The simple consistency of a set theory based on the logic ${\rm CSQ}$.
Ross T. Brady · Notre Dame Journal of Formal Logic · 1983
This paper proves the simple consistency of the set theory CST.CST has the Generalized Comprehension Axiom (GCA), (3jO(V;t)(x e y +-+A), and the Extensionality Rule, x = y => x e w +-* yew, where x -y ~df (Vz)(z e x «-• z e y).CST is based on a logic CSQ, which is semantically described below. CSQ Primitives1.~, &, ->, V (connectives and quantifier) 2. /, g, h, /',... (predicate constants) 3.x, >>, z, x', . . .(individual variables) 4. a lt a 2 , a 3 , α 4 , . . .(individual constants). CSQ Formulas1.An individual variable or constant is a term. 2.If ί l5 . .., t n are terms and /is a predicate constant, then ft 1 . . .^w is an atomic formula.3.If A and 5 are formulas and x is an individual variable then ~A, A & B, A-*B and (\/x)A are formulas.A sentence is a formula with no free variables.A CSQ model structure {CSQ m.s.) consists of ordered triples (Γ, K, R), such that K is a set, T is a member of K, and Z? is a two-place relation on K, with the following postulates holding: For a e K, *I acknowledge help from a referee of this Journal in choosing the logic CSQ, in using the abstract j cy: A\, in setting out the proof of Lemma 4, and in defining and using G(A(a)).