Is there a way to confirm something material is infinite?

Question

Mathematics is full of immaterial examples of infinities. However, is it possible to confirm or prove something material is infinite? Or, can we only conjecture they are?

Answer 1

Paul Corazza is a set theorist known for his interesting, if nonstandard, approach to dealing with an inconsistency between ZFC and the theory of Reinhardt cardinals. Reinhardt cardinals are a form of infinity involving what is called a "nontrivial elementary self-embedding of the universe (of sets)." In ZFC strictly, such cardinals do not exist because they are inconsistent with the general axiom of choice (AC). However, the derivation of their ZFC-nonexistence requires an application of the axiom scheme of replacement to extend the critical sequence of the embedding to its degenerate supremum, so if we waive replacement, we don't get the initial omega-point of the critical sequence (which point is the specific point where AC is violated).

So Corazza is minded to preserve our sense of a nontrivial and universal self-embedding being possible. In a somewhat recent essay, he relates the mathematical dynamics of elementary embeddings to quantum field theory:

We point out that this world view—that diversity is a precipitation of the dynamics of some source—is the current intuitive model underlying quantum field theory: Particles are seen as secondary side effects, precipitations of underlying quantum fields. Seeking to represent this perspective axiomatically, we replace the usual Axiom of Infinity with an axiom asserting There is a Dedekind self-map—that is, an injective function j : A → A having a critical point a (a point a not in [the range of] j). We recall the work of Dedekind who showed how the natural numbers may be viewed as “precipitates” of the interaction of j with its critical point. We propose to use the dynamics by which the natural numbers emerge from j and a as an intuitive model for building a global Dedekind self-map j : V → V from which large cardinals may also be seen to emerge as “precipitates” of j. ... we review recent work that attempts to use the axiomatic system that we propose here as a mathematical foundation for the ontological interpretation of quantum mechanics, due to D. Bohm.

So sometimes in physics, the existence of specific kinds of objects is thought to be indicated when such objects would be solutions to specific equations. For example, black holes were anticipated because they were apparently possible solutions in the relevant theory of gravity. If there were equations that were of use to physicists, and possible solutions to those equations somehow ineliminably required indexes/terms/factors like ∞ or ℵ_a or whatever, with those factors representing substantive QFT phases or states (and not ghost fields), we could take this as some sort of indication that some form of infinity has a substantive manifestation in the physical world as such.

However, the theory and practice of renormalization, which is both standard and well-supported (indeed, standard because well-supported), testifies to some extent against the desirability of trying to accommodate substantial infinities in our physics. More generally, sometimes "infinitizing" a function causes the function's outputs to "break down," whereas there are also cases where doing such a thing trivializes the outputs. This is one of the arguments against multiverse theories, esp. ones in which there are infinitely many universes: that they can be made to account for all possible observations and hence they do not provide a specific explanation for any observation.

Circling back to Corazza's proposal: although it is tantalizingly poetic to think that esoteric concepts in higher category and set theory might be embedded or grounded in our physical world, there is still an easy danger of falling into triviality (if not inchoate emptiness), here. This is one of the problems with similar proposals by Max Tegmark about a "Level IV Multiverse" of all mathematical (read: metrodynamic/topological) structures, or Stephen Wolfram's ruliad as a foundation for physics. (Similarly, David Lewis' conjecture that the number of modal-realist worlds might be exactly ℶ₂ faces the problem of identifying just which transfinite cardinal, more specifically, ℶ₂ is supposed to be, a problem which from the vantage of ZFC admits of a proper class's worth of forcing solutions, or then which, from the adventurous vantage of new-wave set theory, admits of a solution where already ℶ₁ can be forced to be a proper class itself.)

Accordingly, it might be easy to confirm the nonexistence of specific infinite physical objects, or it might be trivial to confirm the existence of others; confirmation of a nontrivially infinite physical presence seems much harder to come by, if possible at all. Even the infinitesimally constituted continuity of time need not be taken for granted.