## Journal of Symbolic Logic

### On Non-Wellfounded Iterations of the Perfect Set Forcing

#### Abstract

We prove that if I is a partially ordered set in a countable transitive model $\mathfrak{M}$ of $\mathbf{ZFC}$ then $\mathfrak{M}$ can be extended by a generic sequence of reals $\mathbf{a}_i$, i $\in$ I, such that $\aleph^{\mathfrak{M}}_1$ is preserved and every $\mathbf{a}_i$ is Sacks generic over $\mathfrak{M}[\langle \mathbf{a}_j : j < i\rangle]$. The structure of the degrees of $\mathfrak{M}$-constructibility of reals in the extension is investigated. As applications of the methods involved, we define a cardinal invariant to distinguish product and iterated Sacks extensions, and give a short proof of a theorem (by Budinas) that in $\omega_2$-iterated Sacks extension of L the Burgess selection principle for analytic equivalence relations holds.

#### Article information

Source
J. Symbolic Logic, Volume 64, Issue 2 (1999), 551-574.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745793

Mathematical Reviews number (MathSciNet)
MR1777770

Zentralblatt MATH identifier
0930.03062

JSTOR