May 5 Information

(Sorry for awkward heading -- I couldn't get it to put the ב before the 2 because of some strange artifact of RTL rendering.)

I've seen in several places the claim that, as there are ${\displaystyle \aleph _{0}}$ natural numbers and ${\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}}$ (sometimes improperly given as ${\displaystyle \aleph _{1}}$) real numbers, there are some greater number of "curves" (sometimes given as f or, again improperly, ${\displaystyle \aleph _{2}}$). Most recently I was reminded of it at our article on George Gamow's (generally excellent) book One Two Three... Infinity.

The usual complaint about these popularizations, a very valid one, is that they uncritically give these cardinalities as aleph numbers in a way that works only if the generalized continuum hypothesis holds. But there's another, quite serious, problem: The claim that there are more "curves" than real numbers is correct only if you have an extremely liberal notion of what constitutes a "curve".

One reasonable notion is that a "curve" is the image of the real line or the unit interval under a continuous function from the reals to Rⁿ (or similar space), but there are only ${\displaystyle 2^{\aleph _{0}}}$ such functions, and therefore the same number of curves.

My best guess is that someone was taking "curve" to mean the graph of an arbitrary function. But these are not typically curves according to any obvious natural-language meaning; they're just scattered points in the plane.

So, question, what's my question? Does anyone know where this idea originated? Was it Gamow, some other popularizer, multiple sources? And what if anything should we do to clean up the text in our One Two Three... Infinity article? I'm thinking an explanatory footnote but ideally I'd want a source directly speaking to the misconception. -- Trovatore ( talk) 20:58, 5 May 2024 (UTC) reply

This Math Stack exchange entry is relevant, but it doesn't seem to cover what you're asking. One problem is that the statement is true by Wikipedia standards; you could cite the book. You would need a reliable source, such as a published article somewhere, to say it was wrong/vague/misleading in order to state that in our article. At the moment the article points out that you'd need the GCH to say what's in the book, but I guess that's supposed to be "common knowledge" (at least among mathies). — Preceding unsigned comment added by RDBury ( talk • contribs) 23:49, 5 May 2024 (UTC) reply

This article aims to classify various subsets of the function space 𝐹(ℝ,ℝ) from a constructive-mathematics perspective. The Introduction states: "mathematicians have made numerous attempts to focus on special subsets of this vast vector space (e.g., all real-valued continuous functions [5])", where the cited text is:

Pugh, C.C. Real Mathematical Analysis, 1st ed.; Undergraduate texts in mathematics; Springer Science Business Media: New York, NY, USA, 2002; pp. 223–225.

The latter is available as a pdf here. The article itself denotes this subset as 𝐶(ℝ,ℝ) and concludes in Proposition 4 that 𝐶𝑎𝑟𝑑(𝐶(ℝ,ℝ)) = 𝑐. But this is of course outside the paradise that Cantor created for you.  -- Lambiam 07:09, 6 May 2024 (UTC) reply

The reply given by a fellow Wikipedian to another Math Stack exchange question appears to imply that this also holds within the paradise.  -- Lambiam 07:23, 6 May 2024 (UTC) reply

Cardinality of the continuum § Sets with cardinality of the continuum also lists, without citation, "the set of all continuous functions from ${\displaystyle \mathbb {R} }$ to ${\displaystyle \mathbb {R} }$".  -- Lambiam 07:30, 6 May 2024 (UTC) reply

Discussed at Stack Exchange. Basically it's because a continuous function from ${\displaystyle \mathbb {R} }$ to ${\displaystyle \mathbb {R} }$ is uniquely determined by its values at rational points. AndrewWTaylor ( talk) 16:11, 6 May 2024 (UTC) reply

I feel a bit of sympathy for him making those mistakes but he should have had a mathematician read through that chapter. NadVolum ( talk) 16:40, 6 May 2024 (UTC) reply

Well, it's kind of the publisher's job to do fact checking. The statement was still in the 2012 Dover edition, so there have been multiple chances to fact check since the original 1947 publication. -- RDBury ( talk) 19:12, 6 May 2024 (UTC) reply

In general I'm skeptical of active attempts to use Wikipedia to correct readers' mathematical misconceptions — too much like righting great wrongs, and can easily become a POV magnet (like the old "What mathematics is not" section that once appeared in our mathematics article).

This one irks me, though, and tempts me to go back on that reasoning. I guess it's slightly personal, because I had internalized this bit about the cardinality of the set of curves, and (embarrassingly) didn't get it corrected till grad school. I had figured out for myself that there were only continuum-many analytic functions, because they're determined by the coefficients of the power series, but I conjectured that there were 2^𝔠 many C^∞ functions, and someone had to set me straight on that.

I think it's not just Gamow (whose book, I want to re-emphasize, is a big net positive). I've been trying to remember where else I might have seen it. I thumbed through Lilian Lieber's Infinity (which is a book that heavily influenced me) and didn't find it there. -- Trovatore ( talk) 20:00, 6 May 2024 (UTC) reply

Just a guess, but the article mentions "What is Mathematics?" by Richard Courant and R. Robbins as a source, and of all the sources it seems the most mathematical. It's a "Text to borrow" on Internet Archive so if you create an account you can view it for free. -- RDBury ( talk) 10:42, 7 May 2024 (UTC) reply

There is a parallel in the treatment of cardinal numbers between Gamow and Courant & Robbins up to the point where the latter write (p. 85): "A similar argument shows that the cardinal number of the points in a cube is no greater than the cardinal number of the segment." After that, they muse briefly on the fact that this is counterintuitive since the correspondence does not preserve dimension, but that this is possible because it is not continuous. That ends their treatment of cardinal numbers. Earlier they note (p. 84): "As a matter of fact, Cantor actually showed how to construct a whole sequence of infinite sets with greater and greater cardinal numbers." They even sketch the proof, but do not pursue the question of mathematical objects of higher cardinality than the continuum that are of interest by themselves.  -- Lambiam 13:21, 7 May 2024 (UTC) reply

Digression: In fact it's challenging to come up with an object of larger cardinality that might naturally be considered by non-set-theorists. One possibility is βN, the Stone–Čech compactification of the natural numbers. I believe this is mentioned in an exercise in Folland's Real Analysis. -- Trovatore ( talk) 18:53, 7 May 2024 (UTC) reply

There's a relevant MathOverflow question about finding cardinalities beyond that of the continuum outside set theory. βN is given as an answer, but maybe the most elementary one offered is the set of all field automorphisms of C. But the answers do kind of make me agree with Gamow's surely intended point that it's difficult to find natural objects of size beyond 2^c, though not with his actual assertion. :) Double sharp ( talk) 15:21, 8 May 2024 (UTC) reply