Does the holographic principle in physics, if true, rule out the existence of infinite sets in mathematics?

Question

The holographic principle sets an upper bound on the number of bits of information in the universe. See for example https://www.scientificamerican.com/article/information-in-the-holographic-univ/

One of the fundamental statements in set theory is that the elements of a set can be assigned distinct symbols. An assignment assumes that the elements' identities can be recorded somewhere, at least in principle. If there is an upper limit on information in the universe, then only sets below a certain finite size can have all of their elements assigned a symbol. This is not an engineering constraint, but a deep fundamental physics constraint on the nature of reality. This would seem to rule out the possible existence of infinite sets. Is there a way around this?

Answer 1

The emphasis below is mine.

One of the fundamental statements in set theory is that the elements of a set can be assigned distinct symbols. An assignment assumes that the elements' identities can be recorded somewhere, at least in principle.

No. That's just not a thing set theory assumes. Or rather, it could be if you interpret your statement sufficiently broadly, but this broad interpretation removes all connection with the physical universe (and is so broad as to be tautological in my opinion).

This issue is seen more clearly in your statement

This would seem to rule out the possible existence of infinite sets.

This relies on a very specific notion of "existence," which in general is not how mathematicians use the word.

The question of what mathematics can be "represented" in the physical universe is a very interesting one, and one might reasonably argue that infinite sets do not have "physical meaning", but since set theory and mathematics in general make no ontological commitment relating mathematical existence to the physical universe this is of limited relevance in general.