In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> > > > But I will answer yours for you. If you create a version of "classical" (= > > "standard" ?) arithmetic with an axiom that sqrt(2) is rational, it would > > > be > > inconsistent, and hence useless. > > Same with the axiom of infinity: ³There exists a complete linear > infinite set² is a self-contradictory similar to ³there exists a pair > of natural numbers, a and b, such that b^2 = 2a^2.
WM keeps claiming this but never manages to prove it. All of his attempts at proofs require assuming a priori and without proof that no infinite set can exist.
> > > > Now how about answering my question. How do you falsify the existence of > > the > > set of all Natural numbers in ZF, as you claimed you could? > > It is simple: ... classical logic was abstracted from the mathematics > of finite sets and their subsets .... Forgetful of this limited > origin, one afterwards mistook that logic for something above and > prior to all mathematics, and finally applied it, without > justification, to the mathematics of infinite sets. [Hermann Weyl, > "Mathematics and logic: A brief survey serving as a preface to a > review of The Philosophy of Bertrand Russell", American Mathematical > Monthly 53: 213]
So from what cathedra was Weyl speaking? Arguing from citing authorities may work in law courts, but carries little weight mathematics. > > Show me a complete finite linear set that does not allow for > quantifier reversal. > En Am: m =< n <==> Am En m =< n [*] .
Show me that every infinite well ordered set does allow it. > > Therefore: Either [*] holds or the set is not complete but allows for > extension.
Or the set is infintie, well-ordered, and has no maximal member.
> Then we have only > En Am: m =< n ==> Am En m =< n [**] > because not all elements are readily available.
Any set in a sane set theory has all members equally "available".
> That is called a potentially infinite set.
That is called nonsense in set theories, as is much of WM's MathUnrealism.