Re: How to do magic with infinity
From: George Greene (greeneg_at_cs.unc.edu)
Date: 10/11/04
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Date: 11 Oct 2004 02:47:46 -0700
Han de Bruijn <Han.deBruijn@DTO.TUDelft.NL> wrote in message news:<ck6517$ndf$1@news.tudelft.nl>...
> >> It *is not* a fact that the set of even naturals, while remaining a
> >> subset of all naturals, can nevertheless acquire the same
> >> cardinality.
> In message <ck61gv$k15$1@south.jnrs.ja.net> Robin Chapman:
> > "Acquire"?
> Yes, acquire.
No, NOT "acquire". What it means for 2 sets to have the
same cardinality is DEFINED. It is NOT "acquired". And
you CAN'T ARGUE with a definition. You can allege that it
is ungrammatical or that it leads to inconsistency, but its
merely having counter-intuitive corollaries does NOT constitute
any sort of refutation or ammunition against it. BY DEFINITION
(of "definition" -- and you can't argue with THAT definition, either).
> Because (in a constructive context) infinity is only
> potential and not actual.
"Constructive" and "constructivism" in mathematics also
involve a definition. Maybe you should look that definition
up instead of embarrassing yourself in public. The "context"
within which you (mistakenly) think you are operating contains
an infinite number of ACTUAL constructed things. All
of them are achieved/extant. NONE of them is potential.
For you to insist that the collection of all of them still
MUST remain "potential and not actual" just proves you're an idiot.
>
> > As I said, it is a fact that the the set of even naturals has the
> > same cardinality as the set of all naturals.
>
> Because "as you said" ? Are you an authority or what ?
No, dumbass, he just has a short PROOF of it, on the basis
of the DEFINITION of what it means for two collections to
have the same cardinality.
> Geez, why is it that I have never heard of you before ?
>
> >> Proof: Start with a finite set of naturals 1 ... N . Count the #
> >> even naturals. And take the limit -> oo . As simple as that.
That IS NOT a proof that "all the naturals" and "the even naturals"
have different cardinalities. You don't even know what you MEAN
by "take the limit". The expression "the number of even naturals
in 1...n" DOES NOT HAVE a limit as n approaches #N. The limit
you are asking people to take is simply undefined, to you. To
those of us who know what a limit ordinal is (and would charitably
assume you mean passage to something like that, if you are going
to take limits in this context, the limits as n -> oo of
#(all the naturals in 1...n) and #(the even naturals in 1..n)
are in fact the same limit. The sequences 0,1,2,3,4,5,... and
0,0,1,1,2,2,3,3,4,4,5,5,... do in fact approach the same oo.
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