For those championing the concept of the actual infinite in mathematics, and physics, set theory comes to their rescue. In this paper, I will refute a major theorem, or alleged theorem purporting to show that there are different levels of infinity regarding sets, Cantor’s power set theorem. Wilfred Hodges, “An Editor Recalls Some Hopeless papers, “Bulletin of Symbolic Logic, vol. 4, no 1, 1998, pp. 1-16, says while gloating about cranks that attempted to refute Cantor, “None of the authors showed any knowledge of Cantor’s theorem about the cardinalities of power sets.” (p.2) So, let’s examine this. Cantor’s power, or the axiom of powers, states that for any set S, there exists a collection of sets Power (S), which contain in its elements, all of the subsets of the given set S. thus, for example, if S = { a, b }, then Power (S) = { ϕ, {a}, {b}, {a, b}}, where “ϕ” is the empty set. As the sets get bigger, so does the power set. In fact:
“If S is a finite set with |S| = n elements, then the number of subsets of S is |P(S)| = 2n. This fact, which is the motivation for the notation 2S, may be demonstrated simply as follows, First, order the elements of S in any manner. We write any subset of S in the format {γ1, γ2, ..., γn } where γi , 1 ≤ i ≤ n, can take the value of 0 or 1. If γi = 1, the i-th element of S is in the subset; otherwise, the i-th element is not in the subset. Clearly the number of distinct subsets that can be constructed this way is 2n as γi ∈ {0, 1}.” “Cantor’s diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (informally the power set must be larger than the original set). In particular, Cantor’s theorem shows that the power set of a countably infinite set is uncountably infinite. The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers.”
https://en.wikipedia.org/wiki/Power_set
That is a fundamental argument for the different infinities of sets. The power set axiom is found in many pedigrees of set theory, especially Zermelo-Fraenkel set theory:
https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory but not in all, such as Kripke-Platek set theory:
https://en.wikipedia.org/wiki/Kripke–Platek_set_theory
https://arxiv.org/abs/1801.01897
Cantor himself realised that a paradox arose from considering the set S, the set of all sets, which would contain the power set, but the power set would also have a higher cardinality than it, or precisely there is no greatest cardinal number, hence the set of infinite sizes is itself greater in cardinality than its power set. The difficulty has been handled by redefining the concept of set, so that collections of this size became proper classes. Nevertheless, while in the 20th century logicians dealt with paradoxes arising from considering sets, there are also more challenging paradoxes involving the elements of sets, or not sets of sets, but sets of infinite elements that are not sets, or even mathematical objects at all. For example, Patrick Grim, “There is No Set of all Truths,” “Analysis,” vol. 44, 1984, pp. 206-208 (also P. Grim, “The Incomplete Universe: Totality, Knowledge, and Truth, (MIT press, Cambridge, MA, 1991)), showed that the set of truths was in conflict with Cantor’s power set theorem. And, we can add the set of facts. The set of facts has the power set Power (F), but for each set of facts F in that power set, there will be another fact, such that F contains those facts which it does. Hence, by Cantor’s power set theorem, there will be more facts in F than in its power set, a contradiction.
Everyone, to date has concluded that these perfectly intuitively meaningful sets do not exist. But, if this is so, then set theory itself is flawed because a general mathematical theory should not generate paradoxes when applied to objects of the world or thought like truths and facts: it defeats the very purpose of a scientific theory. The same paradox comes to exist with other items that could serve as elements of a set, such as the set of all mathematical objects, the set of abstract entities, the set of all items that can be (consistently) contemplated, and so on. Grim was concerned with using this style of argument against the existence of an omniscient being, who knew all facts/truths. We can leave that to one side. More recently he has given some indication that these paradoxes indicate that not all is well with received set theory itself, as detailed in N. Rescher and P. Grim, “Beyond Sets: A Venture in Collection-Theoretic Revisionism,” (Ontos, Verlag, 2011). Rescher and Grim state:
“Set theory was born in paradox, was shaped by paradox, and continues to carry the threat of paradox into its current adolescence. Properly understood … the threat of contradiction is not merely formal and is not to be evaded by merely formal techniques. The fact that there can be no set of all non-self-membered sets might be shrugged aside as a minor logical surprise. Beyond Russell’s paradoxical set, however, there lies the serious philosophical difficulties of coherently conceptualising a set of all things, the realm of unrestricted quantification (or even the sense of restricted quantification), the totality of all events, all facts, all propositions, or al that is true. Sets are structurally incapable of handling any of these.” (p.6)
That is an excellent argument for the rejection of set theory, even in mathematics, since there are clear counter-examples to fundamental axioms, such as the power set axiom. And, at least, the rejection of the power set axiom, via these counterexamples, would severely cripple the Cantorian project of the infinite, that of countable and uncountable sets.