Timeline for Comparison of model-theoretic and axiomatic approaches to NSA
Current License: CC BY-SA 4.0
32 events
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| Nov 5 at 22:16 | comment | added | Gerry Myerson | Editing tags on half-a-dozen old questions in the space of a few minutes drives newer questions off the front page, Mikhail. Please consider doing less of that. | |
| Nov 5 at 14:20 | history | edited | Mikhail Katz |
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| Nov 1 at 10:42 | comment | added | Mikhail Katz | But if you remove such an assumption, then the road is clear to adopt a more expressive set-theoretic foundation, justified by conservativity results, and moreover more faithful to the history of infinitesimal analysis. But you are probably somewhat familiar with my historical papers. | |
| Nov 1 at 10:40 | comment | added | Mikhail Katz | @Noah, A slightly delayed response to your "work" comment: "How do you justify taking an axiomatic approach without first knowing that (under mild assumptions) the relevant theory is consistent? To me it seems that the examples of "axiomatic efficiency" in this question are just hiding the (usually model-theoretic) work elsewhere." In physics, they calculate work from point A to point B. So the real question is: what is your starting point? If you assume that the obligatory starting point is infinitesimal-free ZF, then surely a lot of work will be needed to put infinitesimals back in. | |
| Nov 12, 2023 at 14:41 | comment | added | Mikhail Katz | @James, see comment above. | |
| Nov 12, 2023 at 14:39 | comment | added | Vladimir Kanovei | @ James Hanson Regarding HST and BST see DOI 10.1070/RM2007v062n01ABEH004381 , for sPOT and SCOT see DOI 10.1016/j.apal.2021.102959 and sorry for such a delay | |
| S Sep 23, 2023 at 12:01 | history | bounty ended | CommunityBot | ||
| S Sep 23, 2023 at 12:01 | history | notice removed | CommunityBot | ||
| Sep 19, 2023 at 13:05 | comment | added | Mikhail Katz | By the standard easy yoga with the ultrafilter, a standard set is finite if and only if its natural extension is finite, so one wouldn't need choice here. @JoelDavidHamkins | |
| Sep 19, 2023 at 13:02 | comment | added | Joel David Hamkins | Ah, sorry, indeed I was thinking of standard sets. I don't know anything really about internal sets, as I rarely work in that framework. | |
| Sep 19, 2023 at 12:43 | comment | added | Mikhail Katz | Incidentally, user Rivers seems to share your concerns; see math.stackexchange.com/questions/4770885/… @JoelDavidHamkins | |
| Sep 19, 2023 at 12:37 | comment | added | Mikhail Katz | As far as your objection to item (2) is concerned, I am not sure what you mean. It is true that it boils down to induction, but only internal induction, as I mentioned in my question. In axiomatic frameworks, this is the same old induction (of course modulo the "idiosyncratic, unhelpful, confusing, and incoherent" aspects you mentioned in an earlier comment of yours). @Joel | |
| Sep 19, 2023 at 12:34 | comment | added | Mikhail Katz | Your argument for (1) only works for standard sets, not for internal sets. If you pick a sequence of elements of an internal set, there is no guarantee that $a_N$ will be in the set if it is only known to be internal. Consider for example $S=\{1,2,3,\ldots, K\}$; if $H>K$ your argument fails. A further objection is that choosing a sequence involves choice, whereas axiomatic approaches can do without choice at all. @Joel | |
| Sep 19, 2023 at 12:28 | comment | added | Joel David Hamkins | In items 1 and 2, I wonder whether you are exaggerating the difficulty in using the model theoretic approach. For 1, if $A$ is infinite, including a sequence $\langle a_n\rangle_n$ of distinct elements, and then consider $a^*_N$ for some nonstandard $N$. By assumption this is standard, and so by elementarity it must have a standard index in the sequence, but it doesn't. Contradiction. For 2, overspill is immediate from induction, which is part of the theory we work in. You make it sound like a complicated development? | |
| Sep 19, 2023 at 11:07 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
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| S Sep 15, 2023 at 10:09 | history | bounty started | Mikhail Katz | ||
| S Sep 15, 2023 at 10:09 | history | notice added | Mikhail Katz | Draw attention | |
| Sep 13, 2023 at 11:11 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
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| Sep 13, 2023 at 11:10 | comment | added | Mikhail Katz | I should mention also that adopting Hamkins' "well-foundedness mirage" axiom is helpful in understanding the spririt of axiomatic NSA. | |
| Sep 13, 2023 at 10:36 | history | edited | Mikhail Katz | CC BY-SA 4.0 |
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| Sep 13, 2023 at 10:36 | comment | added | Mikhail Katz | ... the standardness predicate is not some kind of exotic post-modern innovation that must necessarily be viewed as superstructure to the "normal" set theory. My evidence for this is that the assignable/inassignable distinction was already in Leibniz, so it is arguably natural enough to be included as part of the axiomatic background. In other words, once the conservativity of the st-\in-theories are justified by what Schweber refers to as "model-theoretic work", it becomes clear that the st-\in-language is as legitimate as the \in-language rather than being some kind of stepchild. | |
| Sep 13, 2023 at 10:35 | comment | added | Mikhail Katz | @Noah: ""How do you justify taking an axiomatic approach without first knowing that (under mild assumptions) the relevant theory is consistent? To me it seems that the examples of 'axiomatic efficiency' in this question are just hiding the (usually model-theoretic) work elsewhere." This is a good objection, but notice that it only makes sense if you postulate that the appropriate starting point is an \in-theory, rather than a st-\in theory. The point I would like to make is that ... | |
| Sep 13, 2023 at 1:53 | comment | added | James E Hanson | @VladimirKanovei Where are good references for these theories? Google is failing me. | |
| Sep 12, 2023 at 23:45 | comment | added | Noah Schweber | @VladimirKanovei "Once a benchmark thm is established, no reference to it in proofs of different theorems can be considered as hiding anything" But there is no reference to such a benchmark in the arguments in the OP that I see. | |
| Sep 12, 2023 at 23:43 | comment | added | Noah Schweber | @AlecRhea That's the source of my "under mild assumptions." Of course a model-theoretic argument only yields conditional results (but the same is true for anything else, really), but even conditional results are useful for establishing confidence. For instance, speaking only for myself I find that I have a much higher degree of confidence in (say) the consistency of BST after seeing a proof of consistency relative to ZFC. | |
| Sep 12, 2023 at 23:21 | comment | added | Joel David Hamkins | I would find it important to inquire to what extent the nonstandard theories are mutually interpretable or bi-interpretable with the corresponding standard theories. It seems that ZFC is likely bi-interpretable with a suitable nonstandard theory, simply by interpreting via suitable iterated ultrapowers of V. If the theories are bi-interpretable, then they are arguably semantically equivalent variations of same underlying subject. It would be fruitless to argue for one over the other in this case except on style or convenience. So what is the interpretability status of these theories? | |
| Sep 12, 2023 at 22:54 | comment | added | Alec Rhea | @NoahSchweber How does model theory offer reassurance in this area without falling back on axioms? (the models one constructs exist within another axiomatic system) | |
| Sep 12, 2023 at 19:04 | comment | added | Vladimir Kanovei | Noah Schweber: hiding is not an appropriate word here. Once a benchmark thm is established, no reference to it in proofs of different theorems can be considered as hiding anything. | |
| Sep 12, 2023 at 18:57 | comment | added | Vladimir Kanovei | BST, HST, SPOT, and SCOT are nonstandard set theories extending ZFC/ZF in different (but equi-consistent) ways | |
| Sep 12, 2023 at 18:08 | comment | added | Noah Schweber | How do you justify taking an axiomatic approach without first knowing that (under mild assumptions) the relevant theory is consistent? To me it seems that the examples of "axiomatic efficiency" in this question are just hiding the (usually model-theoretic) work elsewhere. | |
| Sep 12, 2023 at 17:20 | comment | added | James E Hanson | What are BST, HST, SPOT, and SCOT? | |
| Sep 12, 2023 at 14:57 | history | asked | Mikhail Katz | CC BY-SA 4.0 |