On 30 Mai, 02:56, "Peter Webb" <webbfam...@DIESPAMDIEoptusnet.com.au> wrote: > >> Grant the existence of two natural numbers m and n such that m/n = sqrt > >> (2). Then falsify it. > >> Grant the existence of a largest natural. Then falsify it. > >> Grant the existence of a largest prime number. Then falsify it. > >> Grant the existence of all natural numbers. Then falsify it. > > >> All these proofs are proofs by contradiction. > > >> ************************* > > >> How do you falsify the existence of the set of all Natural numbers in ZF > >> ? > >> ZF includes an axiom of infinity, which pretty much directly guarantees > >> that > >> there is an infinite set of all finite ordinals. > > > What about classical arithmetics with an axiom that sqrt(2) is a > > rational number? > > This does not answer my question. > > 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: ?There exists a complete linear infinite set? is a self-contradictory similar to ?there exists a pair of natural numbers, a and b, such that b^2 = 2a^2. > > 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: 2?13]
Show me a complete finite linear set that does not allow for quantifier reversal. En Am: m =< n <==> Am En m =< n [*] .
Therefore: Either [*] holds or the set is not complete but allows for extension. Then we have only En Am: m =< n ==> Am En m =< n [**] because not all elements are readily available. That is called a potentially infinite set. But in this case there is no chance to prove uncountability. This had already been recognized by the late Alexander Zenkin, one of the brave scientists who dared to condemn this hypocritical behaviour: Cantor's 'paradise' as well as all modern axiomatic set theory is based on the (self-contradictory) concept of actual infinity. Cantor emphasized plainly and constantly that all transfinite objects of his set theory are based on the actual infinity. Modern AST-people try to persuade us to believe that the AST does not use actual infinity. It is an intentional and blatant lie, since if infinite sets, X and N, are potential, then the uncountability of the continuum becomes unprovable, but without the notorious uncountablity of continuum the modern AST as a whole transforms into a long twaddle about nothing.
Resume: The internal contradiction in set theory is veiled by mixing up potential and actual infinity. That is the reason why set theorists usually refuse to specify which infinity they apply. Most even pretend (or profess) not to know the difference.