Re: Logarithm of transfinite numbers

In article <MPG.1e7b97e77147e4b498aab7@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

matt271829-news@xxxxxxxxxxx said:
What value, if any, can be ascribed to the logarithm of aleph_n?

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.

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

Where did it go wrong?



You paid attention to Cantorian set theory hocus pocus, to begin with. ;)

You make a good point. Let's see what others had to say.......

Logarithms have definitions. For positive real numbers a
the natural logarithm of x is defined to be the area under f(x) = 1/x
between x = 1 and x = a,

ln(a) = Int_1^a 1/x dx.

Then the logarithm of a to base b, where a and b is a positive reals and
b is not 1 is defined by

log_b(a) = ln(a)/ln(b)

Note that the natural log of a number is only defined for finite
positive reals, so that other logs are equally limited.

There is no a priori reason why logs of things which are not real
numbers need make any sense at all, and trying to make sense of logs of
"infinite numbers" is silly. But then TO is!
.

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