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 Tetration of 2 and Aleph_0 sheldonison Long Time Fellow Posts: 641 Threads: 22 Joined: Oct 2008 09/09/2011, 05:54 PM (This post was last modified: 09/09/2011, 08:23 PM by sheldonison.) (09/07/2011, 08:47 PM)sheldonison Wrote: 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 $\text{slog}(\aleph_{\aleph_1})=\aleph_1$ - ShelIt 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 « Next Oldest | Next Newest »

 Messages In This Thread Tetration of 2 and Aleph_0 - by jht9663 - 09/06/2011, 03:47 PM RE: Tetration of 2 and Aleph_0 - by JmsNxn - 09/06/2011, 09:28 PM RE: Tetration of 2 and Aleph_0 - by tommy1729 - 09/07/2011, 03:37 AM RE: Tetration of 2 and Aleph_0 - by jht9663 - 09/07/2011, 03:33 PM RE: Tetration of 2 and Aleph_0 - by sheldonison - 09/07/2011, 08:47 PM RE: Tetration of 2 and Aleph_0 - by sheldonison - 09/09/2011, 05:54 PM RE: Tetration of 2 and Aleph_0 - by jht9663 - 09/07/2011, 03:34 PM RE: Tetration of 2 and Aleph_0 - by tommy1729 - 09/10/2011, 12:22 PM RE: Tetration of 2 and Aleph_0 - by tommy1729 - 11/13/2011, 12:09 PM

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