Journal of Symbolic Logic

Backwards Easton Forcing and $0^#$

M. C. Stanley

Abstract

It is shown that if $\kappa$ is an uncountable successor cardinal in $L\lbrack 0^\sharp\rbrack$, then there is a normal tree $\mathbf{T} \in L \lbrack 0^\sharp\rbrack$ of height $\kappa$ such that $0^\sharp \not\in L\lbrack\mathbf{T}\rbrack$. Yet $\mathbf{T}$ is $<\kappa$-distributive in $L\lbrack 0^\sharp\rbrack$. A proper class version of this theorem yields an analogous $L\lbrack 0^\sharp\rbrack$-definable tree such that distinct branches in the presence of $0^\sharp$ collapse the universe. A heretofore unutilized method for constructing in $L\lbrack 0^\sharp\rbrack$ generic objects for certain $L$-definable forcings and "exotic sequences", combinatorial principles introduced by C. Gray, are used in constructing these trees.

Article information

Source
J. Symbolic Logic, Volume 53, Issue 3 (1988), 809-833.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183742721

Mathematical Reviews number (MathSciNet)
MR961000

Zentralblatt MATH identifier
0657.03036

JSTOR
Stanley, M. C. Backwards Easton Forcing and $0^#$. J. Symbolic Logic 53 (1988), no. 3, 809--833. https://projecteuclid.org/euclid.jsl/1183742721