Timeline for What is the general opinion on the Generalized Continuum Hypothesis?
Current License: CC BY-SA 2.5
22 events
| when toggle format | what | by | license | comment | |
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| May 19, 2020 at 9:51 | comment | added | user76284 | @ThomasBenjamin See plato.stanford.edu/entries/philosophy-mathematics/#Cat. | |
| Dec 10, 2017 at 18:02 | comment | added | Thomas Benjamin | @WillSawin: Wasn't the Universist view that there is a single 'true' universe of sets based on the categoricity of full second-order $PA$ and the partial categoricity of full second-order $ZFC$ and as such, isn't the notion of categoricity (external or internal) an integral part of the Universist point of view? | |
| Dec 7, 2017 at 10:04 | comment | added | Will Sawin | @ThomasBenjamin I don't see why it would require the theory to be categorical. Again, the theory just says that there is some set-theoretic universe, satisfying the axioms of ZFC, which is real, and all other universes are either not real, or contain only some of the really-existing sets. The first-order statements true in this universe form a theory, "True ZFC", which as you say is non-axiomatizable. This theory might or might not gave any other number of models, but the model is the primary thing, not the theory. | |
| Dec 7, 2017 at 9:22 | comment | added | Thomas Benjamin | @WillSawin: So the "universist" view does not require "True $ZFC$" to be categorical? How does Categoricity fit (if at all) into the universist point of view ("True $ZFC$" would still be nonaxiomatizable, though....)? | |
| Dec 7, 2017 at 7:33 | comment | added | Will Sawin | @ThomasBenjamin Saying that Platonists believe in a true theory of ZFC does not imply that's the only thing they believe in. As Joel mentioned, the "universist" view is that there is one unique real set-theoretic universe. Of course whatever axiomatic theory this universe satisfies, there exist other models of the same set of axioms, but, e.g., just because there may be other models of our laws of physics does not imply that those models are as real as our universe. | |
| Nov 19, 2017 at 14:11 | comment | added | Thomas Benjamin | @JoelDavidHamkins: That may be, but wouldn't these Platonists' "True $ZFC$" be in the same boat (so to speak) as "True Arithmetic", i.e., nonaxiomatizable? And since "True Arithmetic" can have nonstandard models, so can "True $ZFC$". Are these Platonists willing to count nonstandard integers and infinitesimals (and whatever other nonstandard sets the nonstandard "True $$ZFC$" might offer) as being in their Ideal realm? | |
| Nov 19, 2017 at 13:58 | comment | added | Joel David Hamkins | No, the "ultimately correct set theory," according to the Platonists I had mentioned in my first paragraph, would view it as something closer to a maximal theory than a minimal theory. The universist, who holds that there is ultimately a unique set-theoretic universe, might believe that the ultimately true set theory is a complete theory, that every set-theoretic assertion has a truth value, even if it is difficult for us to determine. | |
| Nov 19, 2017 at 13:51 | comment | added | Thomas Benjamin | Isn't the "ultimately correct set theory" (for us or for any denizen of any 'universe' of the set-theoreic multiverse) the minimal (in the sense of reverse mathematics) set theory needed to provide a foundation for mathematics at its current state of development? The question now becomes: "How does one advance mathematics so as to make the current set-theoretic foundation inadequate to the task?" . Why would this not be the case? | |
| Oct 11, 2016 at 19:32 | comment | added | Joel David Hamkins | Yes, that is obvious. And? All I'm saying is that some set theorists think that the large cardinal axioms and axioms like PFA, MM and so on are really true. | |
| Oct 11, 2016 at 19:16 | comment | added | Christopher King | @JoelDavidHamkins I don't think there is a computable sequence of effective theories approaching any complete theory, since then you could compute the axioms of your complete theory by dovetailing. | |
| Oct 10, 2016 at 21:38 | comment | added | Joel David Hamkins | I had meant that those theories are taken to be approaching the true theory, not that they are equal to it. | |
| Oct 10, 2016 at 21:37 | comment | added | Christopher King | @JoelDavidHamkins Well you did say "ultimately correct set theory", not "more correct set theory" or "delicious set theory". I get what your mean though. | |
| Oct 10, 2016 at 21:24 | comment | added | Joel David Hamkins | @PyRulez Well, the large cardinal theories and the forcing axioms are computably axiomatizable, and they are held to be true for the philosophical reasons I mentioned. So the view is that they are a larger fragment of the truth than having, say, only the ZFC axioms. But of course, there is no computably axiomatizable complete theory. Your argument is something like: how can any food be delicious, if there is no perfectly delicious food? | |
| Oct 10, 2016 at 19:33 | comment | added | Christopher King | @JoelDavidHamkins "Many set theorists take this as evidence that these large cardinal axioms put us on the right path towards the ultimately correct set theory." But how can anything get us closers to an ultimate set theory, if there is none (at least no effective ones)? For any effective set theory, there will be things true statements that are left out of it. | |
| Feb 12, 2010 at 1:19 | vote | accept | Harry Gindi | ||
| Feb 6, 2010 at 14:03 | comment | added | Joel David Hamkins | I would also suggest the philosophical work of Peter Koellner, as well as Hugh Woodin and Kai Hauser as explaining this view. | |
| Feb 6, 2010 at 5:55 | comment | added | Tom Leinster | Thanks: that's helpful. I've heard of Maddy's work before, and will go and have a look. | |
| Feb 6, 2010 at 5:47 | comment | added | Joel David Hamkins | Yes, I personally agree with you, that it is not ultimately a matter of proof, but one of aesthetics. But my impression is also that many of the set theorists with this view would not characterize it as a mere aesthetic choice, but rather hold that their view is getting at some kind of ultimate set-theoretic truth. Some of the math philosophers, such as Penelope Maddy, are trying to explain what this view means. | |
| Feb 6, 2010 at 5:23 | comment | added | Tom Leinster | Joel, thanks for that. So let's see if I understand correctly. When you (or rather, others) say "ultimately correct set theory", the correctness is judged by aesthetic criteria. I see a picture of the world of sets that looks nice and (to borrow your words) regular and coherent, and gives me results that seem satisfying; so this is the one I choose to call correct. Have I got the right end of the stick? | |
| Feb 6, 2010 at 5:06 | comment | added | Joel David Hamkins | @Tom: We know that the Incompleteness phenomenon means that we will never have a complete axiomatization of set theory. Our every proposed theory will be ultimately inadequate because of the incompleteness theorem. Nevertheless, the large cardinal hierarchy appears to be providing an increasingly regular and coherent picture of the set theoretic universe, particularly for sets of reals and other issues down low. Many set theorists take this as evidence that these large cardinal axioms put us on the right path towards the ultimately correct set theory. | |
| Feb 6, 2010 at 4:46 | comment | added | Tom Leinster | Joel, my mind is totally boggled by talk of "the final, true set theory". I can't even begin to imagine what anyone could mean by this. And it's not the first time I've heard such talk. I understand that you wouldn't use such a phrase yourself, but can you point me to something easy-read that would help me understand what people mean by this? | |
| Feb 6, 2010 at 3:05 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |