What are the requirements of a foundational theory?

There are multiple languages to describe all of mathematics, and there are some equivalences between them, some more successful then others. My question is can we describe some requirements (in some logic system) that any foundational language must satisfy? In praticular, I have been thinking about two requirements: It can describe the logic system it … Read more

Formalization and set-theoretic issues in the definition a functor category

Universes are used in a category theory to handle size-issues. Assuming Grothendieck’s axiom UA that every sets is contained in some universe there are two approaches to U-smallness given a universe U. 1) A set is U-small if it is isomorphic to an element of U (reference: SGA), 2) A set is U-small if it … Read more

Iterated Gentzen: or, a Sith objection to the proof of consistency of PA

$\DeclareMathOperator\PRA{PRA}\DeclareMathOperator\WF{WF}\DeclareMathOperator\Con{Con}\DeclareMathOperator\PA{PA}$Preamble: In the year … in a galaxy far far away, a nasty Sith named Darth Dubious (DD) asks a Jedi, Obi Wan Mathobi (OWM), about the consistency of PA: DD: How do you know that PA is consistent? OWM: Don’t you know that many centuries ago a great Mathematician from Terra by the name … Read more

Can this graph theory serve as a foundational theory of mathematics?

Working in mono-sorted first order logic, add primitives of equality and its axioms, set membership ∈, a partial ternary relation → denoting is the direction from to, and at last a total unary function S dentoing the “structure” of. Axioms for sets: Extensionality: ∀x∀y (∀z(z∈x⟺z∈y)⟹x=y) Membership: x∈y⟹∀z∈x(z=x) Atomicity: ∀x∃y:y∈x Comprehension: [∃y(y={y}∧ϕ)]⟹∃x∀y (y∈x⟺y={y}∧ϕ); for every formula ϕ in … Read more

Class theory with support for self-application of class functions?

To every natural number n, we can assign its Church numeral n_. A formal definition would be: 0_(f)=iddom(f) n+1_(f)=n_(f)∘f where each line is to be understood as implicitly universally quantified over every endofunction f. This gives us nifty formulae like: a_∘b_=a⋅b_ a_(f)∘b_(f)=a+b_(f) Unfortunately, size issues block the existence of Church numerals inside models of ZFC. … Read more

products in a category without reference to objects or sources and targets

Hi, I was thinking about presenting categories with nothing but equations over morphisms. I wondered about products. The definition of a product has its genesis in the following diagram shape A->B A->C You would say that whenever you have this shape, then for any D such that D->A->B and D->A->C blah blah…the axioms for the … Read more

Surreal numbers and large cardinals

This is a question in two parts about the interaction of surreal numbers and large cardinals, in both cases just a request for references on the subject. Part 1 is about foundations. Much of the research that I’ve seen on the surreal numbers typically treats the foundational issues by either working in NBG set theory … Read more

“Surjective cardinals” – using surjections rather than injections to define isomorphism classes of sets

Cantor used the notion of an “injection” to formalize the size of two sets: A is “smaller” than B if A injects into B. Simply put, the question is – how does this situation change if we use surjections instead of injections in our notion of size? And if we use “surjections both ways” to … Read more

linear logic, diagrammatic calculus and foundations

Hi, I have been interested in foundations for a while, especially categories as foundations. I am of the opinion that, as long as we present the theory of categories in SET, we will not be able to give a reasonable justification for categories as a foundation. (that could be a question: does the persistent presentation … Read more

Are the paradoxes of material or strict implication used anywhere to prove theorems in mathematics

In the Stanford Encyclopedia of Philosophy entry “Relevance Logic“, the following inference is listed as classically valid: The moon is made of green cheese. Therefore, it is raining in Ecuador now or it is not. This inference can be tweaked slightly to make it more mathematical: The moon is made of green cheese. Therefore, CH … Read more