Induced Cylindric Algebras of Choice Structures

TL;DR

This paper explores how the extensionality axiom in Hilbert's epsilon calculus induces algebraic structures over choice models, linking semantics with algebraic quotients, especially in Boolean and monadic contexts.

Contribution

It demonstrates that the extensionality axiom induces algebraic structures over canonical models, connecting semantics with algebraic quotients in choice structures.

Findings

The calculus is complete with respect to choice structures.

Induced algebras are isomorphic to quotient algebras of Lindenbaum--Tarski algebras.

Canonical models of sigma complete theories are algebraically characterized.

Abstract

One of the benefit properties implied by the extensionality axiom of Hilbert's epsilon calculus is that the calculus becomes complete with respect to the choice structures as semantics. Another implication of the axiom, discussed in the paper, is that an algebra is induced over the universe of the canonical model of a theory, which is isomorphic to a quotient algebra of the Lindenbaum--Tarski algebra of the theory. Especially, in the case of Boolean or monadic algebras, the canonical model of the theory of a sigma complete model is isomorphic to the algebra induced by the axiom of extensionality.