On 29 Mai, 03:58, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > In article <37a98f77-6bc9-4cae-88e5-fcbadd1d0...@q2g2000vbr.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes: > > On 27 Mai, 15:50, "Dik T. Winter" <Dik.Win...@cwi.nl> 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 [*]. > > > > > > > > > > The above is not contested: [**] implies [*]. > > > > > > > > I said: For complete linear sets [*] is true. > > > > > > Not in the article to which I responded. > > > > But frequently I made use of what you call quatifier exchange and what > > is allowed in case of complete linear sets. > > You think so, but you have to prove that it is valid for infinite complete > linear sets. Note that "classical logic is obtained from finite sets". > Nowhere in that quote the word linear is mentioned.
Nowhere in that quote the word union in mentioned. Nevertheless the logical rules of unions are obtained from unions of finite sets. The logical rules of linear sets are obtained from finite linear sets.
> > > > You said [*] is not true, but [**] is true. > > > > > > That is not what I said. > > > > You said: > > The only thing that can be stated is (symbolically): > > E n A m P(m, n) -> A m E n P(m, n) > > not the reverse, this is just basic logic. > > The reverse of > E n A m P(m, n) -> A m E n P(m, n) > is > E n A m P(m, n) <- A m E n P(m, n) > which is [***], neither [*] nor [**].
I never said so. But [*] is [**] & [***]. Therefore [*] differs from [**] only by the reverse. > > > > I said that for the case involved you have to > > > *prove* that it is true, because it is not generally true. > > > > It is generally true for complete linear sets. You have to prove that > > it is not. > > It is not true for the infinite set of naturals.
That is your claim. It is justified for potential infinity. It is wrong for complete sets.
> (1) define FISON(n) be the set of naturals from 1 to n, that is: {1, ..., n}. > (2) A{m in N} E(n in N} such that FISON(m) subset FISON(n), trivial, take > n = m + 1. > (3) E{n in N} A{m in N} such that FISON(m) subset FISON(n), trivially false, > take m = n + 1. > Which part of this proof is wrong?
The proof is correct for potential infinity. The proof is incorrect for actual infinity. In that latter case you need not take an n that is surpassed by m. Why don't you start with an n that has no greater m? > > It clearly shows that > E n A m P(m, n) <- A m En P(m, n) > is false. Here with: > E n meaning E{n in N} > A n meaning A{n in N} > P(m, n) meaning FISON(m) subset FISON(n). > The first part of the implication is false while the second part of the > implication is true, and so the implication is false (all by classical > logic).
Not at all. By classical logic, a complete linear set has a last element. > > > > But now you include the word "linear". Where did "Weyl" include the > > > word "linear"? > > > > I did never claim that quantifier exchange is allowed in case of non- > > linear sets, like cyclic sets as, for instance, your dice. That would > > be nonsense. A simple example: Every country has a country that lies > > west of it. But there is no country that lies west of all countries. > > But as Weyl did not include "linear" in his words, how can that quote > support your claim?
There are many finite sets with many special properties that follow from classical logic. One of them is that a complete linear set has a lst element.
You drop the completeness condition in certain cases but you assume it in case of Cantor's proof. That is cheating. > > > > > > What is contested is that: > > > > > En Am: m =< n <== Am En m =< n [***] > > > > > implies [*]. And *that* is the form you do use. > ... > > > Because the implication [**] is always true, the only part of the > > > equivalence that is new is the implication [***]. > > > > For complete linear sets both are true, therefore [*] holds. > > You just state without proof. Where in my proof above that it is false > did I go wrong?
State before beginning whether the set that you assume is complete and static, i.e., every element is actually existing, or potentially infinite. > > > And you > > should recognize that actually complete linear sets obey [*]. > > Strange, I give above a proof that it does not hold. I did not use > "actual infinity" nor "potentially infinity"
That is the point! You use the absence of element m when you choose n = m - 1. But you use the non-absence of m when you execute Cantor's proof. Then you do not admit that for every FISON(n) there is an m = n + 1 that is not in the proof.
> > > Only > > potentially infinite sets do not. But you mix up things. You claim the > > existence of a complete linear set but disregard the necessary > > consequence of completeness or linearity, namely the validity of [*]. > > Why is that a necessary consequence? Because you want it to be so? Can > you give a *mathematical* reason?
Every finite linear set obeys [*]. That is the mathematical reason.
