Re: Logarithm of transfinite numbers

In article <1142789037.428545.283610@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
matt271829-news@xxxxxxxxxxx wrote:

imaginatorium@xxxxxxxxxxxxx wrote:
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?

In this case it really just means speculate that such things can be
defined along the lines of aleph[-1] = log(aleph[0]), aleph[-2] =
log(aleph[-1]) etc., with a suitable definition of log of course. My
technical ability is not up to the job of formulating this with any
precision, but the principle is similar, if you like, to x^2 = -1. We
suppose that solutions exist, call them +/-i, and see what happens. In
that case something very fruitful indeed happens. (I am not claiming
that in the aleph case something fruitful *will* happen... it's just an
idea.)

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.

There is nothing wrong with "declaring" something (i.e. saying
something is true by definition) and seeing what happens. You can
declare anything you like in maths, provided it's all done rigorously
and in a non-contradictory manner (which I haven't done, of course).
You might end up with something that obviously doesn't hang together,
or you might end up in la-la-land, or you might, occasionally, end up
with something interesting. Either way you have to start with an open
mind. You can't just say "this system that I'm familiar with doesn't
have such and such a concept; therefore no system can".

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.

I am sure this idea will have been explored before, but I'm nowhere
near knowledgable enough about the subject to know how to map it to
something in existing theory. Possibly it is easily shown to lead
nowhere and therefore easily discarded.

After all, aleph_0 is the first infinite cardinal - how can there
be another infinite cardinal before the first one?

Perhaps there *isn't* a first one!

Perhaps there isn't a first natural either! Makes just as much sense!

A set is (Dedekind) *finite* (and has a finite cardinality) if and only
if there does not exist any injective function from that set to any
proper subset.

A set is (Dedekind) *infinite* (and has infinite cardinality) if and
only if there does exist an injective function from the set to some
proper subset.

If A and B are sets such that there exists an injection from A to B,
(Cantor definition) the cardinality of A is less than or equal to that
of B.

And, of course, if Card(A) <= Card(B) and Card(B) <= Card(A), then
Card(A) = Card(B).

It is easy to show inductively that there is an injection from the
naturals, N, into any infinite set S, so that for *every* infinite set
S, Card(N) <= Card(S).

Since Card(N) = aleph[0], every strictly smaller cardinality must be
finite.
.

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