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Re: Cantor's first proof in DETAILS
Posted:
Nov 17, 2012 7:45 AM


"Ross A. Finlayson" <ross.finlayson@gmail.com> writes:
> On Nov 16, 7:50 pm, Ben Bacarisse <ben.use...@bsb.me.uk> wrote: >> "Ross A. Finlayson" <ross.finlay...@gmail.com> writes: >> > On Nov 16, 8:12 am, Ben Bacarisse <ben.use...@bsb.me.uk> wrote: >> >> "LudovicoVan" <ju...@diegidio.name> writes: >> >> > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote in message >> >> >news:c2ac7c885567408da520df4f143f840f@googlegroups.com... >> >> >> >> <http://www.tikilounge.com/~raf/finlayson_injectrationals.pdf> >> >> >> > I'll see if I can understand it: for now, thanks for sharing. >> >> >> A first step is to remove the indexes from the irrationals. In a >> >> argument about the supposed countability of the irrationals, to refer to >> >> them with indexes (e.g. p_i) looks like begging the question. >> >> >> As it happens, I don't think the indexes do anything but add a layer of >> >> confusion. I think you can rename the various quantities without >> >> altering the meaning, i.e. rather than talk about irrationals p_i and >> >> p_h just use p and r (q is taken). >> >> > It's constructive, that. >> >> The key set, Q_<i (which would then be called Q_<p) is empty though. > > Note the emphasized bit.
I did. That the set in question is empty simply implies that for every rational q < p, there is an irrational r < p with r > q. That the set in question is *not* empty would give any member of it the property of being a rational least upper bound of all the irrationals less than p.
 Ben.



