Re: Epistemology 201: The Science of Science

From: aeo6 (aeo6_at_cornell.edu)
Date: 03/28/05 

Date: Mon, 28 Mar 2005 12:18:56 -0500

robert j. kolker said:
>
>
> Tony Orlow (aeo6) wrote:
> >
> > So, 10-8 in your ordering is the 1015th number, so there are 1014 rationals
> > that are smaller than it quantitatively? Look. Comprehend. I know it is hard
> > but you can do it. What you cannot do is prove that there are as many integers
> > as rationals WITHOUT mangling the number system.
>
> This is true. The order types are completely different. So what? That
> has nothing to do with equicardinality. Nothing.
That's because cardinality doesn't care, not because it doesn't matter. There
are very good reasons to believe, logically, that there are an infinite number
of rationals for every integer. There is only one very shaky reason to believe
there are the same number of each, and it leads to ridiculous implications. Is
there some part of the number line beyond the finite range where there are
infinities of natural numbers with no rationals, that makes up for all the
extra rationals in the finite arena? Come on!! Think about your own thoughts
for a minute. It DOES have to make sense.
>
>
> Forget smaller and larger. The mapping has nothhing whatsoever to do
> with order.
>
> Bob Kolker
>
Then what does it matter what the 1015th number is? If you want to conceptually
extend your mapping to infinity you need to use some kind of formula that mapps
the entire infinite set as a whole to another set. If the numbers are not in
quantitative order, then your formula doesn't work properly. If you do what I
am saying, avoiding this mangling, and taking into account the mapping formula
when it is in proper order, then you get an accurate measure of relative size
for the sets and avoid these erroneous conclusions.

Domain and density measures together form an accurate size measure when
bijections are well-formed, ie in quantitative order for both sets, and the
mapping function is used to determine realtive density over a common domain. 

-- 
Smiles,
Tony

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... >> as rationals WITHOUT mangling the number system. ... The mapping has nothhing whatsoever to do ... > Bob Kolker ... the entire infinite set as a whole to another set. ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... >> as rationals WITHOUT mangling the number system. ... The mapping has nothhing whatsoever to do ... > Bob Kolker ... the entire infinite set as a whole to another set. ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... >> What is wrong with the logic that, if there are infinitely many rationals for ... > that every mapping from A onto B be one to one. ... > Bob Kolker ... between infinite sets using cardinality is infinite realtive size. ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... >> What is wrong with the logic that, if there are infinitely many rationals for ... > that every mapping from A onto B be one to one. ... > Bob Kolker ... between infinite sets using cardinality is infinite realtive size. ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... >> What is wrong with the logic that, if there are infinitely many rationals for ... > that every mapping from A onto B be one to one. ... > Bob Kolker ... between infinite sets using cardinality is infinite realtive size. ...
    (sci.math)
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