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


Re: On the infinite binary Tree
Posted:
Dec 14, 2012 4:13 PM


In article <21cb1f4a0faa4487b1c84b33b5964701@eo2g2000vbb.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> wrote:
> On 13 Dez., 21:02, Zuhair <zaljo...@gmail.com> wrote: > > > Note: definable is short for "definable by parameter free finite > > formula" > > What do you believe to gain by parameters in definitions? > Do you want to define the real numbers by using real numbers as > parameters? > > Otherwise you will not arrive at uncountably many definitions, you may > use all natural numbers and all rational numbers. Or has some great > leading matheologian who bolsters his ego by calling himself a logican > told you that by parametric definitions the real numbers all can be > defined? > > Be informed, that is not the case. If there are uncountable many real > numbers, then they are not real in that nearly all are completely > undefined.
To say that a number is "real", means no more than to say it is a complex number with imaginary part of zero, and is totally unrelated to any nonmathematical meaning of the word "real.
And in that sense, since no numbers exist other than in someone's imagination, no numbers are real in any other sense than in being members of the set of real numbers or some superset.
And all numbers which are embers of the set of real numbers or some superset are equally real, but not all equally accessible.
> Therefore they cannot be used in Cantor lists and cannot > spring off as diagonals. The numbers that can be used in Cantor lists are sufficient to show the uncountability of the set of all reals.
Note that the very definition of countability requires that a set can be declared countable ONLY if one can demonstrate the existence of a surjection from the set of naturals to that set. But Cantor shows, indirectly, that such a surjection is not possible.
Ergo, R cannot be countable.
That this rubs WM's fur the wrong way does not invalidate it 

