Timeline for answer to Can Category theory be founded in set theory using worldly cardinals instead of inaccessibles? by Alec Rhea
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| Jun 27, 2022 at 21:22 | comment | added | anon123 | @AlecRhea interesting. I guess the attitude I’ve absorbed is that everything can be treated as a teeny tiny model sometimes. Then we can sometimes learn something by looking at the possible configurations of such models. | |
| Jun 27, 2022 at 10:51 | comment | added | Alec Rhea | @MonroeEskew For the purposes of category theory I think your suggestion is perfectly sound, and probably the least-effort version for someone already familiar with $ZFC$ and the notion of an inaccessible cardinal. For me, a primary object of interest in any universe of sets is the surreal numbers, so any large cardinals we add are numbers in a number system I want to consider. Adding more cardinals and looking at the universe below them is like 'adding a new number and then ignoring all the numbers equal to larger than that number', which feels very unnatural to me. | |
| Jun 27, 2022 at 0:54 | comment | added | Reid Barton | I don't think the question is whether you can eliminate uses of large cardinal axioms, but whether one should want to do so in the first place. Personally I don't care much but it seems like a reasonable thing for others to want (probably due mainly to tradition). If there was some general metatheorem saying that universes can be eliminated from proofs about (say) Diophantine equations then it would be a moot point, but there can't be such a metatheorem. | |
| Jun 26, 2022 at 14:04 | comment | added | Timothy Chow | @ReidBarton Large cardinal axioms won't introduce new solutions to concrete Diophantine equations. It's just barely conceivable that they might let you prove the nonexistence of solutions to certain concrete Diophantine equations. But if you succeed in doing this and are unhappy to discover that the large cardinal axioms are ineliminable from your argument, then you might take consolation in the fact that you have just revolutionized mathematics, since nothing remotely like that has ever happened before, despite many people trying. | |
| Jun 26, 2022 at 13:35 | comment | added | Timothy Chow | Muller takes it as self-evident that superabundancy is bad: universes are "so ridiculously large in comparison to what we actually need to found category-theory, that is unbelievable it has even been considered seriously." This is a telltale sign that Muller is not an active practitioner, because he fails to recognize that universes are, above all, convenient in practice. They let you sweep set-theoretic technicalities under the rug with the bare minimum of attention. Now if you want to minimize assumptions, that's another story. The best foundation depends on what you want to do. | |
| Jun 26, 2022 at 12:43 | comment | added | Reid Barton | @MonroeEskew Many category theorists are already happy to do just that. But those who are in the business of (per Lenstra/Mazur) "representing functors in order to solve Diophantine equations", say, might reasonably resist adding axioms with consequences for solutions to Diophantine equations... presumably not the ones they were interested in, one hopes? | |
| Jun 26, 2022 at 11:30 | comment | added | Zuhair Al-Johar | @MonroeEskew, if you permit me to answer: part of what you suggest is already running, but this cannot be unleashed since it would be haunted by superabundacy (Muller). The work with proper classes is actually begnine here since it can be interpreted in the standard line of set theory, so there is no trouble relative to set theory. But part of what you say is already done you can have category of larger and larger classes etc.. | |
| Jun 26, 2022 at 8:47 | comment | added | anon123 | Naive question— Why not just use as many large cardinals as you need, building categories of sets of categories of sets, etc, and viewing things as always small from some higher perspective, rather than insisting on working with proper classes, which we know since Russell can get troublesome? | |
| Jun 26, 2022 at 7:02 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| Jun 25, 2022 at 21:49 | history | answered | Alec Rhea | CC BY-SA 4.0 |