> > 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 [*] . > > ************************ > Its your proof. Either provide such a set yourself, or prove that none > exists.
[*] belongs to the basic logic of finite sets. As this logic has been obtained by observing the behaviour of finite linear sets, there is no further proof except that a finite linear set violating [*] has never been observed. > > 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.
Complete means that every element of a set exists. In case of a linear set, complete means that also the last element of the linear order does exist. > > 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.
Every element of a complete set is readily available. If not eery element is readily available, then the set is incomplete, that mean not complete. > > 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.
Potentially infinite is not opposed to infinite but is one kind of infinity. The other one is actually infinite. > > 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? > Potential infinity is possible, actual infinity is not. > > ************************** > Gee, such a big claim you make. Yet no proof. Not even a lame attempt.
Sorry, you are only unable to understand it. Not everybody is able to understand everthing. That is an odl wisdom.
> Even > cranks generally try and produce bad proofs,
Yes, set theorists, for instance. But I have shown that they need to use logic of potential infinity to "prove" their actual infinity.
