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I'm pretty sure the answer is "no", but I've seen this claim a couple of times and want to make sure there isn't anything to it.

According to this Quanta article https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/

For 50 years, mathematicians have believed that the total number of real numbers is unknowable [...] Gödel, for his part, believed that the continuum hypothesis is false — that there are more reals than Cantor guessed.

Granted, Quanta isn't the most reputable source. But a similar claim was made by mathematical physicist John Baez in https://arxiv.org/abs/1609.01421

[H]ow many real numbers are there? The continuum hypothesis proposes a conservative answer, but since this is independent of the usual axioms of set theory, the question remains open: there could be vastly more real numbers than most people think

My understanding is we know exactly how many real numbers there are, $𝔠 = 2^{\aleph_0}$. And within ZFC $\aleph_1 \le 𝔠$, but since the Continuum Hypothesis is independent of ZFC, we can't say whether $\aleph_1 = 𝔠$ or not.

So it seems to me that both sources are making the same mistake where they mix up which quantity is unknown. It's not that $𝔠$ could be bigger than we think, it's that $\aleph_1$ could be smaller than the CH asserts.

It would be like saying we found an exoplanet and we can't tell how its size compares to earth, therefore we don't know how big the earth is.

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    $\begingroup$ I personally disagree with you on this. This is of course subjective, but to me "knowing a cardinal" amounts to knowing the ordertype of the cardinals smaller than it - basically "how big x is" = "how many things <x there are" - and $\aleph_1$ has that built-in but $2^{\aleph_0}$ doesn't. Other examples of ambiguity include the fact that $\aleph_1$ is provably regular, but $2^{\aleph_0}$ might be singular. $\endgroup$ Feb 6, 2023 at 1:36
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    $\begingroup$ Said another way, the claim is exactly that we don't know how large $2^{\aleph_0}$ is. This is the number of subsets of $\mathbb{N}$, but how many of those are there really? Who knows! $\endgroup$ Feb 6, 2023 at 1:42
  • $\begingroup$ Is there a well written but deep explanation of the "forcing" stuff in that article? $\endgroup$
    – Alex K
    Feb 6, 2023 at 2:27
  • $\begingroup$ ^^For details on Forcing, you can refer to Kunen's "Set Theory An Introduction To Independence Proofs" $\endgroup$ Feb 6, 2023 at 2:38
  • $\begingroup$ Saying that $\aleph_1$ is smaller than expected, rather than $\mathfrak{c}$ is bigger, is not wrong per se, I would say. But the $\aleph$-numbers give a nice and structured classification of cardinals that we have a much clearer picture of than of say $2^{\aleph_0}$ and $\aleph_\omega^{\aleph_0}$. So its more a question of perspective and which one makes more sense in practice. It's kind of like the, does the earth revolve around the sun or the orther way round. One clearly is a better frame of reference. $\endgroup$ Feb 6, 2023 at 17:19

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Yes, we know that the cardinality of the Reals is $2^{\aleph_0}$

The problem is that we don't know what that value is. It could be $\aleph_{17}$ or $\aleph_{250}$

Which ordinal is $2^{\aleph_0}$?

If you can't compute the exponentiation, it's hardly reasonable to say we know the value. You just know that it's the result of some exponentiation but what are the laws of cardinal exponentiation? Is $2^λ$ The same as $λ^+$ ? ( where $λ^+$ is the smallest cardinal larger than λ)

If it is, then the generalized continuum hypothesis is true.

So, we have some operation, which produces a value- but we don't know, and cannot know which value.

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    $\begingroup$ "Forbidden knowledge"? Perhaps "unknowable in ZFC", but "forbidden"? $\endgroup$ Feb 6, 2023 at 3:40
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    $\begingroup$ Largely Poetic language. The only way we can get an answer about it in ZFC, is if ZFC turns out to be inconsistent. So, it would be rather disruptive. The same way the forbidden fruit could actually still be eaten, albeit with dire consequence. Of course, if you aren't working with axioms that are independent of CH, then it's free game. $\endgroup$ Feb 6, 2023 at 3:50
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    $\begingroup$ Michael, poetic language, while nice and colourful, can also be somehow misleading to people who have little understanding of logic and set theory. $\endgroup$
    – Asaf Karagila
    Feb 6, 2023 at 9:42
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    $\begingroup$ To make this more on point, it's not even "we don't know" which $\aleph_i$ is the answer: it's that we can construct models of ZFC for whichever one we like best (per Solovay, extending $i=2$ from Cohen). If we have any model of ZFC at all then we can construct a new one where $2^{\aleph_0} = \aleph_{17}$, and nothing in ZFC can stop us. Further, in ZFC terms, knowing "how many of something there are" means that you do or don't have certain bijections available. If ZFC admits many possible equiconsistent choices, in different models, then ZFC alone can't tell us a "true" size. $\endgroup$
    – alexg
    Feb 6, 2023 at 12:52
  • $\begingroup$ To prevent misunderstandings, I have edited my response. Thank you, Asaf for pointing that out.^ Very good point alexg. $\endgroup$ Feb 6, 2023 at 14:20

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