In article <5df917e3-517f-4a38-b28b-3638434966c7@t21g2000yqi.googlegroups.com> WM <mueckenh@rz.fh-augsburg.de> writes: > On 29 Mai, 03:58, "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 [**] ... > > > > > 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.
What is the relevance of this? The logical rules of unions state that when you have a union of sets that union does contain an element if it is in one of the sets. There is no difference between finite unions and infinite unions.
> The > logical rules of linear sets are obtained from finite linear sets.
Aha, here we get to the heart of the matter. You do not believe in infinite sets (or infinite unions or infinite sets of finite linear sets). That is possible, of course, but does not rule out theories in which those things do exist. Quoting philosophers of some time ago does not make that wrong.
> > > 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.
Being incomprehensible again. The reverse of *what*? The reverse of [**] is (as I wrote) [***]. You may never have said so, but it is. To recap: [*] En Am: m =< n <==> Am En m =< n [**] En Am: m =< n ==> Am En m =< n [***] En Am: m =< n <== Am En m =< n
But you ask as a counterexample something where [**] is true but [*] false. But that is the wrong way around. Whenever [**] is true, [*] is also true. What is contested is that there are case where [***] is true and [*] false. And it is the latter implication that you do use.
> > > > 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.
Now you are using words that are again completely incomprehensible. You have still failed to give a definition of "potential infinity" that is valid within ZF. Moreover, you have not proven (within ZF) that the statement is wrong.
> > (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.
Can you provide me with definitions within ZF that shows the difference?
> 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?
Because there is no such n. Remember: the set of natural numbers has no largest element in ZF.
> > 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.
Oh. I think that the term "classical logic" has changed a bit since the last time you looked at it. And, if you refer to Weyl's quote, he stated that classical logic was *derived* from the logic on finite sets, not that it was *identical* to logic on finite sets.
> > > 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.
Not "a complete linear set". But "a complete finite linear set". Why do you drop the word "finite" in the second sentence? To obfuscate?
> You drop the completeness condition in certain cases but you assume it > in case of Cantor's proof. That is cheating.
You again misunderstand the proof completely. There is an assumption that a complete list is provided and that is proven false.
> > > > > > 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.
What in the world is a "potentially infinite element"? And, in ZF all sets are static, there is no place for non-static sets.
I will recap my proof, I assume the axiom of infinity. That is all.
> > (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.
> > 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.
Where in the proof do I use an element m when I chose n = m - 1? In the proof I chose only elements that *follow* given elements, as is assured by the axiom of infinity. Quantifier dyslexia on (3)?
> 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.
This is really incomprehensible.
> > > 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.
Clearly, the only reason is that you want it to be so. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
