A Response to My Question about Gamow

Recently, the physicist Sabine Hossenfelder, who maintains a wonderful blog, posted her thoughts about "infinity," its necessity as a mathematicians' tool, on the one hand, but its unreality from the point of view of the science of physics, on the other. Her point is that a physicist only regards something as real if it is necessary in order to explain an observation. Infinity, she says, will never fill that bill.

She also went a bit into the mathematics of infinity, and the fact that not all infinities are the same. The real numbers are a different sort of infinity from the integers.

I used the comment section under her blog post to ask a question that has long been on my mind.

I remember when I was a kid, many decades ago, some of us nerdy types excitedly read "One, Two, Three, Infinity" by George Gamow. Gamow covered some of the same territory you just did although I'm guessing progress has been made in the last half century or so. If I recall correctly, Gamow said that only three "transfinite" numbers had as yet been identified. The lowest order of transfinite is the number of all integers. The second is the number of all real numbers. The third is the number of curves one can draw through space. Again, I'm relying on memory, but I believe he said nobody had figured out a fourth transfinite. Has that changed? Are there now more transfinite numbers than their used to be?

Hossenfelder had said in the post above this comment that there are an infinite number of types of infinity, although only two especially important ones, Gamow's first two. She hadn't said anything about curves in space, although it makes sense to me that it could be a third order, beyond the real numbers.

Anyway, my question did get a reaction, though not from Hossenfelder. The (first) reaction was from Lawrence Crowell. Not chopped liver. Crowell is affiliated with the Institute for Advanced Study (Albert Einstein's old stomping-ground in Princeton). He is this Lawrence Crowell: (PDF) BMS Supertranslations as Coset States & SLOCC (researchgate.net)

Anyway, Crowell wrote the following. I paste it exactly although I suspect there are typos. I would probably change his meaning if I fooled around with it at all.

"There are the aleph's of George Cantor, where aleph's is countable infinity. Aleph1 is the first uncountable transfinite number. It is also the continuum, or Bernays and Cohen showed it to be unprovably consistent with set theory. It is a form of Godel's theorem. It also turns out a y alephn also works.

"For physics and computation only aleph0 and aleph1 are at all possibly relevant. There many alephs. The beyond those are least inaccessible cardinals, due to Uman. Beyond this, which is beyond classical set theory, are nonstandard set theories. I have little idea of those things."

This seems to have been quickly dashed-off. But I take it that my suspicious was correct, the number of infinities didn't stop at 3, as Gamow suggested (even in the title of the book). There are aleph0, aleph1, and "many" beyond -- I gather the curves-through-space thing turns out not to be special.

I also have resolved, as a new year approaches, that (1) I am not going to look up what BMS supertranslations are, but (2) I am going to find out what Bernays and Cohen [and Uman?] might have said about the manifold infinities.

I pursued the matter a bit further on the comments board, asking Crowell, "to pursue the point, there is nothing special about the infinity of number of curves in space? I believe there was supposed to be a proof, analogous to the 'diagonal proof' but different, that this infinity is of a larger sort than the infinity of real numbers."

Crowell hasn't replied. Someone named Steven Evans, of whom I know nothing else, did reply and says that there has been a recent discussion of this in Stack Exchange.

https://math.stackexchange.com/questions/2550313/is-aleph-3-the-cardinality-of-all-the-surfaces-that-exists

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