On 3 Jan., 22:31, Virgil <vir...@ligriv.com> wrote: > In article > <3d65ff59-bf7e-445b-aad6-77d4ece64...@p17g2000vbn.googlegroups.com>, > > WM <mueck...@rz.fh-augsburg.de> wrote: > > On 2 Jan., 22:19, Zuhair <zaljo...@gmail.com> wrote: > > > > On the other hand Cantor have presented many arguments all of which > > > are rigorously formalized in second order logic under full semantics, > > > and those arguments PROVED that there are uncountably many reals > > > that can be distinguished by their finite initial segments. > > A decimal, or other base, expression is not a actually a number but > merely a numeral, a representation of or name for a number. > > Actually, no real can be distinguished from ALL others by ANY finite > initial segment of its decimal, or other base, representation.
That means that every real that shall be used in mathematics needs a finite representation over a countable alphabet (even a finite alphabet - but that does not matter here). Mathematics of real numbers cannot be done in another way. Examples:
0,25 and only zeros following 0,111... and only 1's following pi e crt(3) and so on.
All distinguishable reals (and all reals that can appear in Cantor's mathematics) belong to this set (as Cantor himself clearly recognized).