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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How can Goodstein's theorem be expressed in PA

I understand Goodstein's Theorem and its proof. I'm trying to understand the proof of why Goodstein's Theorem cannot be proved in PA. However, it's not immediately clear to me that Goodstein's Theorem ...
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501 views

Why wouldn't someone accept Gentzen's consistency proof?

Reading the consistency section of the Peano Axioms wikipedia page, I came across this sentence: The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, ...
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68 views

Question about the incompleteness proof (Theorem V)

Question in short: Where do I find a complete proof of Theorem V from Gödels incompleteness proof? If it does not exists, can someone provide it? Question in detail: I am trying to understand ...
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204 views

Birkhoff's completeness theorem

I have two simple questions. A) Does Birkhoff's completeness theorem follow directly from Gödel's completeness theorem? B) Is Birkhoff's completeness theorem constructive in the following sense: ...
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215 views

Is constructive proof of non-existence possible

Constructive proof construct(indicates) object that satisfies given predicate. Question is whether one can give constructive proof of non-existence of an object with given property e.g. that every ...
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461 views

Weakening and Contraction

I saw this site saying weakening is a structural rule where the hypotheses or conclusion of a sequent may be extended with additional members and that contraction is a rule where two equal (or ...
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123 views

What is really a “complete” deductive system for first-order theories.

Given some first-order language and a set of axioms therefrom one still needs to specify a deductive system to turn it into a full-fledged first-order theory. Currently I'm under the following ...
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65 views

Is a sentence in $\Pi_1$ true given $Q \vdash \lnot\varphi$?

If $Q \vdash \lnot\varphi$ (Q is the Robinson arithmetic), and if I assume that $\varphi \in \Pi_1$; Can I say that $\varphi$ is a true sentence? My thoughts are that, given that Q is $\Sigma_1$- ...
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81 views

A simple question on Gödel's functional interpretation

I've been recently reading the Gödel's functional interpretation (or Dialectica). It is generally defined inductively, as could be found here: http://www.andrew.cmu.edu/user/avigad/Papers/dialect.pdf ...
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308 views

Problems with nesting proof predicates in first order logic.

Whenever I start nesting proof predicates, I always seems to run into these bizarre situations. I was wondering if anyone knows about this and could shed some light on it or provide me with some ...
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Is there a minimal axiomatization of ZFC?

Working in ZFC, does there exist a set $\Sigma$ of sentences which axiomatizes ZFC (i.e. every sentence in $\Sigma$ is provable from your favorite axiomatization of ZFC, and vice versa) and is minimal ...
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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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270 views

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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128 views

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 Y))...
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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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228 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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150 views

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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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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122 views

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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83 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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260 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 ...
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78 views

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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47 views

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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65 views

$HA^{\omega}$ 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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440 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 ...
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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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108 views

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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140 views

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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49 views

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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576 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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400 views

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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255 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 ...
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139 views

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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168 views

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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37 views

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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68 views

$LK-\Phi$ 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
65 views

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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236 views

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
209 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 i'...
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381 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 ...
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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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48 views

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 Curry,...
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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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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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Are geometric proofs less reliable than others?

When I submit a homework with a proof that uses a graph, ball, shape etc., most of the time the professors are not happy with them. They respond with a statement like: "The proof you made seems very ...
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Intuitionistic Linear Logic

I am currently going through some papers that use the "intuitionistic version" of Girard's Linear Logic. Problem is, i seem to find very little literature on it. There is a lot done on Linear Logic ...
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assuming the conclusion

A natural deduction proof goes from premmisses to conclusion, and under normal circumstances you will not assume the conclusion. Sometimes you may assume the negation of the conclusion and do some ...