Re: Logarithm of transfinite numbers


matt271829-n...@xxxxxxxxxxx wrote:
What value, if any, can be ascribed to the logarithm of aleph_n?

If you define it, we can see if anything fits that definition.


Take log_2(aleph_0) as an example.

log_2(aleph_0) can't be finite, and it can't be bigger than aleph_0, so
it has to equal aleph_0.

You're drawing conclusions about something you haven't yet
defined.


But log_2(aleph_0) = aleph_0 implies 2^aleph_0 = aleph_0, whereas in
fact 2^aleph_0 = aleph_1.

Ah, so by "log_2(aleph_0)" you mean "the cardinality of a set
S such that |P(S)|, the cardinality of the powerset of S, is aleph_0".

No such S exists.

Where did it go wrong?

You haven't come up with a definition such that anything
meets that definition. There is no such thing as "log2(aleph_0)".

- Randy

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