Re: Logarithm of transfinite numbers

Randy Poe said:

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


That seems kinda weak, Randy, no offense. "Undefined". Hmph! Sorry.

We have the set of natural numbers, of size aleph_0. Let's represent each one
in binary. How many bits will we need to represent aleph_0 naturals? Well,
given N=S^L, where N=aleph_0 and S=2, aleph_0=2^L, so L=log2(aleph_0), and that
is the length of binary strings you need to represent all the naturals. It's
valid. Is the example concrete enough for you? How can you say this number does
not exist, given this example?

This is a major quantitative shortcoming in transfinite set theory.
--
Smiles,

Tony
.

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