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.