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?
> 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. There was quite a lot of possible mathematics.
> > 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. > > > 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?
> > 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! 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?
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)?