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Re: Distinguishability of paths of the Infinite Binary tree???
Posted:
Dec 26, 2012 2:14 PM
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On Dec 26, 6:49 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
>
> Cantor's and Hessenberg's "proofs" simply show that infinity is never
> finished and a complete infinite set is not part of sober thinking.
>
> Regards, WM
To make the discussion fruitful, lets take all possibilities available
and see what is the response to each.
(1) To say that the formal proof of Cantor is clear and exact in
formal terms, but the distinguishability argument is clear on
intuitive level but has not been verified in formal terms, so
accordingly we have the option of saying that Infinity do not copy
intuitions derived from the finite world, and deem the result as just
counter-intuitive but not paradoxical. I think this is the standard
approach.
(2) To say that the distinguishability argument is so clear and to
accept it as a proved result despite the possibilities of verifying it
at formal level or not, and also maintaining that Cantor's proof is
very clear and valid, and so we deduce that we have a genuine paradox
that resulted from assuming having completed infinity, and thus we
must reject having completed infinity. That's what WM is saying
(3) To consider countability of the finite initial segments FALSE,
i.e. to say that we have uncountably many finite initial segments of
reals and as well we have uncountably many reals. This clearly
preserves congruity of the argument, but it requires justification,
and the justification can be based on the principle of "parameter free
definability of sets", since the alleged bijection between the finite
initial segments of the reals and the set N of all naturals is NOT
parameter free definable, then this bijection does not exist, and it
is false to say that it is. This claim only accepts infinite sets to
exist if there is a parameter free formula after which membership of
those sets is determined, so if there is non then it doesn't accept
the existence sets that are not parameter free definable.
(4) To consider uncountability of the reals to be FALSE by
interpreting the universal quantifier in Cantor's argument over
functions to range over *elements* of the universe of discourse, and
thus it doesn't cover all available functions (which are subsets of
the universe of discourse), so there is a bijection between the reals
and the naturals that is simply missing from being among the
*elements* of the universe of discourse. So the reality of that matter
is that the set of all reals is countable but discourse misses the
necessary bijection. This is the interpretation of having a countable
model of a theory that proves the existence of uncountably many reals.
It is a consequence of Skolem's arguments. The justification for that
is that if we go second order and consider the universal quantifier
over functions in Cantor's argument to range over all subsets of the
universe of discourse, then we are at second order logic which is not
known to support a proof system and so it can barely be seen as a kind
of logic or something that we prove things after. So we are better
with at first order. If we use Henkin semantics to explain the second
order quantifier then we'll end up by the same argument of skolem. So
Henkin semantics for second order or first order semantics both ensure
having a countable model of any theory, and since proof theory by
ordinal analysis depends on constructiveness within countable limits,
then we only need to stipulate the existence of countable models since
it is those the ones that are both provable and also they provide a
reductionist approach reminiscent of Ockam's razor, that is if we can
do the job with less so why demand more. And since there is no logical
argument to force us to adhere to uncountability without assuming it
first, so why adhere to it?
It is interesting to know the responses to all those options.
Zuhair
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