Wikipedia:Reference desk/Archives/Mathematics/2022 April 5

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April 5

Power set of the real numbers

I am curious. Have mathematicians already developed what the power set of the real numbers is?
Is this a thing for the mathematicians at all?--2A02:908:426:D280:C4BC:2A5E:F3E8:85DF (talk) 11:39, 5 April 2022 (UTC)

It is what it is, ie. by definition it is the power set of the reals. Its cardinality is denoted as aleph 2 $(\aleph _{2}\,)$. See the article on aleph number for details--- Abdul Muhsy talk 13:05, 5 April 2022 (UTC)

Doesn't this assume the generalized continuum hypothesis?  --Lambiam 16:07, 5 April 2022 (UTC)

Well, you don't need the full GCH. Nevertheless, I'm afraid Abdul Muhsy is quite wrong. The cardinality is not denoted $\aleph _{2}$. It is denoted $2^{2^{\aleph _{0}}}$, or sometimes $\beth _{2}$. That may or may not equal $\aleph _{2}$, but even if you think it does, it's still useful to distinguish the notation. (See Hesperus is Phosphorus.) --Trovatore (talk) 17:38, 5 April 2022 (UTC)

The powerset of the reals is a very interesting structure. Abdul Muhsy got one thing right — it is what it is by definition; there's not that much more to say about what it literally is not that I can't go on about it if you let me :-).

However there's a lot more to say about its internal structure. In some sense that's the entire subject matter of descriptive set theory, so I can't summarize it in a single refdesk response.

But the powerset of the reals is the set of all sets of reals, and there's an ordering on different sets of reals, called the Wadge hierarchy. You can think of that hierarchy as stratifying sets of reals, at least for a while, into something very close to a wellordering.

Then if the axiom of choice holds, which it does in the "real world", at some point that breaks down, when you get to sets of reals that code games that are not determined.

The existence of large cardinals implies that more and more complicated pointclasses contain only determined games, and that there are lots of interesting inner models of the axiom of determinacy. In these models, the Wadge hierarchy goes "all the way up", and the powerset of the reals in the sense of those models is stratified into Wadge degrees.

Does that help at all? I wouldn't expect it to be understandable, on its own, to someone who hasn't studied quite a bit of set theory, but maybe it gives you a bit of "flavor", and the links can point to more opportunities to learn. --Trovatore (talk) 23:18, 5 April 2022 (UTC)