Re: Epistemology 201: The Science of Science

stephen_at_nomail.com
Date: 03/04/05 

Date: 4 Mar 2005 22:32:14 GMT

In sci.math Tony Orlow (aeo6) <aeo6@cornell.edu> wrote:
: stephen@nomail.com said:
:> In sci.math Tony Orlow (aeo6) <aeo6@cornell.edu> wrote:
:> : And, aren't transcendentals reals that are not rationals or true
:> : irrationals?
:>
:> Transcendentals are irrationals, however not all irrationals are
:> transcendental. I have no idea what you mean by "true irrationals".
:> The reals consist of the rationals and the irrationals.
:>
:> : Seems like it might be more than the general idea, but good
:> : enough! I wouldn;t mind the whole thing about transcendentals not lying
:> : on the same line as rationals or irrationals, but I already have a lot
:> : to learn as it is.....sigh!
:>
:> The definitions for these terms are all freely available on the web.
:> http://mathworld.wolfram.com/TranscendentalNumber.html
:> http://mathworld.wolfram.com/IrrationalNumber.html
:>
:> Lester is using his own private definitions that do not
:> correspond to the standard usage and who knows if his
:> definitions are even consistent.
:>
:> Stephen
:>
: My whole point was that cardinal numbers are numbers on the number line.
: I was using Lester's definition of transcendentals as other than
: irrationals to avoid having that conversation again, which I don't fully
: understand. Do you disagree, Stephen, that transcendentals reside on the
: number line conceptually?
: --
: Smiles,

: Tony

Sure. The real number line contains all the reals. Transcendentals
are real numbers. Of course "the real number line" is just one way
to think about the real numbers. Defining "transcendental" to mean
"other than irrational" would be equivalent to defining "transcendental"
as "rational" which would be absurd, but who knows with Lester.

You need to be careful about saying that cardinal numbers are numbers
on the number line. In common usage, cardinal numbers refer to
numbers that are used in counting (i.e. the positive integers),
and mathematically the cardinal numbers include the transfinite cardinals
which are not on the real number line. So in other words, lots
of numbers on the number line are not cardinals, and not all cardinals
are on the number line.

Stephen

Relevant Pages

  • Re: Rational/Irrational Numbers
    ... The consideration of rationals and irrationals, ... other sets that are NCD2 in the reals from "On Well-Orderingof Sets ... number line, or algebraics and transcendentals, or rationals, ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... aren't transcendentals reals that are not rationals or true ... :> Transcendentals are irrationals, ... :> The reals consist of the rationals and the irrationals. ... and mathematically the cardinal numbers include the transfinite cardinals ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... aren't transcendentals reals that are not rationals or true ... :> Transcendentals are irrationals, ... :> The reals consist of the rationals and the irrationals. ... and mathematically the cardinal numbers include the transfinite cardinals ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... Do you disagree, Stephen, that transcendentals reside on the ... The real number line contains all the reals. ... Rationals, irrationals, and trancendentals are real numbers. ... >and mathematically the cardinal numbers include the transfinite cardinals ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... Do you disagree, Stephen, that transcendentals reside on the ... The real number line contains all the reals. ... Rationals, irrationals, and trancendentals are real numbers. ... >and mathematically the cardinal numbers include the transfinite cardinals ...
    (sci.cognitive)