Virgil
Posts:
8,833
Registered:
1/6/11


Re: Joel David Hamkins on definable real numbers in analysis
Posted:
Jun 22, 2013 1:33 PM


In article <91f1cfb700404ebf866058b12ce4eb1e@googlegroups.com>, mueckenh@rz.fhaugsburg.de wrote:
> On Saturday, 22 June 2013 17:44:42 UTC+2, dull...@sprynet.com wrote: > > Consider the following statement: (*) "In order to order the reals, you > > must be able to distinguish them. But it is impossible to distinguish more > > than countably many." _Is_ (*) correct? > > Yes, that is correct. > > > If (*) is correct it follows that there's no _order_ on the reals, which > > seems curious, to say the least, since there _is_ a perfectly simple, > > standard order on the reals, nothing to do with AC or anything else > > problematic. > > There is an order on the rationals. In order to put an irrational into that > order, usually its rational approximations are chosen. By the way, this is > another reason that not more irrationals than rationals can exist. Every pair > of irrational numbers must be distinguished by a rational between them.
And every pair of distinct irrationals have MORE irrationals between them than they have rationals between them. > > I think
WRONG! 

