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As I have been studying real analysis, I am wondering if there is a meaningful use of cardinal number in analysis. For example, given a set $S$, I treat the cardinality of $S$ being $\infty$ if it is not finite; it doesn’t account for whether $|S| = \aleph_k$. My background in analysis is the first course in functional analysis, so there can be a result that I am not aware of. Is there any example of analysis that involves cardinal number $\aleph_1$ and beyond?

One example that I can think of is a complete measure space $(2^{\mathbb R}, \mathcal A, \mu)$, but I don’t think this example gives me much information.

James C
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    Surely you are aware that the distinction between $\aleph_0$ and $2^{\aleph_0}$ is hugely important in analysis? – Alex Kruckman Apr 29 '21 at 00:08
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    You seem to be implicitly assuming the continuum hypothesis when you speak of $\aleph_2$ specifically as something you're unlikely to meet. For all you know you might be in a universe where $|\mathbb R|=\aleph_2$ ... – Troposphere Apr 29 '21 at 00:10
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    One perhaps-too-simple example is the usual counting argument that there are Lebesgue measurable sets that are not Borel-measurable, since there are $2^{2^{\aleph_0}}$ (provably $\ge\aleph_2$) of the former and only $2^{\aleph_0}$ (consistently $\ge$ $\aleph_2$) of the latter. – spaceisdarkgreen Apr 29 '21 at 00:11
  • @AlexKruckman between $\aleph_0$ and $\aleph_1$, yes. However, I am unsure if there is a distinction beyond this. – James C Apr 29 '21 at 00:12
  • @Troposphere Yes, I am assuming continuum hypothesis. – James C Apr 29 '21 at 00:13
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    @Troposphere : And $\aleph_1$ is also something one would not often meet. It is defined as the cardinality of the set of all countable ordinals. On the other hand, $2^{\aleph_0}$ certainly comes up. – Michael Hardy Apr 29 '21 at 00:13
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    The distinction between $\aleph_0$ and $2^{\aleph_0}$ is essential in several basic facts in analysis, including the Baire category theorem and the countable additivity of Lebesgue measure. – Andreas Blass Apr 29 '21 at 02:46
  • In the 1930's Kurt Godel showed that if the axiom system now called ZFC is consistent then it cannot disprove CH, since his "constructible class" $ L$ satisfies ZFC+CH. in the 1960's Paul Cohen developed "Forcing" to show that if Con(ZFC) then ZFC cannot prove CH. – DanielWainfleet Apr 29 '21 at 14:54
  • For results that extend beyond the cardinalities of the natural numbers and the real numbers (or slightly beyond, as in these existence proofs using $c < 2^c$), see Krzysztof Ciesielski's 1997 survey article Set theoretic real analysis. – Dave L. Renfro May 01 '21 at 08:56

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The cardinality of the set of integers, and hence of the set of indices of an infinite sequence, is $\aleph_0,$ and the cardinality of the real line is $2^{\aleph_0},$ so those two numbers are considered in analysis.

Probably few analysts ever encounter $\aleph_1$ in doing analysis. Ever since the time of Georg Cantor in the 19th century, $\aleph_1$ has been defined as the cardinality of the set of all countable ordinals. And $\aleph_2$ is the cardinality of the set of all ordinals whose cardinality does not exceed $\aleph_1,$ and so on. Alephs other than $\aleph_0$ are not often seen in analysis, but "beths", or "beth numbers", $2^{\aleph_0},$ $2^{2^{\aleph_0}},$ etc. can come up. I seem to recall that that last-mentioned number is the cardinality of the Stone–Cech compactification of $\mathbb N.$

Michael Hardy
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    I'm no analyst, but I seem to recall transfinite induction up to $\omega_1$ being used from time to time (sometimes, but not always in conjunction with an assumption of CH). (e.g. when proving there are $2^{\aleph_0}$ Borel sets.) – spaceisdarkgreen Apr 29 '21 at 00:14
  • I, so far, have mistakenly thought that $\aleph_1 = 2^{\aleph_0} = |2^{\mathbb Q}|$. However, it says more involved than that. Thanks for the answer. – James C Apr 29 '21 at 00:18
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    $\omega_1$ is the length of the Borel hierarchy. – Andreas Blass Apr 29 '21 at 02:44
  • How much of the analysis currently used in science or engineering depends on the existence of cardinal numbers greater than $2^{\aleph_0}$? – Dan Christensen Apr 29 '21 at 13:39
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    For an infinite discrete space $D$, the Cech-Stone compactification $\beta D$ can be identified with the Wallman extension $wD$, whose points are the ultrafilters on $D,$ which, by a theorem of set theory, has cardinal $|wD|=2^{2^{|D|}}.$ See General Topology by Engelking, or Terence Tao's nice exposition on $\beta \Bbb N.$ – DanielWainfleet Apr 29 '21 at 15:29