Timeline for Where in ordinary math do we need unbounded separation and replacement?
Current License: CC BY-SA 3.0
31 events
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Jun 21, 2022 at 15:13 | comment | added | Toby Bartels | Agreed. I'm not familiar enough with models of various set theories to say that it can't be done, but I still don't see how to do it. | |
Jun 20, 2022 at 10:12 | comment | added | user21820 | @TobyBartels: Thanks, yes I saw that post, so we know that if there are no other 'dangerous moves' then all we need is to be able to construct (Pow^k)(ℕ) for any countable ordinal k. But I don't know whether we can do that in BZFC, nor whether we can obtain the ordinal rank of any Borel set in BZFC. Even KP^P+AC (where KP^P is as defined by Mathias in "The Strength of Mac Lane Set Theory" (Section 6.1)), which has inbuilt Powerset, is not obviously sufficient. | |
Jun 20, 2022 at 9:50 | comment | added | Toby Bartels | @user21820 : I don't know, but this post (by Tom Leinster) argues that BD doesn't require very much in the way of Replacement. | |
Jun 19, 2022 at 6:19 | comment | added | user21820 | @TobyBartels: Yes I have always viewed bounded ZFC as having both bounded specification and replacement. By the way, do you know if bounded ZFC proves Borel determinacy? | |
Jun 19, 2022 at 5:03 | comment | added | Toby Bartels | @user21820 : I meant that unbounded Replacement implies unbounded Separation. But it would make sense to interpret BZFC as incorporating only bounded Replacement, like in Kripke–Platek set theory, so my parenthetical remark is wrong for that reason. | |
Jun 17, 2022 at 13:31 | comment | added | user21820 | @TobyBartels: Your last comment replacement seems to say that bounded replacement implies unbounded separation. But I don't think so. | |
Jul 22, 2021 at 2:17 | comment | added | David Roberts♦ | I've been vocal in this space, but these days I prefer the term "generic mathematics" (and "generic mathematician"), rather than "ordinary mathematics". The term is less loaded, and conveys a better idea for me of what is meant. The allusion to mathematical genericity is entirely intentional. | |
Oct 29, 2019 at 17:24 | comment | added | Toby Bartels | My last comment above should say BZC, not BZFC. (The F is what adds Replacement, so it's exactly what we do not want. Hopefully, nobody was confused, since there is no BZFC; Replacement implies unbounded Separation.) | |
Sep 1, 2013 at 16:27 | answer | added | Adam Epstein | timeline score: 6 | |
Mar 5, 2013 at 2:10 | answer | added | Adam Epstein | timeline score: 12 | |
Feb 13, 2013 at 16:22 | comment | added | Toby Bartels | In my mind, I usually define ‘ordinary mathematics’ as mathematics that can be formulated in ETCS (or BZFC if you prefer). Of course, that begs the question, and I try not to take that definition too seriously. | |
Feb 11, 2013 at 19:15 | vote | accept | Andrej Bauer | ||
Feb 11, 2013 at 19:15 | vote | accept | Andrej Bauer | ||
Feb 11, 2013 at 19:15 | |||||
Feb 11, 2013 at 19:15 | history | made wiki | Post Made Community Wiki by Andrej Bauer | ||
Feb 11, 2013 at 8:24 | comment | added | Zhen Lin | @David – My example does not make sense in structural set theory by design. I am thinking of the fact that $V_{\omega + \omega}$ is a model of Mac Lane set theory (hence of $\Delta_0$-separation). $V_{\omega + \omega}$ believes there exist uncountable well-ordered sets (using the Hartogs construction, say), but obviously it does not believe in the von Neumann ordinal $\omega + \omega$. | |
Feb 11, 2013 at 6:03 | comment | added | Andrej Bauer | Should this be a wiki? | |
Feb 11, 2013 at 1:23 | answer | added | arsmath | timeline score: 6 | |
Feb 10, 2013 at 22:45 | comment | added | David Roberts♦ | @Zhen - what is the codomain of this function? Isn't it just $\omega + \omega$? So if one thinks structurally, i.e. isomorphism invariantly, one is trying to define a function whose codomain is isomorphic (as a set) to $\mathbb{N}$ But also, it makes no sense structurally to define a function with a codomain you cannot prove exists (one could of course say $\exists \omega + \omega \Rightarrow \exists (\n \mapsto \omega + n)$). | |
Feb 10, 2013 at 22:08 | history | edited | Andrej Bauer |
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Feb 10, 2013 at 21:09 | comment | added | Jason Rute | Add a logic tag? | |
Feb 10, 2013 at 21:03 | history | edited | Andrej Bauer | CC BY-SA 3.0 |
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Feb 10, 2013 at 20:08 | comment | added | user30035 | It's very difficult to say what "ordinary mathematics" is -- I would even say that this sort of phrase could be interpreted as being "subjective and argumentative" :-) Let me suggest a related, perhaps more precise, question, which might capture the flavor of what you're after -- you could ask whether the proofs of the Poincare Conjecture or FLT use things like replacement. In the case of FLT I'm pretty sure that Wiles' proof does not use replacement. | |
Feb 10, 2013 at 20:06 | answer | added | user30035 | timeline score: 4 | |
Feb 10, 2013 at 19:51 | answer | added | Todd Trimble | timeline score: 20 | |
Feb 10, 2013 at 19:14 | history | edited | Andrej Bauer | CC BY-SA 3.0 |
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Feb 10, 2013 at 19:11 | comment | added | Andrej Bauer | I mean unbounded quantifiers. @Zhen: I added a clarification which excludes your example. | |
Feb 10, 2013 at 19:01 | comment | added | Asaf Karagila♦ | Zhen Lin: Can you modify that and talk about the various hierarchies (Borel, arithmetical, analytical, projective, etc.) which are constructed by induction, and/or transfinite inductions? | |
Feb 10, 2013 at 18:13 | answer | added | Cody Dance | timeline score: 21 | |
Feb 10, 2013 at 18:08 | comment | added | Zhen Lin | Without unbounded replacement, it may be impossible to define a function on the natural numbers recursively if one does not know in advance that the values of the function are all contained in some set. For example, the function $n \mapsto \omega + n$ cannot be constructed using only $\Delta_0$-replacement, even though each von Neumann ordinal $\omega + n$ exists. However, perhaps this specific example does not count as ordinary mathematics... | |
Feb 10, 2013 at 17:27 | comment | added | Asaf Karagila♦ | What is ordinary mathematics? By unbounded do you mean any unbounded quantifier, or something else (e.g. arguments whose proof uses formulas of unbounded complexity)? | |
Feb 10, 2013 at 17:06 | history | asked | Andrej Bauer | CC BY-SA 3.0 |