Timeline for What can be preserved in mathematics if all constructions are carried out in ZF?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 22, 2023 at 18:01 | comment | added | Andrés E. Caicedo | @Timothy I didn't mean any specific text, but yes, you can find details in volume 5 of the book you mention, particularly Chapter 56. | |
| Aug 21, 2023 at 14:42 | comment | added | Timothy Chow | @AndrésE.Caicedo When you say "Fremlin," I take it you mean his multi-volume treatise, Measure Theory? That's a dauntingly long text. Can you point to the most relevant sections for the present topic of discussion? | |
| Aug 5, 2023 at 21:27 | comment | added | Martin Väth | @TimothyChow: In nonstandard analysis of Robinson/Luxemburg, the transfer principle applies, roughly speaking, only to standard sentences; but there are more. You can “explicitly” define non-measurable sets there. For instance, let $f\colon[0,1)\times\mathbb N\to\{0,1\}$ be such that $f(x,n)$ is the $n$-th digit of the binary expansion of $x$. Then for any infinite hypernatural $h$ the set $\{x\in[0,1):{}^*f({}^*x,h)={}^*1\}$ is not Lebesgue measurable. It is surprising that this set cannot be defined in the restricted languages of internal set theory from the cited paper of Hrbacek-Katz. | |
| Aug 2, 2023 at 1:09 | comment | added | Timothy Chow | @MartinVäth I realize that I'm confused about your comment regarding nonstandard analysis. Shouldn't there be some kind of transfer principle that means that the apparent use of stronger axioms isn't really necessary? Admittedly, I'm not very familiar with nonstandard analysis. | |
| Oct 9, 2022 at 22:31 | history | edited | Martin Väth | CC BY-SA 4.0 |
Reference to nonstandard analysis without choice
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| Sep 25, 2022 at 15:02 | comment | added | Martin Väth | @TimothyChow ”almost impossible” was probably a bit too harsh, but you have to be extremely careful which definitions you choose. So there is not “the” integration theory anymore in ZF, but several such theories, usually preserving different properties (compared to the ZF+DC situation which has all properties and in which all approaches are equivalent, at least for the Lebesgue measure). | |
| Sep 24, 2022 at 22:16 | comment | added | Andrés E. Caicedo | Without any choice what you get does not really resemble the classical setting. Lebesgue "measure" is not $\sigma$-additive, for instance. It is true you can "recover" more than one would expect. Fremlin has an interesting approach using "codes" of sets rather than the sets themselves. | |
| Sep 24, 2022 at 20:57 | comment | added | Timothy Chow | I'm not sure that "In pure ZF ... it is almost impossible to do a reasonable measure or integration theory" is quite correct. There are difficulties, to be sure, but Bishop already took a stab at developing measure theory without choice. See also Constructive algebraic integration theory without choice by Spitters and Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory by Petrakis and Zeuner. | |
| Sep 24, 2022 at 13:49 | history | answered | Martin Väth | CC BY-SA 4.0 |