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Cantor's theorem states that the cardinality of a set's powerset is strictly greater than that of the set itself. This clearly applies to the reals also; if I'm not mistaken, the cardinality of the power set of the reals would be $\beth_{2}$.

Is there any literature on the powerset of the reals? I'm rather interested in reading about what properties it would have, since mathematics seems to rarely consider cardinalities beyond $\beth_{1}$. I have done some searching around Google, but have so far had no luck. It doesn't help that I'm unaware of any formal name which this set may or may not possess.

Thanks in advance!

Andrés E. Caicedo
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Dan M. Katz
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    What kind of questions are you interested in finding answers to? – Qiaochu Yuan Dec 25 '11 at 04:50
  • Anything really...Topological, algebraic, how this set is like and unlike the reals...I just haven't been able to really find any literature on it, I was hoping for some links. Plenty of stuff on large(r) cardinals themselves, not so much on sets of those cardinalities. – Dan M. Katz Dec 25 '11 at 04:52
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    @Dan: For most purposes, there is no difference between $\mathscr{P}(\mathbb{R})$ and any other set of the same cardinality. In general power sets do not have much structure beyond being complete atomic boolean algebras, and there aren't many structures on a set $X$ which can be transferred naturally to $\mathscr{P}(X)$. – Zhen Lin Dec 25 '11 at 05:02
  • Well, it's known that there exists a model of ZF in which every subset of the reals is measurable, so I would guess that the properties of $P(\mathbb{R})$ depend strongly on what additional axioms you want to use. – Qiaochu Yuan Dec 25 '11 at 05:03
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    Well...I mean the reals have the same cardinality as the powerset of the naturals, and given their properties, it's a bit unfair to say "do not have much structure"...I'm not so much looking for properties of the powerset of R itself per se, I'm moreso looking for any literature about sets whose cardinalities are strictly greater than that of R. Does that make sense? – Dan M. Katz Dec 25 '11 at 05:07
  • Maybe he means how could we express an element? When we write a real number we express it in decimal notation, a countably infinite string from our "alphabet" ${0,1,\dotsc,9}$. Maybe there is some analogue to this? This could sound ridiculous, but maybe its expression is somehow uncountably infinite? – mathmath8128 Dec 25 '11 at 05:17
  • That's an interesting thought. I'm not really looking for any particular information, just any material at all that's been published/written on the topic haha – Dan M. Katz Dec 25 '11 at 05:21
  • Large cardinals is a technical term for very large cardinals, which in comparison the continuum and its power set are almost finite. :-) – Asaf Karagila Dec 25 '11 at 06:38
  • Yeah, but it seems that it depends on who you're talking to :-) – Dan M. Katz Dec 25 '11 at 06:46
  • It doesn't seem to apply here, but you might be interested in http://en.wikipedia.org/wiki/Easton's_theorem. – sdcvvc Dec 25 '11 at 07:06
  • @sdcvvc: But what if the continuum is singular? You cannot use Easton's theorem on singular cardinals. :-) – Asaf Karagila Dec 25 '11 at 07:11
  • The collection of intersections of arbitrary closed unit intervals $[a,a+1]$ where $a\in \mathbb{R}$. Is that a power set for $\mathbb{R}$? – user398843 Feb 24 '18 at 15:50
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    @nph: Avoid, at any reasonable cost, putting the [set-theory] and [elementary-set-theory] tags together. When in doubt, use the elementary one. – Asaf Karagila Apr 23 '20 at 18:37
  • @user398843 The intersection of (finitely many) closed intervals is a closed interval (possibly empty: $[a,b] = \emptyset$ if $a > b$), so the collection you mention contains only intervals and has size continuum. 'Arbitrary' sets of reals can be arbitrarily complex, and there are many more of them. – BrianO Oct 29 '21 at 03:31

2 Answers2

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Non-set theoretical mathematics indeed stay within the limits of the continuum (and rarely its power set, e.g. Lebesgue measurable sets is a collection of size $\beth_2$).

In set theoretical aspects the power set of the continuum is the number of ultrafilters on $\mathbb N$ (assuming the axiom of choice and whatnot); there are a few theorems discussing $\beth_2$ under additional axioms and CH; and when you decide to toss away the axiom of choice and take the axiom of determinacy instead you find yourself in a strange model in which every countable partially ordered set can be embedded into cardinalities below the continuum.

It is also not true at all that there is no talk about cardinalities above $\beth_2$. It appears in PCF theory (and the PCF theorem itself), large cardinals are usually strong limit cardinals, so they dwarf the continuum much like the universe dwarfs an electron, and of course the choice-less contexts often throw a fit and head out to a whole other direction with cardinals (i.e. cardinals which cannot even be compared with the continuum).

In the rest of mathematics, the concrete object $\mathcal P(\mathbb R)$ forms a Boolean-algebra, it does not have a natural linear ordering, and it is mainly used as an example of an underlying set for structures of that size. However such objects are not very common (yet, anyway) since they tend to be a little bit too big to handle once you add structure which was not natural (in the sense that the real numbers have).

The more structure you want, the harder it is to handle larger and larger objects. Once you go beyond $\beth_1$ you cannot have both Hausdorff and separable topologies. One of my teachers once explained this very question to me with the answer that we can grasp finite things, and we can approximate countable (and thus separable) things. However beyond that it becomes very hard to work with things. There are objects which are very large, in modern fields such as C*-algebras you get to meet them from time to time, and slowly in other fields. However it is still convenient to work with separable/countably generated/finitely generated objects for most people. If you wait a century or two then I'm certain that larger constructions will seep through the cracks and become mundane.

Asaf Karagila
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  • Very interesting, especially the part about Hausdorff+separable. You noted that some large objects (presumably larger than $\beth_{1}$?) appear in Operator Theory. I'm not very familiar with the field, but can you give any examples of such objects? – Dan M. Katz Dec 25 '11 at 07:39
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    @Dan: I am not coming from operator theory, alas I attended a seminar on Friday where the topic was C*-algebras and set theory. Apparently there is some use of Stone-Cech compactification spaces, which are of size $\beth_2$ or larger. – Asaf Karagila Dec 25 '11 at 07:46
  • Ah, gotcha. Well, thanks a lot! Some of the things you mentioned above are pretty interesting. Please feel free to post anything here (or message me or whatever) if you happen across anything else related to the topic! :-) – Dan M. Katz Dec 25 '11 at 07:50
  • @Dan: I'll be sure to do that. – Asaf Karagila Dec 25 '11 at 07:52
  • @Asaf-Karagila, wikipedia says $\mathbb{R}^\mathbb{R}$ has this cardinality. Then for $f, g: \mathbb{R} \rightarrow \mathbb{R}$ we have $f + g$ and $fg$. Ring. Can we represent each function with decimals 0,1,2,...,9? $h = \Sigma_{\alpha \in \mathbb{R}} f_\alpha 10^{-g_\alpha}$ Or, is there another basis? – Vinícius Ferraz Jul 07 '15 at 14:28
  • @Vinícius: I'm not sure that I understand your question, or its relevance to a nearly-four years old answer. You should probably ask a new question. – Asaf Karagila Jul 07 '15 at 14:30
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http://www.earlham.edu/~peters/writing/infapp.htm

The power set of the rational numbers can be represented by a table like the one below, where each row shows a set, and a 1 in a column means that the corresponding number is in that set. Then each set is a binary string, which could represent a real number.
$$\begin{array}{ccccc} 1 & 2 & 3 & 4 & 5\\\hline 0 & 1 & 1 & 0 & 0\\ 0 & 0 & 1 & 1 & 1\\ 1 & 1 & 1 & 0 & 1 \end{array}$$ A similar table can be done for the reals, with each column heading a real number. Therefore each real number can be written with countably many decimal places, and the power set of the reals has the same cardinality as the set of all numbers which can be written with as many decimal places as there are real numbers.

Asaf Karagila
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Angela Pretorius
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  • I tried to add an array command, I hope it didn't screw with your content too much. – Asaf Karagila Dec 25 '11 at 07:24
  • @Angela-Richardson, Asaf-Karagila, how is it possible to define sum and product over "uncountable decimal places"? e.g. 9 + 1 in a position carries 1 to the next position. Define "next". Alternatively, in the real line, complete ordered field, we have proprierties of sum and product of "supremum", "infimum", "limits" or "series". is this possible in the power set of $\mathbb{R}$? – Vinícius Ferraz Jul 03 '15 at 17:35