Timeline for Topological space (or math structure more generally) without encoding as set
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Jul 18, 2017 at 14:09 | comment | added | Mike Shulman | @AsafKaragila Coding-agnosticism (i.e. isomorphism-invariance) has nothing to do with consistency strength, so bi-interpretability doesn't tell you anything about it. When ETCS is formulated properly in a dependently typed language, then there is a theorem that everything you can say in it is invariant under isomorphism, and in particular invariant under replacing any cartesian product by any other cartesian product. | |
| Jul 17, 2017 at 17:04 | comment | added | Asaf Karagila♦ | @Mike: Maybe I don't know enough about ETCS, then. But it seems to me that if it is bi-interpretable with Z, and Z is missing Replacement, then perhaps it is not as-coding-agnostic-as-you'd-think, but rather the language used to talk about sets is more limited than that of material set theory. But again, I am completely ignorant about structural set theory issues, so I will take your word on that issue. | |
| Jul 17, 2017 at 0:04 | vote | accept | Brandon Brown | ||
| Jul 16, 2017 at 22:01 | comment | added | Gerhard Paseman | When I talk about category theory, I usually remark about my relative discomfort with it, not to disparage the subject but to indicate my preference for a certain environment in which to think about mathematics. However, some have seen these remarks as disparaging or insulting. I suspect the poster is in a somewhat similar situation, where (as a poor analogy with no insult intended anywhere ) he is looking for a more comfortable/appropriate development environment than COBOL for his efforts. Gerhard "Imagine Mathematics Developing From RATFOR" Paseman, 2017.07.16 | |
| Jul 16, 2017 at 21:46 | answer | added | Mike Shulman | timeline score: 13 | |
| Jul 16, 2017 at 21:41 | comment | added | Mike Shulman | @AsafKaragila While I agree it's not fair to disparage ZFC, it's disingenuous to suggest that the absence of replacement in ETCS somehow makes it less coding agnostic than ZFC. In fact coding agnosticism is built into ETCS at a much more basic level, so there is no need for a complicated argument using a powerful axiom such as replacement. | |
| Jul 16, 2017 at 20:19 | comment | added | Asaf Karagila♦ | Then what does set theory even has to do with it? Just study topological structures. People think that ZFC is this sort of rigid and concrete thing that is completely inflexible on how you code things. It's not. In fact, it's even more coding agnostic than category theory in some extent, since the Replacement schema is exactly equivalent to saying that any coding that satisfies the ordered pair property can be used to code ordered pairs (and Replacement is absent from ETCS, for example). So yeah, pretty much go Nike and "Just Do It". Why even bother about your underlying set theory? | |
| Jul 16, 2017 at 20:10 | comment | added | Brandon Brown | @AsafKaragila: I recognize your concern about seemingly disparaging set theory. That is not my point at all. ZFC set theory is but one of many possible universes for mathematics, and the sets of that theory to be one of many possible objects to study. As someone with a more computer science inclination, I'm more interested in encoding-agnostic structures. For example, if I want to study the topology of images, I don't want to have to think in terms of sets, I just want to think in terms of abstract relations between images, irrespective of the encoding. | |
| Jul 16, 2017 at 19:35 | comment | added | Benjamin Steinberg | Perhaps you want to look at locales? | |
| Jul 16, 2017 at 19:27 | review | Close votes | |||
| Jul 17, 2017 at 4:00 | |||||
| Jul 16, 2017 at 19:15 | answer | added | Ivan Di Liberti | timeline score: 2 | |
| Jul 16, 2017 at 18:57 | comment | added | Philippe Gaucher | What's wrong with "the sets of ZFC" (what a weird expression !), and with topological spaces ? Topological spaces are precisely designed to study all these notions of connectedness, continuity, etc... It is not the starting point, but the conclusion of a very long historical process to formalize real analysis. What you need is a good book about the history of mathematics. The notion of topological space has various generalizations, Grothendieck topologies, locales more adapted to other parts of mathematics. | |
| Jul 16, 2017 at 17:58 | comment | added | Asaf Karagila♦ | I somehow feel that this question is the embodiment of the whole "scared of ZFC" philosophy that we are seeing more and more by people who were not exposed to set theory or classical foundations properly in their studies. This is very much how I read the remark "especially just the sets of ZFC". It's as though sets form some psychological barrier in some people. And that is truly a shame. I'm not saying that everyone should be a set theorist or even know set theory. But it seems like a particularly ignorant comment to make, in a setting that feels more and more "normal" in today's mathematics. | |
| Jul 16, 2017 at 17:38 | comment | added | Steven Landsburg | You might want to look at Peter Johnstone's paper on "The point of pointless topology". | |
| Jul 16, 2017 at 17:10 | comment | added | Tyler Foster | Topoi might be what you're looking for. Here's the wiki page: en.wikipedia.org/wiki/Topos | |
| Jul 16, 2017 at 17:09 | history | edited | Andrés E. Caicedo |
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| Jul 16, 2017 at 17:06 | history | asked | Brandon Brown | CC BY-SA 3.0 |