Timeline for Philosophy of forcing and ctm
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38 events
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| Apr 6, 2022 at 10:54 | answer | added | Neil Barton | timeline score: 4 | |
| Feb 1, 2022 at 17:51 | comment | added | Julia Williams | @TimothyChow For example, à la Fujimoto (see the linked paper below), one might be a predicativist about classes, seeing them as coming from predicates of sets and thereby distinct from sets. Analogously, a finitist might not think there are any infinite mathematical objects, but still think it's sensible to talk about predicates like "being prime" or "being the index of a Turing machine which halts" or "being a true sentence of arithmetic". (philpapers.org/rec/FUJPAC-2) | |
| Feb 1, 2022 at 17:29 | comment | added | Timothy Chow | @KamerynWilliams What accounts of classes are you referring to? I don't think my objection depends delicately on any particular conception of classes. As I understand it, the skeptic regards the powerset axiom as introducing some qualitatively new "jump" (just as someone might regard the leap from finite to infinite as a qualitatively new jump). If the response is, "Okay, we'll call the new things classes instead of uncountable sets" (or, we'll call the new things classes instead of infinite sets) then what is gained? | |
| Feb 1, 2022 at 15:55 | comment | added | Julia Williams | @TimothyChow The question that seems to underlie your comment is, why should we think of (proper) classes as anything other than sets but bigger? Classes as sets under a different name is an unsatisfying answer and, following what you said, would undermine the Barton & Friedman argument. But there are other accounts for what classes are in the literature and if one holds to something like them then I think the Barton & Friedman paper has more argumentative force. | |
| Feb 1, 2022 at 10:15 | comment | added | Timothy Chow | @KamerynWilliams That's an interesting paper. But I would think that a true skeptic of uncountability would not be satisfied by their approach, according to which there are legitimate uncountable things out there, but we're just not going to call them "sets." For comparison, what if we were to say that "all sets are finite," but then do mathematics pretty much as usual, except that we call the infinite things "proper classes" and not "sets"? I can't see skeptics of infinite sets being happy with that. | |
| Feb 1, 2022 at 9:37 | answer | added | Corey Bacal Switzer | timeline score: 9 | |
| Feb 1, 2022 at 5:19 | history | edited | Kushi | CC BY-SA 4.0 |
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| Feb 1, 2022 at 2:02 | history | edited | Kushi |
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| Jan 31, 2022 at 21:25 | comment | added | Kushi | @KamerynWilliams Thanks for the reference! Glad to see that they also quoted Scott in the paper. I'm a countabilist now! | |
| Jan 31, 2022 at 17:16 | comment | added | Julia Williams | "another doubt of mine is that the von-Neumann hierarchy picture is often used to justify the consistency of ZFC, but the idea of forcing seems to say that we actually never reach the "true" power set of a set, so the hierarchy cannot proceed in the first place." This is an idea that has been taken seriously in the literature, leading the view that all sets are countable and so the von Neumann hierarchy doesn't get off the ground. See e.g. this paper by Barton & Friedman: philpapers.org/rec/BARCAM-5 | |
| Jan 31, 2022 at 5:51 | comment | added | David Roberts♦ | @Kushi yes, I mean the topos-theoretic approach, which has a very different ontology. It's not about "the" universe V, or CTMs, but more flexible (and, I note, it can stand on its own foundations). I can't do the topic justice in a comment, though. | |
| Jan 31, 2022 at 3:13 | history | edited | Kushi | CC BY-SA 4.0 |
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| Jan 31, 2022 at 2:41 | history | edited | Kushi | CC BY-SA 4.0 |
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| Jan 31, 2022 at 2:31 | history | edited | Kushi | CC BY-SA 4.0 |
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| Jan 31, 2022 at 2:27 | comment | added | Farmer S | @Kushi Yes, it's in $L$. I interpreted 3 as saying "can consistently have countable transitive models without uncountable ones". | |
| Jan 31, 2022 at 2:27 | history | edited | Kushi | CC BY-SA 4.0 |
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| Jan 31, 2022 at 2:17 | comment | added | Kushi | @FarmerS I don't know enough to understand your construction, but it seems you are talking about truth in $L$? I am looking for $\phi$ s.t. ZFC+"there exists an uncountable transitive model of ZFC+$\phi$" is inconsistent, while it is consistent (relative to some hypothesis) if the word uncountable is changed to countable. | |
| Jan 31, 2022 at 2:10 | comment | added | Farmer S | An example as in 3: If there is a measurable cardinal plus an inaccessible above it, then $L$ satisfies "there are countable transitive models of ZFC + "there is a measurable cardinal", but no uncountable such models". Moreover, for every $\alpha<\omega_1^L$ there is such a model in $L$ which contains $\alpha$ and thinks that $\alpha$ is countable. The existence of the countable models is by Lowenheim-Skolem and Shoenfield absoluteness. The non-existence of uncountable ones is because otherwise the model's version of $0^\#$ would be iterable, hence be the true $0^\#$, but $0^\#\notin L$. | |
| Jan 31, 2022 at 1:40 | history | edited | Kushi | CC BY-SA 4.0 |
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| Jan 31, 2022 at 1:24 | comment | added | Kushi | @DavidRoberts But Dana Scott wrote this in the context of forcing, so I suppose he is imagining some axiomatic system strong enough to encapsulate all current set theory (in contrast to predicative mathematics) and yet the continuum is a class? Also by "analogue of forcing" do you mean the sheaf-theoretic version? Is it possible to explain to a layman what its intuition is? | |
| Jan 31, 2022 at 1:20 | answer | added | Timothy Chow | timeline score: 16 | |
| Jan 30, 2022 at 15:06 | comment | added | Nik Weaver | Let us continue this discussion in chat. | |
| Jan 30, 2022 at 14:22 | comment | added | David Roberts♦ | @NikWeaver interesting! But surely this is not quite the same as the sort of predicativism as eg CZF, where powerset is out the window ("depends on all sets" seems to me to be a weak sort of predicativism, compared to those that think the reals form a proper class). And I find it hard to link Skolem's paradox with this quote. | |
| Jan 30, 2022 at 13:52 | comment | added | Nik Weaver | @DavidRoberts I might be getting a little off-topic, but it seems worth noting that Skolem apparently did approve of predicativism. I think one can infer this from the following quote: “But the difficulty [for the consistency of ZFC] is that we have to form some sets whose existence depends on all sets. We then have what is called a nonpredicative definition” ("Some remarks on axiomatized set theory") (implying that impredicative definitions are untrustworthy). | |
| Jan 30, 2022 at 9:04 | comment | added | David Roberts♦ | As far as that Dana Scott quote goes, I don't think it should be considered in the same breath as CTMs, since it's more a statement about predicative foundations, whereby the powerset axiom is dropped, other axioms are posited so we have the constructions we like, and we still have the existence of countable infinite sets. In this viewpoint, one can talk about the continuum, but just as a proper class. | |
| Jan 30, 2022 at 9:02 | comment | added | David Roberts♦ | @Kushi ok, just checking. You should be aware that there are other foundations of equivalent strength, where the analogue of forcing proves the same theorems, but one might consider the resulting philosophical implications completely different. So I'm a bit reserved on stating that there is one specific interpretation. | |
| Jan 30, 2022 at 5:56 | comment | added | Nik Weaver | "Since within any axiomatization of set theory or any formal logical system one reasons in such a way that the absolutely uncountable does not exist, the statement that uncountable sets exist can only be considered as a play on words, hence this absolutely uncountable is just a fiction." --- Skolem (Sur la portee du theoreme de Lowenheim-Skolem) | |
| Jan 30, 2022 at 5:55 | comment | added | Nik Weaver | "does it mean actually all sets are countable, and uncountability is just an illusion?" --- Skolem argued exactly this. | |
| Jan 30, 2022 at 5:31 | comment | added | Kushi | @DavidRoberts Changed the wording a little bit. I understand how forcing works and how it proves relative consistency statements in the metatheory. I am more curious about the interpretation of it, hence the tag soft-question. | |
| Jan 30, 2022 at 5:28 | history | edited | Kushi | CC BY-SA 4.0 |
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| Jan 30, 2022 at 5:18 | comment | added | Kushi | @HanulJeon I mean a transitive model of ZFC with the usual membership relation. Boolean-valued models are nice but they are Boolean-valued, just like you can have a model of ZFC+$\lnot$Con(ZFC) but it's ill-founded. | |
| Jan 30, 2022 at 5:13 | history | edited | Kushi | CC BY-SA 4.0 |
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| Jan 30, 2022 at 5:08 | comment | added | David Roberts♦ | People in my experience don't say something is true if it can be forced, but that it is relatively consistent. If you are worried about the process, people have formalised things like the independence of CH, which includes formalising forcing. | |
| Jan 30, 2022 at 5:01 | comment | added | Hanul Jeon | What is the 'standard' model, and what is the difference between standard models and Boolean-valued models? | |
| Jan 30, 2022 at 4:10 | history | edited | Kushi | CC BY-SA 4.0 |
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| Jan 30, 2022 at 1:52 | history | edited | Kushi | CC BY-SA 4.0 |
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| S Jan 30, 2022 at 0:30 | review | First questions | |||
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| S Jan 30, 2022 at 0:30 | history | asked | Kushi | CC BY-SA 4.0 |