Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!watmath!clyde!burl!hou3c!hocda!houxm!ihnp4!inuxc!pur-ee!uiucdcs!uiuccsb!grunwald 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 Lines: 33 #N:uiuccsb:9700020:000:1555 uiuccsb!grunwald Feb 17 21:33:00 1984 [ This line does not exist ] 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?