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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Decidability of certain first-order statements

Is it possible to construct an algorithm that can formally prove any statement in some countable first-order theory except for exactly those which aren't provable in the theory? Why or why not? Edit: ...
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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 ...
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251 views

What are those “things that cannot be proved using only ordinary rules of inference”?

The online edition of the book Introduction to Logic by Michael Genesereth and Eric Kao, has a detail that left me confused. CHAPTER 4 [...] 4.2 Linear Proofs [...] The ...
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Curry-Howard Correspondence (Proof Theory)

As you all know, the Curry-Howard correspondance provides a link between type theory and predicate logic. Concepts featured in the former, such as $\Pi$-type and $\Sigma$-type can, by the ...
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48 views

What is the proof-theoretic ordinal of $PA+TI(\epsilon_0)$?

what is the proof-theoretic ordinal for $PA+TI(\epsilon_0)$, where $PA+TI(\epsilon_0)$ is Peano arithmetic where transfinite induction up to $\epsilon_0$ was added? Is it known? Thank you
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Infinitely many proofs?

While compiling a list of my favorite proofs of the infinitude of primes, the following came to mind; Proposition: There are infinitely many non-isomorphic proofs of the infinitude of primes. I'm ...
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47 views

Inversion lemma for G3ip

I'm following the book Structural Proof Theory by Negri and others. In it, they claim on page 32 about G3ip that if $⊢ _ n A \& B, Γ ⇒ C$, then $⊢ _ n A, B, Γ ⇒ C$. But, given that the only ...
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The ethics of Borel determinacy

I was speaking with a friend the other day, and I happened to say "morally, Borel determinacy is as strong as ZF." I was riffing on the well-known result of Harvey Friedman, that we need ...
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What are some arguments/counterarguments for Zeilberger's “proof certificates”?

Here is the quote I wish to ask about: "I speculate that similar developments will occur elsewhere in mathematics, and will 'trivialize' large parts of mathematics, by reducing mathematical ...
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61 views

Proof by reflection and Homotopy Type Theory

I have been looking into various proof assistants, and came across the method of proof by reflection in Coq, which (from what I understand) allows one to verify a program provides the correct "answer" ...
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460 views

Difference between soundness and correctness

Is there any actual semantic difference between soundness and correctness? Can I use these words interchangeably when talking about formal reasoning, proof, logics, etc.? Otherwise, is there a ...
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What to call a term-in-context whose context contains exactly the variables occurring in the term?

In type theory, a term-in-context $\Gamma \vdash t : \tau $ is only well-formed when $\Gamma$ contains all the variables occurring in $t:\tau$. Is there a name for when it contains exactly the ...
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Can every mathematical proof be seen as the verification of some algorithm's action?

Put another way: Can every mathematical proof be reformulated to be about some class of Turing Machines? Example Any proof of the existence of infinite prime numbers is equivalent to the statement: ...
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Is the following derivation of Gödel's Paradox generalizable to an intuitionistic one using provability?

I will follow Goedel's original proof of Goedel's paradox located on pg. 264 of Hao Wang's A Logical Journey: From Gödel to Philosophy. With this in sight, and to also avoid confusion, I will also ...
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What is the proof-theoretic strength of the predicative second-order theory of real numbers?

The first-order theory of real numbers, AKA the theory of real closed fields, is obtained by added to the axioms for ordered fields an axiom schema of completeness, which states that for each formula ...
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23 views

Why don't the quantifiers split in linear logic?

Every presentation of linear logic I've seen seems to either omit or treat quantifiers as an after-thought. Even Girard says that there is "little to say" about them. However, if we view universal ...
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How far can-I rewrite in lambda functions?

I am quite new with the lambda calculus. I am experimenting lambda-calculus proofs through the coq proof assistant, but the question I have is not related to coq (I guess). However, I'm going to use ...
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54 views

Girard's System $F$ (also named Polymorphism)

I have been studying Girard's Polymorphism and a question came to my mind: why is it (also) called system $F$? Where does the $F$ come from? (i searched it online but didn't get any luck...)
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90 views

How much arithmetic can Predicative Second-Order EFA do?

As discussed in this MathOverflow question, I'm trying to find what the result would be of applying a Feferman-Scutte-like analysis to the predicativism of Edward Nelson and Charles Parsons, who ...
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61 views

What is the relationship between Realizability and the Curry-Howard isomorphism?

I have recently been studying the Curry-Howard isomorphism/correspondence. My sources have primarly been Sørensen [1] and Girard [2]. Realizability is introduced here in the form of Kleene's ...
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32 views

Unprovable identity over the integers

I was thinking about Tarski's problem, and was wondering what happens if we have a theory $T$ with two sorts $N,Z$ with intended interpretations $\def\nn{\mathbb{N}}$$\def\zz{\mathbb{Z}}$$\nn,\zz$ ...
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46 views

Do we sometimes prove things based on the assumption that mathematics is self-consistent?

Do we sometimes prove things based on the assumption that mathematics is self-consistent? I recently started to be dubious about proofs by contradiction. It seems to me that it is somehow based ...
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29 views

Equivalence between sequent calculi with different cut rules

Let G4' be a sequent calculus G4 for classical logic with the addition of the following pair of "left" and "right" cut rules: Let now be G4'' a second calculus G4 for classical logic with the ...
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79 views

Incompleteness theorems in encoding schemes other than Gödel numbering

Gödel's proof of his incompleteness theorems makes use of Gödel numbering, which is a device that allows a theory of arithmetic $S$ (e.g. PRA) to express and reason about metamathematical statements ...
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55 views

Show that if A and B are strictly convex, then A + B is strictly convex or provide a counter example.

We have: If A is open: $\exists x,y \in A,$ $x \neq y$ such that $\lambda x+(1-\lambda y)\in \dot A $ (the interior) and $\exists u,v \in B,$ $x \neq y$ such that $\lambda u+(1-\lambda v)\in ...
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33 views

Is the full strength of first-order logic needed for dealing with equational theories?

More specifically, if we have an equational theory $T$ (a set of equations understood as being implicitly universally quantified), are the (equational) consequences of $T$ that can be proved with ...
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24 views

Independence of FLT over weak systems

It is known that Fermat's last theorem can be proven in finite-order arithmetic (e.g. accoridng to this site). This is still an extremely high upper bound on proof complexity (for example, compared to ...
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64 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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112 views

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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46 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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63 views

Proof of $(\forall x. \varepsilon(x)) \Rightarrow \bot $ in $\lambda\pi $ calculus $\equiv$

What is the right representation of the proof of $(\forall x. \varepsilon(x)) \Rightarrow \bot $ in simple type theory as a term of $\lambda\pi $ calculus $\equiv$? Note on notation: The epsilon ...
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proof checking machine vs. provability checking machine

Let M be a proof-checking Turing machine which takes two inputs, A and B. : M(A,B) = 0 if A codes a valid proof of the sentence coded by B in ZFC. M(A,B) = 1 if A does not code a valid proof of the ...
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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 ...
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26 views

What are some active areas of research in proof theory?

Is there any research activity going on in the field of proof theory today? If so, what are some of the most active areas, what types of questions do they deal with, and where can I go to find out ...
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27 views

Proof tree of $[(\phi\lor \psi)\land (\phi \lor \chi)] \to [\phi \lor (\psi \land \chi)]$

I need to construct a proof tree of: $$[(\phi\lor \psi)\land (\phi \lor \chi)] \to [\phi \lor (\psi \land \chi)]$$ Could someone check the following proof tree? I first proved the following: ...
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67 views

Profane Model Theory, sacred Proof Theory

Dirck van Dalen starts the Preface to his Logic and Structure with the following words: "Logic appears in a ‘sacred’ and in a ‘profane’ form; the sacred form is dominant in proof theory, the ...
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Show that $σ(i_1, i_2, . . . , i_k)σ^{−1} = (σ(i_1), σ(i_2), . . . , σ(i_k))$

Here's the full question: If $σ ∈ S_n$ is any permutation and $i_1, . . . , i_k $ are $k$ distinct elements of $\{1, . . . , n\}$, show that $σ(i_1, i_2, . . . , i_k)σ^{−1} = (σ(i_1), σ(i_2), . . . , ...
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In what formal proof systems is the deduction theorem taken as a primitive rule of inference?

Wikipedia's article on the deduction theorem states: Although the deduction theorem could be taken as primitive rule of inference in [Hilbert-style] systems, this approach is not generally ...
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45 views

Is there a standard notation for coding finite sets of numbers as numbers?

Hajek and Pudlak Metamathematics of First-Order Arithmetic use the Ackermann encoding of hereditarily finite sets, but they use no notation for codes. They let the reader see from context when a ...
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11 views

Proving weak simulation

I want to prove something but I am not sure if it is the right way to do it. I have two LTS that define different semantics. A=($Q_a,Λ,\to)$, and B=$(Q_b,Λ\cup\{\beta\},\leadsto)$, where $\beta$ is ...
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46 views

Prereqisites for: Subsystems of second order arithmetic

As the title suggests, im wondering what the prerequisites for Simpsons book, Subsystems of... are? Unfortunately I cant find it in the preface. My background is a Bachelor in Philosophy and ...
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37 views

Algebraic embeddings and isomorphisms in formalized ZFC

Example: It is always said that we can embed $\mathbb{Z}$ within $\mathbb{Q}$ by identifying $z \in \mathbb{Z}$ as $(z,1) \in \mathbb{Q}$. This is because there is an injective ring homomorphism $\phi ...
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Proof of cut elimination

I am reading Proofs and types and am blocked at the proof of cut elimination in sequent calculus (chap 13). I don't see either how the cuts are being pushed up above the preceding steps to the top of ...
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A provability puzzle

This is a problem I came up with on my own, and it has me stumped, so I am going to pose it as a kind of puzzle. Let $F$ be a formal proof system, recursively axiomatizable, with an acceptable ...
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Are proofs for many-sorted first order logic shorter than single sorted first order logic?

I understand that the expressive power of first order logic with one sort is the same as any many sorted first order logic, and that higher order logic with general semantics is the same as a many ...
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126 views

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 ...