Re: Why? [was Re: Cantor`s powerset theorem is false?]

Daryl McCullough wrote:
Don't tell me that transfinite mathematics is simple.
It is a completely meaningless formal construct.

On the contrary, the assumption that the reals are *countable*
assumes the existence of something for which there is no evidence
(namely a function enumerating the reals) and which is inconsistent
with much of standard mathematics. I would consider that assumption
a meaningless construct.

I'm not taking sides or anything, but from my understanding the
issue is not so simple. Let T_n be the n-th Turing machine with
alphabet {blank,0,1}, and suppose I were to cite as an enumeration of
[0,1] the following partial function:

f : N --> [0,1]

f(n) = the real number whose k-th digit is obtained by running T_n on
input k and looking at the rightmost non-blank character on the tape
once T_n stops. (Leave f(n) undefined in case there are only blanks
left over, or T_n never stops.)

Now, I don't claim that f is injective, but suppose I did claim it
was surjective. A function, defined only on a subset of N, surjective
into [0,1]! Crazy talk. However, in a very strong sense, you can't
*show* me a number which lies outside its range.Whichever effective
procedure you give me to produce the successive digits of a real number
in [0,1], I'll give you a Turing machine that simulates my work, thus
showing your purported counterexample to be enumerated by f after all.

That leaves us in the position of asserting the existence of real
numbers we can't get any sort of handle on. For which there is "no
evidence", so to speak.

I'm not suggesting that there are not solutions one way or another,
only that they're not obvious.

--Pietro

.

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