Tetration of 2 and Aleph_0

[sheldonison]

I'm no expert on set theory, but on a humorous note (not mathematically sound), assuming the generalized continuum hypothesis, then what happens if we take the slog of an aleph number?
\( \aleph_1=2^{\aleph_0} \) which implies \( \text{slog}_2(\aleph_1) = \text{slog}_2(\aleph_0)+1=\aleph_0 \)

And for any integer n where \( \aleph_{n+1}=2^{\aleph_n} \), then \( \text{slog}(\aleph_n)=\aleph_0 \)

Perhaps \( \text{slog}(\aleph_{\aleph_1})=\aleph_1 \)
- Shel

It turns out aleph and beth numbers should be indexed by ordinal numbers. The ordinal number equivalent to \( \aleph_0=\omega \) and the ordinal number equivalent to \( \aleph_1=\omega_1 \) But I have no idea whether slog or sexp have any meaning for \( \aleph \) numbers. The other possibility would be to see if sexp/slog would be more applicable to ordinal numbers. But the exponentiation rules for ordinal arithmetic say that \( 2^\omega=\omega \) I'm unsure of what \( \text{sexp}(\omega) \) would be; the result might just be \( \omega \).
http://en.wikipedia.org/wiki/Ordinal_arithmetic
http://en.wikipedia.org/wiki/Aleph_number