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.
> > 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
> > 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. > > > 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.
> 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! Similarly omega is not a natural number and the number of natural numbers is not omega.
> > 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. > > > 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.
