## Journal of Symbolic Logic

### The Model of Set Theory Generated by Countably Many Generic Reals

Andreas Blass

#### Abstract

Adjoin, to a countable standard model $M$ of Zermelo-Fraenkel set theory (ZF), a countable set $A$ of independent Cohen generic reals. If one attempts to construct the model generated over $M$ by these reals (not necessarily containing $A$ as an element) as the intersection of all standard models that include $M \cup A$, the resulting model fails to satisfy the power set axiom, although it does satisfy all the other ZF axioms. Thus, there is no smallest ZF model including $M \cup A$, but there are minimal such models. These are classified by their sets of reals, and there is one minimal model whose set of reals is the smallest possible. We give several characterizations of this model, we determine which weak axioms of choice it satisfies, and we show that some better known models are forcing extensions of it.

#### Article information

Source
J. Symbolic Logic, Volume 46, Issue 4 (1981), 732-752.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183740884

Mathematical Reviews number (MathSciNet)
MR641487

Zentralblatt MATH identifier
0482.03022

JSTOR