"WM" <mueckenh@rz.fh-augsburg.de> wrote in message news:4f20ac0a-a6b7-4a20-8eca-84a63b1f69b6@c19g2000yqc.googlegroups.com... 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.
***************************** I understand your claim. I was asking fir a proof of it, not you restating it.
> > 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]
******************************* Well, Weyl never claimed or proved your assertion, so starting off with a quote from him is a piss-poor starting point. I assume you have no proof, which is why you are just in random quote mode.
Show me a complete finite linear set that does not allow for quantifier reversal. En Am: m =< n <==> Am En m =< n [*] .
************************ Its your proof. Either provide such a set yourself, or prove that none exists.
Therefore: Either [*] holds or the set is not complete but allows for extension.
***************************** You better define "extension", and show its relationship to "complete". Just in case, you better define "complete" as well, in case you are using it differently to everybody else.
Then we have only En Am: m =< n ==> Am En m =< n [**] because not all elements are readily available.
************************** You better define "readily available" while you are at it.
That is called a potentially infinite set. But in this case there is no chance to prove uncountability.
************************* Define "potentially infinite" as opposed to "infinite". Show that this does not allow uncountability to be proved.
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.
************************* What *is* the difference?
Regards, WM
************************** Gee, such a big claim you make. Yet no proof. Not even a lame attempt. Even cranks generally try and produce bad proofs, you don't even bother trying to do that. You are actually sub-crank. To become merely crank, define some terms, and try and make your rant at least look like a proof.
