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    $\begingroup$ I might be missing something, but the setting you are proposing seems to me a bit uncomfortable to do category theory (and higher category theory): with this type of definition of "small" the category of small set no longer have small colimits. If you want to have a category of small set whose class objects is a set, which has small limits and small colimits and which is indeed a full subcategory of the category of sets, then its set of object is an inaccessible cardinal. $\endgroup$ Commented Jan 27, 2021 at 0:17
  • 27
    $\begingroup$ Have you read Feferman's paper “Set-theoretical foundations of category theory"? He describes a modification of ZFC that does this sort of "reflection" automatically in the background. I wrote a bit about this idea myself in arxiv.org/abs/0810.1279 and golem.ph.utexas.edu/category/2009/11/feferman_set_theory.html. $\endgroup$ Commented Jan 27, 2021 at 1:08
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    $\begingroup$ @TimCampion: It's a theorem schema. If it were a theorem of ZFC, I think that would violate undefinability of truth. $\endgroup$ Commented Jan 27, 2021 at 3:51
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    $\begingroup$ We use the real numbers and infinite dimensional Hilbert spaces in physics and other sciences and engineering not because it's really needed (the observations and measurements we make in real life are always very very very finite), but because it simplifies a lot of things. Mathematicians tend to shun annoying logic where we need to carefully make sure that all our formulas are suitable for our purposes (this is one of the reasons Quine's NF never fully caught on). This here is a real nice example for large cardinals alleviating this issue as well. It saves time and effort on bookkeeping. $\endgroup$ Commented Jan 28, 2021 at 14:03
  • 9
    $\begingroup$ Yes, I understand. I was merely trying to give motivation as to why we should be more relaxed (for lack of a better term) about large cardinals in our assumptions. I tend to agree with the idea that people mistrust LCA because they haven't really worked with them, and they don't understand them properly. This can be easily remedied, all it requires is that (1) people around the world stop saying that set theory is dead weight and will soon be over; (2) people start hiring more set theorists; (3) people learn about set theory from actual expert and from the very beginning. Everyone's a winner. $\endgroup$ Commented Jan 28, 2021 at 15:18