[Vincent Lepore]

I believe the following applies.

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Theorem 4. The cardinality of the power set of an arbitrary set has a greater cardinality than the original arbitrary set, or |*A| > |A|.
This is called simply Cantor's Theorem. It generalizes the previous theorem, in which we proved that the power set of a particular set, N, had a greater cardinality than the original. The present theorem is trivial for finite sets, but is fundamental for infinite sets.
Proof. Let A be an arbitrary set of any cardinality, finite or infinite. Again we supply a negative proof, and assume that the members of A can be put into one-to-one correspondence with the subsets of A. Take any one of the supposed ways of pairing off the members of A with the subsets of A. Let us say that if a member of A is paired with a subset of A of which it happens to be a member, then it is happy; otherwise it is sad. Let S be the set of sad members of A. Clearly S is one of the subsets of A. Therefore S is paired off with one of the members of A, say, x. Is x happy or sad? If x is happy, then x is a member of the set to which it is paired, which is S, but that would make it sad. If x is sad, then x is not a member of the set to which it is paired, which is S, but that would make it happy. So if x is happy, then it is sad, and if it is sad, then it is happy. Our assumption implies a contradiction and is therefore false. So the members of A cannot be put into one-to-one correspondence with the subsets of A.
But if A and *A cannot be put into one-to-one correspondence, then they cannot have the same cardinality. If so, then the larger one must be *A, for A can be put into one-to-one correspondence with a proper subset *A. For example, if the members of A are A1, A2, A3..., then they can be put into one-to-one correspondence with this subset of *A: {{A1}, {A2}, {A3}...}.


Many profound consequences follow directly from Cantor's theorem. But we make the most important of them explicit in the next theorem.

Theorem 5. If S is a set of any infinite cardinality, then its power set has a greater infinite cardinality, or |*S| > |S|.
This follows directly from Cantor's Theorem (Theorem 4). Cantor's theorem applies equally to finite and infinite sets; this corollary focuses on the important consequence for infinite sets.
If we follow the notation for finite sets, and say that a set of cardinality a has a power set of cardinality 2a, then this theorem asserts that 2a > a, for each transfinite cardinal a.
This theorem asserts that for any infinite cardinality, there is a larger infinite cardinality, namely, the cardinality of its power set. Hence, there is an infinite series of infinite cardinal numbers. We will meet some of the infinite cardinals larger than N shortly.
This theorem also implies that, for every set, there is a greater set. It follows that there is no set of all sets, or no set of everything.

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I believe that an infinite set of wins is a subset of the infinite set of events of wins and losses.

Vince