Re: Logarithm of transfinite numbers

On 12 Mar 2006 10:27:48 -0800, matt271829-news@xxxxxxxxxxx wrote:


David C. Ullrich wrote:
On 12 Mar 2006 04:49:31 -0800, matt271829-news@xxxxxxxxxxx wrote:


David C. Ullrich wrote:
On 11 Mar 2006 09:07:31 -0800, matt271829-news@xxxxxxxxxxx wrote:


Dave Rusin wrote:
In article <1142021663.636830.285410@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<matt271829-news@xxxxxxxxxxx> wrote:
Shmuel (Seymour J.) Metz wrote:
In <1142014707.444947.118750@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, on
03/10/2006
at 10:18 AM, matt271829-news@xxxxxxxxxxx said:

What value, if any, can be ascribed to the logarithm of aleph_n?

None, unless you define it.

Yes, that is the question. How, if at all, can this logarithm be
"sensibly" defined? The answer may of course be "not at all".

Why would you expect the log of that complicated cardinal to be
defined when there isn't even a logarithm of the cardinal "3" ?
Seems to me to make perfect sense to decree log(X)=Y if X=2^Y
(there's a problem if 2^Y and 2^(Y') have the same cardinality
of course) but most cardinals are not of this form at all and we can't
expect them to have "logarithms"; not just aleph_0, but 3 and 5 too!

dave

This is a potential glitch, but since aleph-0 = 2*aleph-0, somewhere
"within the scope of aleph-0" we "ought to" be able to find an "exact"
log_2(aleph-0). Ditto for any other base.

(In case not obvious, I am just playing around with ideas, not trying
to make rigorous mathematical statements!)

Uh, you can call it playing around with ideas if you want, but
the statements you're making are not just less than rigorous,
they're simply _false_. Very easily seen to be false -

Please indicate which statements I have made that are "very easily seen
to be false"

Uh, maybe it was the statement I was replying to? That we ought to
be able to find an exact log_2(aleph_0)? You _really_ have a hard
time figuring out that's the statement I was referring to? Remarkable.

You said statements. Plural. Unremarkably, I assumed you were talking
about several of the comments I have made in this thread, or possibly
all of them.


and how so.

Haven't you read any of the thread so far?

Yes, and my conclusion is that some (including you, presumably) dismiss
the idea of such a logarithm of out of hand as nonsense,

I haven't seen anyone dismiss the idea out of hand. I've seen
people give simple explanations why there is no such thing
as log_2(aleph_0).

while others
seem not to be so instantly dismissive, even quoting published works
that mention the idea.

ReallY? I missed that - who quoted published works on the topic?
(And did the publizations have anything to say other than that
there's no such thing?)

************************

David C. Ullrich
.

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