Re: Logarithm of transfinite numbers

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.
.

Relevant Pages

  • Re: Logarithm of transfinite numbers
    ... there is no representation with less than aleph_0 bits ... Aleph_0 bits gives us vastly more strings than we actually need. ... You are mistakenly thinking that the set of all binary sequences is the ... Chas Brown's explanation of "finite support" is interesting, ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... there is no representation with less than aleph_0 bits ... Aleph_0 bits gives us vastly more strings than we actually need. ... You are mistakenly thinking that the set of all binary sequences is the ... Chas Brown's explanation of "finite support" is interesting, ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... there is no representation with less than aleph_0 bits ... Aleph_0 bits gives us vastly more strings than we actually need. ... You are mistakenly thinking that the set of all binary sequences is the ... same as the set of all binary sequences with finite support. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... there is no representation with less than aleph_0 bits ... You are mistakenly thinking that the set of all binary sequences is the ... same as the set of all binary sequences with finite support. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... there is no representation with less than aleph_0 bits ... You are mistakenly thinking that the set of all binary sequences is the ... same as the set of all binary sequences with finite support. ...
    (sci.math)