Date: Apr 23, 2013 4:01 AM
Subject: Matheology § 258

Matheology § 258

So what about Cantor?s much celebrated non-denumerable real? Where is
it? Did Cantor produce such a real number? No, he merely sketched out
the logic for a nonterminal procedure that would produce an infinitely
long digit string representing a real number that would not be in the
input stream of enumerated reals. Cantor?s procedure, and with it his
celebrated nondenumerable, infinitely long real number, will appear
with 100% certainty in the denumerable list of procedures. {{That's
the point: Every diagonal number can be distinguished at a finite
position from every other number. But if all strings are there to any
finite dephts, as is easily visualized in the Binary Tree, then there
is no chance for distinction at a finite position - and other
positions are not available.}}
There is no non-denumerable real, and every source of real numbers
is denumerable [...] Implications throughout mathematics that build
upon Cantor?s Diagonal Proof must now be carefully reconsidered.
So Who Won? Professor Leopold Kronecker was right. Irrationals are
not real {{ - at least they have no real strings of digits, and only
countably many of them can be defined in a language that can be
spoken, learned and understood}}. God made all the integers and Man
made all the rest {{and in addition something more - unfortunately.}}
[Brian L. Crissey: "Kronecker 1, Cantor 0: The End of a Hundred Years?

Regards, WM