## Journal of Symbolic Logic

### Spector Forcing

J. M. Henle

#### Abstract

Forcing with $\lbrack\kappa\rbrack^\kappa$ over a model of set theory with a strong partition cardinal, M. Spector produced a generic ultrafilter $G$ on $\kappa$ such that $\kappa^\kappa/G$ is not well-founded. Theorem. Let $G$ be Spector-generic over a model $M$ of $ZF + DC + \kappa \rightarrow (\kappa)^\kappa_\alpha, \kappa > \omega$, for all $\alpha < \kappa$. 1) Every cardinal (well-ordered or not) of $M$ is a cardinal of $M\lbrack G\rbrack$. 2) If $A \in M\lbrack G\rbrack$ is a well-ordered subset of $M$, then $A \in M$. Let $\Phi = \kappa^\kappa/G$. 3) There is an ultrafilter $U$ on $\Phi$ such that every member of $U$ has a subset of type $\Phi$, and the intersection of any well-ordered subset of $U$ is in $U$. 4) $\Phi$ satisfies $\Phi \rightarrow (\Phi)^\alpha_\beta$ for all $\alpha < \aleph_1$ and all ordinals $\beta$. 5) There is a linear order $\Phi'$ with property 3) above which is not "weakly compact", i.e., $\Phi' \nrightarrow (\Phi')^2$.

#### Article information

Source
J. Symbolic Logic, Volume 49, Issue 2 (1984), 542-554.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183741554

Mathematical Reviews number (MathSciNet)
MR745382

Zentralblatt MATH identifier
0586.03046

JSTOR