```Date: Jan 22, 2013 3:28 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: ZFC and God

On 21 Jan., 19:07, Zuhair <zaljo...@gmail.com> wrote:> Doesn't that say that mathematics following ZFC is only grounded in> Mythology driven principles!>> Doesn't that mean that ZFC based mathematics is too imaginary that> even if consistent still it is based and rooted in fantasy that cannot> really meet reality!ZFC is not consistent unless inconsistencies are defined to be noinconsistencies, distinctions need not be distinguishable,incomletenesses need not be incomplete, and so on.Consider, for instance, all terminating binary fractions b_n0.00.10.000.010.100.110.000where some numbers are represented twice (in fact each one appearsinfinitely often). Constructing the diagonal d we find that d differsfrom *every* b_n *at a finite place*.Since the above list is complete, which is possible because allterminating fractions, as a subset of all fractions, are countable, itis impossible that the diagonal differs from all entries b_n at afinite place. If this was possible, the list would have a gap, namelya finite initial segment of d. That means, the diagonal up to everybit can be found in the list. And after every finite place there isnothing that could distinguish two numbers.Therefore the diagonal does not increase the cardinal number of thelisted entries b_n.The diagonal may be infinitely long. But what does that mean? Everygiven number of bits is surpassed. But the same holds for the entriesof the list. The only difference could be a bit of the diagonal thathas no finite index. But such bits are not part of mathematics and ofCantor's argument.Regards, WM
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