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> > > Mark-T wrote:
> > > > You hypothesize the countable list of all reals, encoded
> > > > in binary, then diagonalize out of it, by complementing the
> > > > diagonal. Sounds good.
> > > >
> > > > Except... isn't there a slight problem here, in the sense of
> > > > 'computable', or 'effective procedure'?
> >
> > Aatu Koskensilta wrote:
> > > Cantor's theorem says nothing about computability or
> > > effective procedures.
> george wrote:
> > That's his PROBLEM with it (that it says nothing).
Randy Poe wrote:
> No, it's not a problem.
Please. You can't refute the objection if you don't understand it.
Mark-T has a problem with it. It IS a problem for Mark, regardless
of whether it should or shouldn't be. The question is, WHY shouldn't
it be a problem?
> > His point is that it NEEDS to say something.
> No, it doesn't need to say something about computability.
That is NOT the point. Given the hypotheses of the indirect proof,
the anti-diagonal IS computable AND constructible, given a proper
understanding of those notions. It is that notion that Mark-T lacks.
What you SHOULD be trying to do is explain it to him, NOT whining
that it is irrelevant.
> The question it is trying to answer is about existence. So it has
> to show existence.
Have you SEEN the proof??? (i.e., do you KNOW WHAT you are
talking about?) It is true that it would suffice "merely" to show
existence (as OPPOSED to constructibility or computability),
but AS PRESENTED, the proof ACTUALLY DOES construct
"a real not on the list" or "a subset not mapped to any element".
So, if you ARE constructing something, you DO have to show that
you are constructing it reasonably/rationally as opposed to
erroneously.
> > Otherwise it is talking about something it can't do.
> There are many cases in mathematics where things are
> known to exist but there may not be a procedure to find
> them in finite time.
That is not the issue.
Does ADDITION exist (as a binary function on the natural numbers)?
You certainly cannot cannot compute all the infinity x infinity
instances
of addition using any finite amount of time. The question of what it
means for ANYthing infinite to be "computable" is, for, him, the prior
question.
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