Apart from Cantor's diagonalization argument, there are a number of ways to show that cardinality of R is greater than that of N (eg: Baire Category theorem, path connectedness of R and so on).

Are there any arguments outside Set Theory which naturally lead us to sets of higher cardinality than R ?

In other words, if Set Theory never existed, would we be happy to conclude that there are only three types of cardinality - finite, countably infinite and uncountably infinite - or would we have gone beyond ?

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    $\begingroup$ What do you mean by outside of set theory? The family of functions $\mathbb R \to \mathbb R$ has cardinality $2^{\mathfrak c}$. If you restrict to continuous functions or nicer, then yes, it goes down to $\mathfrak c$, but if you restrict to merely Lebesgue measurable functions, which I think are of reasonable interest outside of set theory, you still get $2^{\mathfrak c}$. $\endgroup$ – Dustan Levenstein Aug 25 '15 at 21:48
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    $\begingroup$ So how are you defining $\mathbb{R}$ "outside set theory"? And what argument based on path-connectedness do you have in mind? $\endgroup$ – Rob Arthan Aug 25 '15 at 21:56
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    $\begingroup$ There is a computability theory version. If a real number is a Turing machine that produces a Cauchy sequence of rationals, you can show that there is no computable function from the natural numbers onto the reals. This is a case where you are really talking about a aubset of the natural numbers having no enumeration, but the proof is at heart Cantor's diagonal proof. It gives another view about the meaning of Cantor - that the reals are more complex, rather than bigger than, the natural numbers. $\endgroup$ – Thomas Andrews Aug 25 '15 at 22:25
  • $\begingroup$ And to drive home the point made by @ThomasAndrews that it's about complexity not size, it's easy to write down a computable partial function from the natural numbers onto the computable reals. The trick is that you can't compute the domain of the partial function. $\endgroup$ – Hurkyl Aug 25 '15 at 22:47
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    $\begingroup$ Maybe it's worth recalling here that Cantor's first proof of uncountability of the reals was not by means of his diagonal argument, but by a different method ${}\qquad{}$ $\endgroup$ – Michael Hardy Aug 25 '15 at 23:31

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