Re: Logarithm of transfinite numbers

In article <MPG.1e824979c752875e98aaf4@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Virgil said:
In article <MPG.1e820b41d42e0d8698aaea@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Shmuel (Seymour J.) Metz said:
In <1142386560.427503.17760@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, on
03/14/2006
at 05:36 PM, matt271829-news@xxxxxxxxxxx said:

To represent all the reals you need aleph_0 bits.

To represent all the natural numbers you need how many bits?

Aleph_null. Of course, there is a constraint on which
combinations represent integers.

What is that constraint?

The constraint that no string contain more that a finite number of
non-zero bits. So that each may be replaced by a finite string
indicating the same natural, or integer.


That's the same as saying all strings of finite length, meaning bit
positions in N, which would be, according to your theory, aleph_0 bit
positions, and 2 ^aleph_0 strings. That's not an additional
constraint at all.

Does TO claim that restricting strings to having only finitely many
non-zero characters allows one to have infintely many on-zero characters?
If not, it is indeed a constraint on the set of all strings which
eliminates most of them.



You can't cull an uncountable number of strings
from this uncountable set and leave the countable set of strings that
you evision.

On the contrary, I can do just that, as can anyone. Create an
appropriate predicate defining a subset of a given set, and that subset
exists. At least outside of TOmatics.
.

Relevant Pages

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    ... What is that constraint? ... Depends on your exact usage of bit strings to represent naturals. ... The second is as "normal" as any other representation scheme. ...
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