In article <email@example.com>, firstname.lastname@example.org wrote:
> On Sunday, 23 June 2013 18:06:06 UTC+2, dull...@sprynet.com wrote: > > > > Oh! You say that every pair of irrationals "must" be distinguished by a > > rational between them. You say we need to do this in order to define the > > standard order on the reals. And you agree that there is no problem > > defining the standard order on the reals. > > > So you must agree that any two reals _can_ be distingushed by a rational > > between them. > > Otherwise they cannot be proven to be different. > > > In particular, any two reals _can_ be distingushed. > > If the reals can be identified and are not the same, then they can be > distinguished by a rational. > > > Now what the heck are you talking about when you say there's no well-order > > on the reals because we cannot distinguish them? > > I say there could not be a well-order on *uncountably many reals*, because > most of them could not be identified by a finite amount of bits.
OUTSIDE of WM's wild weird world of WMytheology, there is no requirement expressed or even expressible in any standard real number system that any real number be identified by a finite number of bits.
WM is referring to numerals, not numbers, a distinction that he does not seem to be able to comprehend. Nothing in any standard axiom system for the reals requires that there be any numeral for any number, except possibly those named by 0 and 1. --