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GENERIC CODING WITH HELP AND AMALGAMATION FAILURE

Part of: Set theory

Published online by Cambridge University Press:  05 October 2020

SY-DAVID FRIEDMAN  and
DAN HATHAWAY
SY-DAVID FRIEDMAN
Affiliation:
KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC UNIVERSITY OF VIENNAVIENNA, AUSTRIAE-mail: [email protected]
DAN HATHAWAY
Affiliation:
MATHEMATICS DEPARTMENT UNIVERSITY OF VERMONTBURLINGTON, VT05405, USAE-mail: [email protected]
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Abstract

We show that if M is a countable transitive model of $\text {ZF}$ and if $a,b$ are reals not in M, then there is a G generic over M such that $b \in L[a,G]$ . We then present several applications such as the following: if J is any countable transitive model of $\text {ZFC}$ and $M \not \subseteq J$ is another countable transitive model of $\text {ZFC}$ of the same ordinal height $\alpha $ , then there is a forcing extension N of J such that $M \cup N$ is not included in any transitive model of $\text {ZFC}$ of height $\alpha $ . Also, assuming $0^{\#}$ exists, letting S be the set of reals generic over L, although S is disjoint from the Turing cone above $0^{\#}$ , we have that for any non-constructible real a, $\{ a \oplus s : s \in S \}$ is cofinal in the Turing degrees.

Keywords

forcinginner modelsconstructibilitymodels of set theory

MSC classification

Primary: 03E35: Consistency and independence results 03E40: Other aspects of forcing and Boolean-valued models 03E45: Inner models, including constructibility, ordinal definability, and core models
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 4 , December 2021 , pp. 1385 - 1395
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Copyright
© Association for Symbolic Logic 2020

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