Forcing Over The Language , j
by
Paul Corazza
Boise State University

If one performs a fine analysis of Kunen's famous inconsistency proof by working in the language {\epsilon, j}, using as axioms ZFC together with additional axioms about j asserting that j is a nontrivial elementary embedding, one discovers that inconsistency arises in this context by assuming Sigma_1 (or Pi_1) Replacement for j-formulas. Dispensing with Replacement for j-formulas, one then has a spectrum of theories in the language {\epsilon, j} whose models are universes M equipped with an elementary embedding j: M --> M. In this talk I will discuss what happens when one attempts to do forcing arguments for such theories over such models. I will point out differences in the forcing methodology and mention several relative consistency results.

Date received: March 26, 2002