Timeline for Topological space (or math structure more generally) without encoding as set
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 17, 2017 at 11:28 | comment | added | Mike Shulman | @JoelDavidHamkins By "esoteric" I didn't mean to be disparaging, or imply that it isn't interesting or doesn't have lots of people studying it. All of pure mathematics is pretty esoteric to the man on the street. (-: I just meant that point-set topology is used by lots of people doing algebraic topology, low-dimensional topology, differential geometry, etc., and I don't think any of them have ever had any occasion to worry about large cardinal axioms. | |
| Jul 17, 2017 at 11:20 | comment | added | Joel David Hamkins | I like your second answer much better, and that is one I can agree with. My point wasn't about ZFC specifically, but rather with the claim you seemed to make that topological questions don't depend on one's foundation, since we know that they definitely do. See en.wikipedia.org/wiki/Set-theoretic_topology for a start. This is not esoteric, but rather an active field of mathematical research, with its own conferences and so on and connected with other fields. As you say, the extra hypotheses can of course be expressed in any sufficiently robust foundational system. | |
| Jul 17, 2017 at 11:13 | comment | added | Mike Shulman | The second answer is that such "extra-ZFC axioms" are not specific to ZFC: most of them have equally natural formulations in any other foundational set theory like ETCS. So while point-set topology is not independent of the axioms of your set theory, it is independent of how you choose to formalize that set theory, and in particular of whether you use global-membership-based sets, categorical-structural sets, h-sets in HoTT, etc. | |
| Jul 17, 2017 at 11:09 | comment | added | Mike Shulman | @JoelDavidHamkins Two answers. The first is that I was not thinking about such questions at all, which I regard as rather esoteric. Certainly I believe that if you spend enough time studying point-set topology as a field in its own right you will run into them soon enough, but for someone who primarily wants a tool to study "notions of nearness, connectedness, continuity" in "naturally occurring" spaces (which is what I assumed the OP wanted) I think they will not play much role. | |
| Jul 17, 2017 at 10:57 | comment | added | Joel David Hamkins | In other words, what we've learned in point-set topology is that many fundamental topological questions do depend on one's set theoretic background. | |
| Jul 17, 2017 at 10:12 | comment | added | Joel David Hamkins | For statement 1, could you clarify what you mean by saying "ordinary point-set topology is..already independent of any particular axiom system for set theory", in light of the fact that so much of the (serious) theory of point-set topology makes use of extra-ZFC principles such as diamond, CH, MA or large cardinals. We now know that many questions, concerning whether every such-and-such kind of space is also that kind and so on, are independent of ZFC and require these extra axioms to settle them. So even the ZFC-style versions of point-set topology seem to depend on the set-theoretic system. | |
| Jul 17, 2017 at 0:04 | vote | accept | Brandon Brown | ||
| Jul 16, 2017 at 21:46 | history | answered | Mike Shulman | CC BY-SA 3.0 |