# 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 $$| P(\mathscr{S})| > |\mathscr{S}|$$.

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