On 28 Mai, 05:21, Virgil <virg...@nowhere.com> wrote: > > > > The problem boils down to the following: > > > > > En Am: m =< n <==> Am En m =< n [*] > > > > En Am: m =< n ==> Am En m =< n [**] > > > > > You know: Classical logic was obtained from finite sets ... > > > > Show me a finite set that obeys [**] but not [*].
> > I said: For complete linear sets [*] is true. > > You said [*] is not true, but [**] is true. > > Weyl said: Classical logic was obtained from finite sets. > > Therefore I asked you: Show me a linear complete finite set, that > > makes your claim [**] right and my claim [*] wrong. > > What Weyl may have said in no way limits logic, classical or otherwise, > to finite sets. Or does WM claim that Weyl was speaking ex cathedra?
Weyl has recognized, by the end of his life (when obtaining his doctorate under Hilbert he was preoccupied with Hilbert's opinion), that logic is not included into the 10 commandments, but has to be constructed or obtained as an abstraction from the reality. Therefore we have no guarantee that it is suitable for sets that cannot belong to reality. And I added: We cannot use logical theorems that are counterfactual, i.e., contrary to what we obtained from reality. There is no model of a complete linear set in reality that makes [*] false and [**] true. But there are many models showing that [*] is true whenever [**] is true. > > > No. I do not use the implication only, I use the full equivalence. Of > > course the equivalence includes the implication [***] as well as the > > implication [**] > > Then WM is not operating in the same world as everyone else.
Most of your ilk intermingle the worlds in which they work. They work in potential infinity if the question is put whether there is a largest natural number. But they work in actual infinity if the Cantor- diagonal or any other irrational number is concerned. Only the possibility to complete it by a limit process, leads to the erroneous assumption of 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.
> > > Or if I > > taught, contrary to fact, Cantor's claim that a real number is the > > limit of its finite initial segments but all real numbers are not the > > limits of all their finite initial segments? > > That may be WM's claim, but it is certainly not Cantor's. > Cantor may have viewed reals as equivalence classes of Cauchy sequences, > or, more likely, as Dedekind cuts, but where did WM get his delusion.
Cantor did not hold Dedekind's work in high esteem. Here is one example: Die Schrift von Dedekind ?Was sind und was sollen die Zahlen" ist, wenn auch ihrer Tendenz nach, die Arithmetik rein logisch zu begründen, lobenswerth, nicht nach meinem Geschmack. ... Das künstliche System der 172 sich nur um das Elementarste und zum Theil Trivialste drehenden Dedekindschen Sätze scheint mir mehr geeignet, die Natur der Zahlen zu verdunkeln als sie aufzuhellen. [Letter from Cantor to Vivanti, 1888, April 2]
Cantor used his Fundamentalreihen:
Bemerkungen mit Bezug auf den Aufsatz: Zur Weierstraß-Cantorschen Theorie der Irrationalzahlen. [Math. Annalen Bd. 33, S. 476 (1889).]
Es möge mir gestattet sein, nur ganz kurz auf die Bedenken zu antworten, welche Herr Illigens in bezug auf meine Theorie der Irrationalzahlen ausgesprochen hat. ... sqrt(3) ist also nur ein Zeichen für eine Zahl, welche erst noch gefunden werden soll, nicht aber deren Definition. Letztere wird jedoch in meiner Weise etwa durch
(1,7, 1,73, 1,732, ...)
Cantor talks about "his" theory of irrational numbers, neither Cauchy's nor Dedekind's.