Re: Logarithm of transfinite numbers

matt271829-news@xxxxxxxxxxx wrote:
imaginatorium@xxxxxxxxxxxxx wrote:
matt271829-news@xxxxxxxxxxx wrote:
Virgil wrote:
In article <1142680417.743195.20160@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
matt271829-news@xxxxxxxxxxx wrote:

cbrown@xxxxxxxxxxxxxxxxx wrote:
matt271829-news@xxxxxxxxxxx wrote:
Randy Poe wrote:

<snip>

The number of bits can't be finite, since for any finite
value, only finitely many natural numbers can be represented.
Yet the smallest infinite cardinal is aleph_0. Therefore
there is no representation with less than aleph_0 bits
which is large enough for all the natural numbers.


Yes I know, and this is what, to me, "doesn't seem quite right".
Aleph_0 bits gives us vastly more strings than we actually need. Does
it not, even just a *little* bit, seem "not quite right" to you too? Or
do you not see any scope at all for seeing a problem here?

You are mistakenly thinking that the set of all binary sequences is the
same as the set of all binary sequences with finite support.


What do you mean by "with finite support"?

The set of bit positions at which there are non-zero bits is finite.

<snip>

If so, then no, I am not thinking this.

Do tell us what you _are_ thinking <g>!

I'm thinking that we should introduce aleph[-1], aleph[-2], aleph[-3],
.... But where that would lead I don't know. Maybe quickly to a
contradiction.

What does "introduce" mean? I could define aleph_0 (for example) as the
equivalence class under the bijection relation of sets that can be
counted against a ditty without ever ending. (You recite the ditty
"un-deux-trois" whatever, and are guaranteed to get to every element of
the set eventually, but the process never stops.) This isn't a standard
formal definition, but I think it is enough to support aleph_0 on the
basis of elementary set theory. Where would this "aleph_(-1)" come
from? It's a rather Orlovian approach to go round "declaring" this that
or the other, just because it looks like helping to produce the local
result you want at the moment. Of course if you managed to define
something in the surreals or whatever that appeared to be log(aleph_0),
hmm, you could call it what you like, but if I (semi-)understand
correctly, such a thing would be in ordinal arithmetic, not cardinal
arithmetic. So it wouldn't really be _reasonable_ to give it an 'aleph'
name. After all, aleph_0 is the first infinite cardinal - how can there
be another infinite cardinal before the first one?

Brian Chandler
http://imaginatorium.org

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