Forcing over choiceless models and generic absoluteness

TL;DR

This paper develops a new method for forcing over models of set theory without the axiom of choice, allowing for the manipulation of the universe's theory and exploring principles of generic absoluteness.

Contribution

It introduces a variant of the countable chain condition applicable to choiceless models and proves an iteration theorem for classical forcings, enabling new results on the theory of the universe.

Findings

Adding Cohen and random reals can produce different theories.

Forcing with certain classes can change the universe's first-order theory.

Principles of generic absoluteness imply all infinite cardinals have countable cofinality.

Abstract

We develop a toolbox for forcing over arbitrary models of set theory without the axiom of choice. In particular, we introduce a variant of the countable chain condition and prove an iteration theorem that applies to many classical forcings such as Cohen forcing and random algebras. Our approach sidesteps the problem that forcing with the countable chain condition can collapse $\omega_1$ by a recent result of Karagila and Schweber. Using this, we show that adding many Cohen reals and random reals leads to different theories. This result is due to Woodin. Thus one can always change the theory of the universe by forcing, just like the continuum hypothesis and its negation can be obtained by forcing over arbitrary models with choice. We further study principles stipulating that the first-order theory of the universe remains the same in all generic extension by a fixed class of forcings. Extending a result of Woodin, we show that even for very restricted classes such as the class of all finite support products of Cohen forcing or the class of all random algebras, this principle implies that all infinite cardinals have countable cofinality.