Is the cardinality of the real numbers $\mathfrak{c}$ a real number?

Question

This may be a stupid question, but I was learning some set and group theory and it just made me think. Clearly the continuum is an infinite quantity $\mathfrak{c}$, but the set of all reals is also infinitely long. Or is it that $\sup(\mathbb{R})=\mathfrak{c}$ and $\mathfrak{c}\notin\mathbb{R}$. Regardless this is a theoretical question, but if someone who likes set theory could help me out it'd be very appreciated!

Answer 1

Assume there is a real number r, of real numbers.
Clearly there is a positive number of real numbers.
Also the number of real numbers is not fractional.
So r has to be a positive integer.

Let R be the set of all r real numbers.
Since r is finite, R has a maximum m.
Since m + 1 is not in R, a contradiction ensues.
Thus there is not a real number of real numbers.

Hence there be an unreal number of real numbres.

Answer 2

No. But the first uncountable ordinal, $\omega_1$, does lie on the long line...