On intermediate extensions of generic extensions by a random real
The paper is the second of our series of notes aimed to bring back in circulation some bright ideas of early modern set theory, mainly due to Harrington and Sami, which have never been adequately presented in set theoretic publications. We prove that if a real $a$ is random over a model $M$ and $x\i...
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ftdatacite:10.48550/arxiv.1811.10568 2023-05-15T18:12:06+02:00 On intermediate extensions of generic extensions by a random real Kanovei, Vladimir Lyubetsky, Vassily 2018 https://dx.doi.org/10.48550/arxiv.1811.10568 https://arxiv.org/abs/1811.10568 unknown arXiv arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Logic math.LO FOS Mathematics 03E15, 03E35 Preprint Article article CreativeWork 2018 ftdatacite https://doi.org/10.48550/arxiv.1811.10568 2022-04-01T08:50:40Z The paper is the second of our series of notes aimed to bring back in circulation some bright ideas of early modern set theory, mainly due to Harrington and Sami, which have never been adequately presented in set theoretic publications. We prove that if a real $a$ is random over a model $M$ and $x\in M[a]$ is another real then either (1) $x\in M$, or (2) $M[x]=M[a]$, or (3) $M[x]$ is a random extension of $M$ and $M[a]$ is a random extension of $M[x]$. This is a less-known result of old set theoretic folklore, and, as far as we know, has never been published. As a corollary, we prove that $Σ^1_n$-Reduction holds for all $n\ge3$, in a model extending the constructible universe $L$ by $\aleph_1$-many random reals. : 11 pages Report sami DataCite Metadata Store (German National Library of Science and Technology) |
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DataCite Metadata Store (German National Library of Science and Technology) |
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Logic math.LO FOS Mathematics 03E15, 03E35 |
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Logic math.LO FOS Mathematics 03E15, 03E35 Kanovei, Vladimir Lyubetsky, Vassily On intermediate extensions of generic extensions by a random real |
topic_facet |
Logic math.LO FOS Mathematics 03E15, 03E35 |
description |
The paper is the second of our series of notes aimed to bring back in circulation some bright ideas of early modern set theory, mainly due to Harrington and Sami, which have never been adequately presented in set theoretic publications. We prove that if a real $a$ is random over a model $M$ and $x\in M[a]$ is another real then either (1) $x\in M$, or (2) $M[x]=M[a]$, or (3) $M[x]$ is a random extension of $M$ and $M[a]$ is a random extension of $M[x]$. This is a less-known result of old set theoretic folklore, and, as far as we know, has never been published. As a corollary, we prove that $Σ^1_n$-Reduction holds for all $n\ge3$, in a model extending the constructible universe $L$ by $\aleph_1$-many random reals. : 11 pages |
format |
Report |
author |
Kanovei, Vladimir Lyubetsky, Vassily |
author_facet |
Kanovei, Vladimir Lyubetsky, Vassily |
author_sort |
Kanovei, Vladimir |
title |
On intermediate extensions of generic extensions by a random real |
title_short |
On intermediate extensions of generic extensions by a random real |
title_full |
On intermediate extensions of generic extensions by a random real |
title_fullStr |
On intermediate extensions of generic extensions by a random real |
title_full_unstemmed |
On intermediate extensions of generic extensions by a random real |
title_sort |
on intermediate extensions of generic extensions by a random real |
publisher |
arXiv |
publishDate |
2018 |
url |
https://dx.doi.org/10.48550/arxiv.1811.10568 https://arxiv.org/abs/1811.10568 |
genre |
sami |
genre_facet |
sami |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.1811.10568 |
_version_ |
1766184666320076800 |