In article <557d4c53-cba7-4fca-aec2-aa62770788ec@l32g2000vba.googlegroups.com> WM <mueckenh@rz.fh-augsburg.de> writes: > On 4 Jun., 04:04, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > > The word union and the word linear are not mentioned in the quote. > > > Nevertheless the due logica rules were obtained from unions of finite > > > sets and linear finite sets. > > > > Yes, obtained from. But that does not mean they are identical. Moreover, > > does what Weyl wrote a long time ago still have validity now? > > It is 50 years younger than Canto's writings.
What is the relevance?
> > > But it does rule out theories which are contradicted by the > > > fundamental logical rules. And one of these rules is that a complete > > > linear set has a last element. > > > > What fundamental rules of logic are you using? I have never seen such a > > logical rule, because logic does not talk about sets. > > Logic states that the union of a *complete* set of finite linear sets > is a finite linear set
As I said, logic is not talking about sets. So where in logic is such stated?
> > > No. I use the fact that for complete linear sets always both > > > implications are true : > > > [**] & [***]. This means that [*] is true. > > > > You just state so without proof. > > A proof is a derivation of theorems from axioms or basic truths by > means of rules of logical inference. These rules themselves cannot be > proven but can only be obtained from the behaviour of existing (i.e., > finite) sets.
So you are not using mathematical logic?
> > > If you disagree, then you should come up with a finite linear set for > > > which only one implication is true. > > > > Why should I show that for a *finite* linear set? Why not for an > > *infinite* linear set? You are *still* thinking that wat is valid > > for finite things is also valid for infinite things. > > There is no reason to believe that always the contrary is true.
I do not believe so, so what is the relevance?
> > So in your opinion: > > sum{i = 0..oo} 1/(i!) = lim{n -> oo} sum{i = 0..n} 1/(i!) > > is rational, because it is rational for each n, and so your logic > > requires that. Also: > > lim{n -> oo} 1/n > 0 > > because it is > 0 for each n, and so should also be > 0 in the infinite > > case. > > The limit does not belong to the series!
According to your reasoning it does belong to the complete series.
> Similarly omega is not a natural number and the number of natural > numbers is not omega.
What is in this context the relevance of the second part of the statement?
> > > ZF uses potential infinity whenever the validity of > > > En Am: m =< n <== Am En m =< n [***] > > > for linear sets is denied. > > > > And you still do not answer my question. You fail to give a definition > > that is valid in ZF, so I have no idea what you are talking about. > > Potential means not complete. There always appears another number once > you have found the last one.
Your definition of complete is not the standard definition. In N there is always one after the other, but the complete set does exist.
> > > ZF claims that this denial is correct for complete linear infinite > > > sets, but this is a wrong claim, as we can obtain from logic. > > > > What logic? Not the logic as discussed in sci.logic. > > No, I mean the correct logic.
I still do not see the logic through which you obtain it. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
