Re: Cantor Confusion

In article <1165345832.735910.255620@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"MoeBlee" <jazzmobe@xxxxxxxxxxx> wrote:

Bob Kolker wrote:
There is no such thing as a countable integer, countable rational or
countable real.

In the sense you're trying to get across to the other poster, I
understand your point. But, just for the record, in a technical sense
in set theory, as integers, rational numbers, and real numbers are
themselves sets, it does make sense to say whether one of them is
countable or not. For example, where integers are defined as
equivalence classes of natural numbers, each integer is itself a
denumerable set. I am not necessarily endorsing anything the other
poster has said; I'm just adding the technical note that in a strict
set theoretic sense, even numbers are sets and thus it is meaningful to
talk about the cardinality of a number.

MoeBlee

In that particular sense, each real number as a Dedekind cut, being a
partition of the rationals into two sets, always has cardinality 2,
while each member of a Dedekind cut is countably infinite.

But each real number as a set of equivalent Cauchy sequencesis
uncountable.
.

Relevant Pages

  • Re: Cantor Confusion
    ... In the sense you're trying to get across to the other poster, ... talk about the cardinality of a number. ... while each member of a Dedekind cut is countably infinite. ... But each real number as a set of equivalent Cauchy sequencesis ...
    (sci.math)
  • Re: Cantor Confusion
    ... In the sense you're trying to get across to the other poster, ... But, just for the record, in a technical sense ... talk about the cardinality of a number. ... while each member of a Dedekind cut is countably infinite. ...
    (sci.math)
Loading