Fifth

Claim: there are other types of infinity that are larger than universe infinity. There are an infinite number of sizes of infinity, fitting into a hierarchy where each is infinitely larger than the previous. Or, \(| P(\mathscr{S})| > |\mathscr{S}|\).

What we will show is this: given any set \(\mathscr{S}\), we can construct a set, denoted \(P(\mathscr{S})\), such that \(P(\mathscr{S})\) is larger in size than \(\mathscr{S}\).

To do this, we need to explain how to construct \(P(\mathscr{S})\) from \(\mathscr{S}\), and we need to explain how we know that the size of \(P(\mathscr{S})\) is larger than the size of \(\mathscr{S}\).

But once we have done that, then taking the example where \(\mathscr{S}\) is universe infinity, the set \(P(\mathscr{S})\) will be an infinite set that is larger than universe infinity.

Furthermore, we can iterate the process. For any infinite set \(\mathscr{S}\), we construct \(P(\mathscr{S})\), but then we can also construct \(P(P(\mathscr{S}))\), which will be an infinite set larger than \(P(\mathscr{S})\). And \(P(P(P(\mathscr{S})))\) is an infinite set that is larger than \(P(P(\mathscr{S}))\), which is larger than \(P(\mathscr{S})\), which is larger than \(\mathscr{S}\). And on and on. Thus we will have a never-ending hierarchy of ever-larger infinities.

Given a set \(\mathscr{S}\), the definition of \(P(\mathscr{S})\) is easy to state: \(P(\mathscr{S})\) is the set of subsets of \(\mathscr{S}\). (This is a bit hard to think about, but we’ll say more about it in the short and in-depth explanations.)

Then the crucial (and challenging) mathematical result is that, for any set \(\mathscr{S}\), we always have \(| P(\mathscr{S})| > |\mathscr{S}|\).

You can read a short explanation of why \(|P(\mathscr{S})| > |\mathscr{S}|\), or an in-depth explanation (a rigorous proof), or you can take our word for it, and go read the story or the meta-discussion.

Short Version Meta Discussion In-Depth Version True Story

Fifth

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