Class-Set Theory

Class-Set Theory:

This is a first order theory that defines classes and their elements (sets).

 Language: Mono-sorted Classical First Order Logic with Equality and extra-logical primitives of Membership “∈”, and ‘‘W”; the last is a constant symbol.

 Axioms: ID axioms +

 Define: set(y) ⟺ ∃x (y ∈ x)

1.      Class comprehension schema: if φ is a formula in which x is not free, then all closures of:

∃x ∀y [y ∈ x ↔ φ ∧ set(y)]

            ; are axioms.

In English: for any formula φ there exists a class of all and only sets that satisfy φ

2.     Set Comprehension schema: for n=0,1,2,3,...; if φ is a formula in the pure language of set theory (i.e., doesn't use the symbol ‘‘W”), all free variables of which are among symbols “y, x1,..., xn”;  then:

x1,..., xn ∈ W→[∀y (φ → y ⊆ W) → ∀y (φ → y ∈ W)]

             is an axiom.

In English: any pure set theoretic formula from parameters

in W, that only holds of subclasses of W;

also only holds of elements of W.

 Theory definition finished.

Statement: This theory is stronger than ZFC. However, its axiomatization is very clean, elegant, natural, and simple. I’d propose such a theory as a foundational theory about sets and classes.

Zuhair Al-Johar

March 23. 2019