Cantor Confusion

Cantor's proof is one of the most popular topics on this NG. It
seems that people are confused or uncomfortable with it, so
I've tried to summarize it to the simplest terms:

1. Assume there is a list containing all the reals.
2. Show that a real can be defined/constructed from that list.
3. Show why the real from step 2 is not on the list.
4. Conclude that the premise is wrong because of the contradiction.

The steps are simple except for a possible debate about defined /
constructed. I don't think anyone believes the proof is invalid
because of that debate however.

There seems to be another area that seems to be a problem
though. The problem is that step #2 doesn't seem valid. If we
assume the list contains all the real numbers, then defining or
constructing a real number in terms of that list would be
self-referential. The number from step #2, that is normally defined
digit-by-digit along the diagonal, must have its digits (or at least
one of them) defined as not equal to itself, if we are to assume the
list contains all the real numbers. Certainly the conclusion in that
case is that the premise is wrong or that the construction is not
valid, but the conclusion can't be simply that the premise is wrong.

This same problem appears in the "power-set" theorem, where we
have a definition of a set, S, which is a subset of N, defined in
terms of the image of a function, f, whose image is assumed to be
the power-set of N. If the image of f is assumed to be the power-set
of N and S is defined in terms of f, then S must necessarily be
defined in terms of itself. Again, if we assume that the image of
f is P(N), then defining S as a set whose elements are defined
to be elements not in the image of f is a self-referential definition
of S because S is also a subset of N, making it meaningless.
Certainly a meaningless definition can't be used to prove a
contradiction.

I'm guessing that the "discussions" that occur stem from the fact
that mathematicians disagree that the seemingly self-referential
definitions are a problem but it's not intuitively obvious why that is
so, therefore many people feel the need to try to refute the proofs.
The problem is that it really isn't clear why mathematicians seem to
accept the self-referential definitions.

.

Relevant Pages

  • Re: Cantor Confusion
    ... Assume there is a list containing all the reals. ... constructing a real number in terms of that list would be ... not, misses at least one real", and thus the absurd conclusion ("the ... the power-set of N. If the image of f is assumed to be the power-set ...
    (sci.math)
  • Re: Could you help me with complex numbers?
    ... math and try to study a bit more thn required by my school. ... particular way of constructing the complex numbers.) ... the rationals are _constructed_ starting ... from the reals. ...
    (sci.math)
  • Re: Could you help me with complex numbers?
    ... math and try to study a bit more thn required by my school. ... particular way of constructing the complex numbers.) ... the rationals are _constructed_ starting ... from the reals. ...
    (sci.math)
  • Re: Question about a minor recasting of Dedekind cuts.
    ... Now I'm constructing the reals in two ways. ... Your old exercise has been perpetuated onto the human nearly ... is already true for the rationals. ...
    (sci.math)
  • Re: Question about a minor recasting of Dedekind cuts.
    ... Now I'm constructing the reals in two ways. ... that if A and B intersect then the intersection ... is already true for the rationals. ...
    (sci.math)