In article <278180e4-8675-40bc-871c-a84af38f6635@k20g2000vbp.googlegroups.com> WM <mueckenh@rz.fh-augsburg.de> writes: > On 4 Jun., 04:21, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > In article <5b9b7d5b-11d3-4125-8052-ad45e6e7a...@x5g2000yqk.googlegroups.= > com> WM <mueck...@rz.fh-augsburg.de> writes: > > > On 3 Jun., 04:25, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > > > There is no difference between finite unions and infinite > > > > unions. > > > > > > There should not be, but there is --- at least if ZF is taken to be > > > true. > > > > You are again wrongly interpreting things. There is no question about > > ZF being true or not. ZF is only one of the many possible theories. > > It is an impossible theory because finished infinity is impossible.
Again giving opinion only.
> > Mathematics is not a science that tries to find truth, > > There is a mathematics, namely basic arithmetic of integers, that is > right. And there are some logical rules that are right. And from these > foundations some theorems can be obtained that are right. There is no > need and no space for any axiom at all.
Oh. I would have thought that in that model the basic arithmetic of the integers forms the axioms...
> There is no need , in > particular, for "a set N" with some "binary relations". It is > impossible, to change that stuff. And Hilberts famous model including > 1 + 1 = 0 does nothing but express the display of the last wheel of a > binary calculator (or Leibniz's calculator that all time long had > problems with the digit transfer).
I think you are confusing a few things here. In your model you may not need a set N with binary relations, but mathematics tries to find the minimum that is required to get something done. From that follows fields like group theory, etc.
> > it is a science where > > it is determined what a given set of axioms (i.e. presupposed valid > > statements) leads to. > > A given set of axioms is sometimes impossible. This is so with the > axiom of infinity that Zermelo created, misunderstanding Dedekind's > idea of potential infinity.
There is no misunderstanding. And that it is impossible is just opinion, nothing more. Like earlier people thought that non-Euclidean space was impossible. And some people thought that irrational numbers were impossible, or negative numbers, or imaginary numbers.
> > > The union of FISONs (of natural numbers or of other finite linear > > > sets) is the last FISON. > > > > ZF does agree, when there is a last FISON. Your problem is that the > > axiom of infinity states that here is no last FISON of all FISONs. > > That is not my problem. For potentially infinite sets it is not a > problem at all.
ZF does not know about the impossible set.
> For a completely existing set ZF is wrong.
Again, just opinion.
> But that is > usually veiled by applying potential infinity whenever the problem > occurs.
ZF does not apply something that is not defined in 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/
