Tetration of 2 and Aleph_0

[jht9663]

So essentially [\(\aleph_{\aleph_{0}}+1=\aleph_{\aleph_{0}}\)],
[\(2*\aleph_{\aleph_{0}}=\aleph_{\aleph_{0}}\)],
[\(2^\aleph_{\aleph_{0}}=\aleph_{\aleph_{0}}\)], and
2^^[\(\aleph_{\aleph_{0}}=\aleph_{\aleph_{0}}\)].

However I do not agree that [\(\aleph_{\aleph_{1}}] does not exist.

My heuristic reasoning is:

1 (the first integer past the addition identity) + 0 = 1 (the first integer past 0)

(assuming the Continuum Hypothesis)
2 (the first integer past the exponentiation identity) ^ [\)\aleph_{0}\(] = [\)\aleph_{1}\(] (1 being the first integer past 0)

if these are true then

2 (the first integer past the pentation identity) ^^^ [\)\aleph_{\aleph_{0}}\(] = [\)\aleph_{\aleph_{1}}\(] (1 being the first integer past 0)

and you could extend the pattern.

Of course I have no other reasons to believe that the third statement is true, as one would have to prove that there does not exist a bijection from [\)\aleph_{\aleph_{0}}\(] to 2^^^[\)\aleph_{aleph_{0}}$].

Also, where would be a place I could go to on the internet to find more discussion on this topic?

Thanks,
Hassler Thurston