In article <firstname.lastname@example.org> WM <email@example.com> writes: > On 8 Dez., 16:07, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > > > You have seen the axiom of infinity. It say that an infinite set > > > exists and that implies that infinitely many elements of that set > > > exist. That is actual infinity. > > > > Oh, so actual infinity means that a set with infinitely many elements > > exists? > > Yes. > > > In that case you should reject the axiom of infinity. You are allowed > > to do that, and you will get different mathematics. But you can not > > claim that mathematics with the axiom of infinity is nonsense just > > because you do not like it. But go ahead without the axiom of infinity, > > I think you have to redo quite a bit of mathematics. > > Before 1908 there was quite a lot of mathematics possible.
Yes, and since than quite a lot of newer mathematics has been made available. Moreover, before 1908 mathematicians did use concepts without actually defining them, which is not so very good in my opinion.
> > > The definition of an actually infinite set is given in set theory by > > > the axiom of infinity. > > > > You are wrong, the axiom of infinity says nothing about "actually > > infinite set". Actually the axiom of infinity does not define anything. > > It just states that a particular set with a particular property does > > exist. > > That is just the definition of actual infinity.
I see no definition, so what *is* the definition?
> > > The definition of a potentially infinite set is given by > > > 1 in N > > > n in N then n+1 in N. > > > > That does not make sense. Without the axiom of infinity the set N does > > not necessarily exist, so stating 1 in N is wrong unless you can prove > > that N does exist or have some other means to have the existence of N, > > but that would be equivalent to the axiom of infinity. > > N need not exist as a set. If n is a natural number, then n + 1 is a > natural numbers too. Why should sets be needed?
Ok, so N is not a set. What is it?
> > > The complete infinite binary tree can be constructed using countably > > > many finite paths (each one connecting a node to the root node), such > > > that every node is there and no node is missing and every finite path > > > is there and no finite path is missing. > > > > Right. > > > > > Nevertheless set theory says that there is something missing in a tree > > > thus constructed. What do you think is missing? (If nothing is > > > missing, there are only countably many paths.) > > > > And here again you are wrong. There are countably many finite paths. > > There are not countably many infinite paths, and although you have tried > > many times you never did show that there were countably many infinite > > paths. > > There is not even one single infinite path!
Eh? So there are no infinite paths in that tree?
> But there is every path > which you believe to be an infinite path!! Which one is missing in > your opinion? Do you see that 1/3 is there?
If there are no infinite paths in that tree, 1/3 is not in that tree. Otherwise 1/3 would be a rational with a denominator that is a power of 2 (each finite path defines such a number).
> What node of pi is missing in the tree constructed by a countable > number of finite paths (not even as a limit but by the axiom of > infinity)?
By the axiom of infinity there *are* infinite paths in that tree. So your statement that there are none is a direct contradiction of the axiom of infinity. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/