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From: grunwald@uiuccsb.UUCP
Newsgroups: net.math
Subject: Pedantic Question - (nf)
Message-ID: <5710@uiucdcs.UUCP>
Date: Fri, 17-Feb-84 22:33:22 EST
Article-I.D.: uiucdcs.5710
Posted: Fri Feb 17 22:33:22 1984
Date-Received: Mon, 20-Feb-84 06:55:59 EST
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#N:uiuccsb:9700020:000:1555
uiuccsb!grunwald    Feb 17 21:33:00 1984

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   This note is intended for my further illumination and is spurred by a deep
lack of intuitive understanding of the nature of infinities.

  The question turns to Cantors diagonalization proof for the existence of
uncountably infinite numbers. In particular, the question arises "Why can I
not show that the natural numbers can be demonstrated to be uncountably
infinite", although I realize this runs rather counter to the definition of
the naturals.

Let us define a bijection from the naturals such that we reverse the digits
of the index, e.g. since 6 = 110 in binary, F(6) = 011. Since we can have
leading zeros in the naturals, let us represent 6 = ....000000110, and hence
F(6) = 01100000000...

Applying the diagonalization proof to these numbers is equivalent to the way
one constructs the proof for fractional numbers in binary.

The problem is that we construct a "member" of the naturals which is 
....1111111xxxx (e.g. an infinite number of leading ones followed by
something). Although I realize that this is not a proper member of the
naturals, how can this be demonstrate outside the realm of this proof? E.g, 
it the property of naturals not having an infinite number of digits derived
from the Cantor proof or does one have to presuppose it?

Sorry if this is rather obvious, but I've not had formal set theory and thus
this seemed a little more puzzling than it should be. I've been assured that
naturals must be finite length can be demonstrated by other means, but I
haven't seen the means. Any takers?