**616 proof-theory questions.**

Defining the new proof system $N$ as this:
We have 2 Axioms -
$$A \rightarrow (A \lor B)$$
$$A \rightarrow (B \rightarrow A)$$
A new deduction rule:
$$\bullet \frac{(A \rightarrow B)}{A \rightarrow (...

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 ...

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 ...

I've been reading on type theory, and it being a possible foundation of mathematics. I have to say it is all very abstract to me. I'm very used to using set theory to think about mathematical proofs, ...

I am working on an assignment in discrete structures and I am blocked trying to prove that $\sqrt{11}-1$ is an irrational number using proof by contradiction and prime factorization.
I am perfectly ...

From my understanding, for proofs to be considered "formal" as opposed to social/casual, they need to be computable or at least be a set of transformations of strings of symbols. By algorithm I mean ...

In Realizability: A Historical Essay [Jaap van Oosten, 2002], it is said that recursive realizability and HA provability do not concur, because although every HA provable closed formula is realizable, ...

Say we have some theory $T$ such that $Th(A_E) \subseteq T$ where $A_E$ are the axioms of arithmetic. How do I show that
(1) there are sentences $\varphi_1$ and $\varphi_2$ such that $Th(A_E) \vdash ...

I am confused about Church's simply typed lambda calculus and the Curry-Howard isomorphism.
Church's simply typed lambda calculus in the paper cited above is given a classical proof theory, in that ...

Related questions: Formally what is a mathematical construction? and What is a Universal Construction in Category Theory?
Backstory: A question arose in a seminar that concluded with the statement ...

In a sequent, on the left and right-hand side of the turnstile operator, does the comma denote disjunction or conjunction?
$$\frac{...}{\Delta_1,\Delta_2 \vdash \Gamma_1,\Gamma_2}$$
I think it's one ...

The Goldbach Conjecture asserts:
It is possible to write every even number greater that 2 as the sum of
two primes.
Assume I can prove that the Goldbach Conjecture is unprovable from the Peano ...

Def 1: $M$ is a $\underline{theory}$, if it is a collection of first order statements.
Def 2: Given theory $M$ and first order statement $\phi$, then $M$ can $\underline{prove}$ $\phi$ if, starting ...

For over 8 months, I have been actively looking for a resource that rigorously tackles mathematical logic - at the same level described in this lecture, timestamp 56:09. A resource which minds the ...

I have been getting confused thinking about non-constructive proofs.
Several axioms of ZFC imply existence of a set with certain properties, and for each axiom except foundation, infinity, and ...

Note: I'll be putting letters before each statement I discuss in order to easily reference them.
If I'm proving a statement with induction, can I use induction on a statement that I derive inside ...

I am curious of why fundamental sequences of limit ordinals were invented? Was it only to be able to define a function (e.g. fast-growing) hierarchy?
For instance:
Zero ordinal 0: $f_{0}(n) = n + 1$
...

Linear logic is a certain variant of sequent calculus that does not generally allow contraction and weakening. Sequent calculus does admit the cut rule: given contexts $\Gamma$, $\Sigma$, $\Delta$, ...

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

In Linear logic (LL), it is unprovable but when considering the resource interpretation it seems to me that from the resources $A, B$ we can produce the resources $A, B$.
By $A, B \vdash A, B$ I mean ...

In a Gentzen system (i.e. sequent calculus) for Intuitionistic Linear Logic (from now, ILL), given the usual rules for ILL ($\wedge L, \wedge R, \circ L, etc.$), I want to prove that the Identity $A \...

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 (...

The positive exponential ! has a very satisfying interpretation in terms of the standard resource interpretation of linear logic. Given a resource $a$, we know that $!a$ means an infinite supply of $a$...

A tableaux method for linear logic is briefly discussed in
https://www.academia.edu/6591354/TABLEAU_METHODS_FOR_SUBSTRUCTURAL_LOGICS?auto=download
D'Agostino writes (p.418-9):
''It is ...

I know the notions "satisfiablity", "validity" and "consequence" as applied to the logic, e.g. First Order logic https://en.wikipedia.org/wiki/First-order_logic#Validity,_satisfiability,...

While I am discussing over the definition of the proof with my friends, one says
"proof is definitely a mathematical object which maps a formal representation of mathematical objects into two ...

In natural deduction, what says that the following is correct?
$\Gamma \Rightarrow B$ then $ \Gamma, A \Rightarrow B$
I saw a proof that uses this rule without mentioning it and I can't find the ...

Here Presburger arithemtic is given by a set of axioms over the signature with binary operation $+$ and two constants $0$ and $1$. Similarly in Presburgers original paper he gives the arithmetic in ...

Given a language $\mathcal L$ and a structure $\mathcal A$ compatible with it, then we call $\mbox{Th}(\mathcal A) = \{ \varphi \mid \mathcal A \models \varphi \}$ the theory of $\mathcal A$; where $\...

The only problem I could see is that there should be two more cases: $x \in A$ and $y \in D$; and $x \in C$ and $y \in B$. Please suggest.

If someone can prove that one cannot prove from the axioms statement that is opposite to some true statement in the system, does it mean that this system is consistent ? For example, is proving that ...

GÃ¶del's incompleteness theorem says that formal arithmetic can't prove its own consistency, but how can formal system even state its own consistency with its own language/semantics ?

I the book Computational Complexity by C. Papadimitriou he introduces for first order logic the following axioms:
AX0: Any expression whose Boolean form is a tautology.
AX1: Any expression of ...

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