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A long time ago a similar question was asked on math.stackexchange.

There are many sets which we know to be either finite or infinitely countable but do not know which cardinality specifically.

An example that comes to mind is the set of twin primes. But comparing countably infinite vs cardinality of the continuum I can't really come up with any "natural looking" examples (or any examples for that matter to be frank).

As a result I'm interested in organizing a community wiki to keep track of sets for which we haven't yet settled if the cardinality is countably infinite or the same as the continuum but we do know that it must be one of the two.

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    $\begingroup$ Non-natural example: the closure in $\mathbf{Z}_p$ of the union of the set of twin primes. $\endgroup$
    – YCor
    Commented Sep 26, 2023 at 14:27
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    $\begingroup$ $\mathbb N\cup\{x\in\mathbb R:\text{the Riemann hypothesis holds}\}$. $\endgroup$
    – Emil Jeřábek
    Commented Sep 26, 2023 at 14:29
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    $\begingroup$ Your title is about the cardinality of $\mathbb{N}$ vs. the cardinality of $\mathbb{R}$, while your main text is about "countable vs. uncountable". These are not the same thing, so please clarify your question. $\endgroup$
    – GH from MO
    Commented Sep 26, 2023 at 14:41
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    $\begingroup$ I have clarified the question, when I had written “countable” earlier I meant “countably infinite” or “cardinality of the natural numbers”. Similarly when I had written uncountable earlier I meant “cardinality of the continuum” $\endgroup$
    – Sidharth Ghoshal
    Commented Sep 26, 2023 at 14:51
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    $\begingroup$ Borel's conjecture on Strong Measure Zero Sets comes to mind. Is that the kind of answer you want? $\endgroup$
    – François G. Dorais
    Commented Sep 26, 2023 at 23:13

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