Axioms to be extended?

Sorry if this is going to be a really loaded question. I was told several times that for virtually all theorems/corollaries/propositions of mathematics (except those cases not compatible with ZFC ...
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1answer
123 views

Provable formulas in everyday Mathematics

Basically all statements ( lemmas, theoremas, corollaries ) in Mathematics can be expressed as a conditional statement in first-order language, or existential statement ( existence proofs ). Here ...
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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 ...
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How to quantify this statement

How do I quantify this statement? Let $x \in \mathbb{R}$. $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. When I am trying to prove this, I am led (in the course of ...
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The approximation rule implies the equality rule in systems of type assignments

I'm reading Barendregt's Lambda calculi with types (1992). In Proposition 4.1.4.1., he "proves" a lemma which shows the approximation rule implies the equality rule in typed lambda-calculi à la ...
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Proof by contradiction and order of statements

Order matters, take sequences tending towards a limit: $$\forall\epsilon>0\exists N\in\mathbb{N}:n>N\implies|x_n-L|<\epsilon$$ "For all $\epsilon$ there exists an $N$" is totally different to ...
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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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Show that if $3\mid(a^2+1)$ then $3$ does not divide $(a+1)$.

Show that if $3\mid(a^2+1)$ then $3$ does not divide $(a+1)$. using proof of contradiction can someone prove this using contradiction method please