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