Timeline for Martin's "Philosophical Issues about the Hierarchy of Sets"
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| May 1, 2011 at 22:21 | comment | added | Joel David Hamkins | I take Martin's argument (of my first comment above) as an anti-multiverse argument, leading one to the conclusion that there is at most one complete concept of set, since any two of them must agree. (Ultimately, I don't agree with this position, as Marc mentions, and I have defended a multiverse position pointing to diverse incompatible set concepts.) | |
| May 1, 2011 at 19:56 | comment | added | Marc Alcobé García | Asaf, I think Joel refers in his slides to the staggering diversity of models of set theory as giving rise to different set concepts (please correct me if I am wrong, I didn't check). I am not sure one can make this identification the other way round. I would not, a priori, think of an instantiation of a set concept as a model of a first-order theory of sets. | |
| May 1, 2011 at 6:21 | comment | added | Asaf Karagila♦ | @Joel: So this philosophical argument is really like "Set theory behaves as you would expect it to behave" in terms of model isomorphisms? | |
| Apr 30, 2011 at 21:26 | comment | added | Joel David Hamkins | Yes, this is a philosophical argument, rather than a mathematical one, but the idea as I understand it is that if each set concept is thought to include absolutely all the subsets of any set that it has, then they will have to agree increasingly throughout the hierarchy and thus ultimately give rise to the same universe of sets. | |
| Apr 30, 2011 at 20:08 | comment | added | Marc Alcobé García | I suppose that something counts as an instantiation of the full set concept if its powersets contain every conceivable subset (if this really means something), but also if it contains every conceivable ordinal (again, if this really means something). | |
| Apr 30, 2011 at 19:46 | comment | added | Asaf Karagila♦ | Joel, as usual a very informative and interesting answer. As for your first comment, what do you mean by "full set concept"? | |
| Apr 30, 2011 at 18:52 | comment | added | Joel David Hamkins | I use the axiom $V_\delta\prec V$ in my article on the Maximality Principle (J. D. Hamkins, "A simple maximality principle," Journal of Symbolic Logic, vol. 68, pp. 527--550, June 2003), where I give a brief account of it. It is necessary in the forcing construction that is used to obtain the Maximality Principle. Also, I believe that Solomon Feferman has used this axiom in some of his work, in order to provide an alternative weaker foundation for the use of universes in category theory. One can have a whole proper class club of such $\delta$, still just with consistency strength ZFC. | |
| Apr 30, 2011 at 18:07 | comment | added | Marc Alcobé García | Thank you vey much, Joel. Do you know where could I read more about $V_\delta\prec V$ and its properties? Also, I have googled for Martin's articles and the most recent that I have found is "Multiple Universes of Sets and Indeterminate Truth Values" (2001). | |
| Apr 30, 2011 at 17:52 | vote | accept | Marc Alcobé García | ||
| Apr 30, 2011 at 16:28 | comment | added | Joel David Hamkins | I remember also an argument that he made or considered (and I've heard him make this argument in other forums) that any two instantiations $V$ and $\bar V$ of the full set concept must agree; the idea is that one inductively shows that they agree at every level of the hiearchy, essentially since if they agree up to $V_\alpha$ and each is claiming to have all of the subsets of $V_\alpha$, then they agree up to $V_{\alpha+1}$. | |
| Apr 29, 2011 at 16:07 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |