616 proof-theory questions.

10 Is it a paradox if I prove something as unprovable?

1 answers, 1.376 views logic proof-theory peano-axioms
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 ...

How to formally prove $\forall x: x11=x$ in the theory of groups.

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 ...

1 Reference request for a formalized and organized take on the foundations (logic and set theory)

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 ...

1 Must non-constructive existential proofs use axioms of foundation or choice?

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 ...

1 Is an induction inside of an induction allowed in a proof?

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 ...

2 Original purpose of Fundamental Sequences of limit ordinals

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: $f_{0}(n) = n + 1$ ...

4 In linear logic sequent calculus, can $\Gamma \vdash \Delta$ and $\Sigma \vdash \Pi$ be combined to get $\Gamma, \Sigma \vdash \Delta, \Pi$?

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 $\Gamma$, $\Sigma$, $\Delta$, ...

4 Intuitionistic Linear Logic

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 ...

2 How can we interpret that $A, B \vdash A, B$ is unprovable with resource interpretation in Linear Logic?

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 ...

2 Finding a flaw in a proof that $(A\cup C)\times (B \cup D)\subseteq(A\times B)\cup(C\times D)$

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.

1 Is such a proof enough for proving the system is consistent?

2 answers, 69 views logic proof-theory
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 ...

1 How can a system state its own consistency within its own language?

1 answers, 44 views logic proof-theory
Gödel'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 ?

1 Hilbert style axiom system without generalisation

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 ...