## Journal of Symbolic Logic

### Souslin Forcing

#### Abstract

We define the notion of Souslin forcing, and we prove that some properties are preserved under iteration. We define a weaker form of Martin's axiom, namely $\mathrm{MA}(\Gamma^+_{\aleph_0})$, and using the results on Souslin forcing we show that $\mathrm{MA}(\Gamma^+_{\aleph_0})$ is consistent with the existence of a Souslin tree and with the splitting number $s = \aleph_1$. We prove that $\mathrm{MA}(\Gamma^+_{\aleph_0})$ proves the additivity of measure. Also we introduce the notion of proper Souslin forcing, and we prove that this property is preserved under countable support iterated forcing. We use these results to show that ZFC + there is an inaccessible cardinal is equiconsistent with ZFC + the Borel conjecture + $\Sigma^1_2$-measurability.

#### Article information

Source
J. Symbolic Logic, Volume 53, Issue 4 (1988), 1188-1207.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183742790

Mathematical Reviews number (MathSciNet)
MR973109

Zentralblatt MATH identifier
0673.03039

JSTOR