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Reflection Principles, Generic Large Cardinals, and the Continuum Problem Read chapter Read first chapter

2021 | OriginalPaper | Chapter

Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds

Author : Toshimichi Usuba

Published in: Advances in Mathematical Logic

Publisher: Springer Nature Singapore

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Abstract

In this paper, without the axiom of choice, we show that if a certain downward Löwenheim–Skolem property holds then all grounds are uniformly definable. We also prove that the axiom of choice is forceable if and only if the universe is a small extension of some transitive model of \(\mathsf {ZFC}\).

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previous chapter Reforming Takeuti’s Quantum Set Theory to Satisfy de Morgan’s Laws
next chapter Irrational-Based Computability of Functions
Footnotes
1
In [4], our M(X) is referred to M[X].
 
2
Actually Kripke-Platek set-theory is sufficient.
 
3
Of course this definition cannot be formalized within \(\mathsf {ZF}\), so we will use it informally.
 
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Metadata
Title
Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds
Author
Toshimichi Usuba
Copyright Year
2021
Publisher
Springer Nature Singapore
Book
Advances in Mathematical Logic

Print ISBN: 978-981-16-4172-5

Electronic ISBN: 978-981-16-4173-2

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-981-16-4173-2_8

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