An improper arithmetically closed Borel subalgebra of P(ω) mod FIN

Ali Enayat; Saharon Shelah

doi:10.1016/j.topol.2011.08.006

An improper arithmetically closed Borel subalgebra of P(ω) mod FIN

Ali Enayat^*, Saharon Shelah

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We show the existence of a subalgebra A⊆P(ω) that satisfies the following three conditions: A is Borel (when P(ω) is identified with 2^ω). A is arithmetically closed (i.e., A is closed under the Turing jump, and Turing reducibility). The forcing notion (A,⊆) modulo the ideal FIN of finite sets collapses the continuum to א₀.

Original language	English
Pages (from-to)	2495-2502
Number of pages	8
Journal	Topology and its Applications
Volume	158
Issue number	18
DOIs	https://doi.org/10.1016/j.topol.2011.08.006
State	Published - 1 Dec 2011

Keywords

Borel structure
Completely separable family
Forcing
Tree indiscernible

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abstract = "We show the existence of a subalgebra A⊆P(ω) that satisfies the following three conditions: A is Borel (when P(ω) is identified with 2ω). A is arithmetically closed (i.e., A is closed under the Turing jump, and Turing reducibility). The forcing notion (A,⊆) modulo the ideal FIN of finite sets collapses the continuum to א0.",

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AB - We show the existence of a subalgebra A⊆P(ω) that satisfies the following three conditions: A is Borel (when P(ω) is identified with 2ω). A is arithmetically closed (i.e., A is closed under the Turing jump, and Turing reducibility). The forcing notion (A,⊆) modulo the ideal FIN of finite sets collapses the continuum to א0.

KW - Borel structure

KW - Completely separable family

KW - Forcing

KW - Tree indiscernible

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An improper arithmetically closed Borel subalgebra of P(ω) mod FIN

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Keywords

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