In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> 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. > There was quite a lot of possible mathematics.
Most of which has been extended considerably since.
Even if WM wants to restrict himself to only those parts of mathematics which precede 1908, he has not the right, or the power, to impose such restrictions on anyone else, except possibly his poor misled students. > > > > 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.
That may be WM's definition of "actual infinity", but his definitions carry weight only in Wolkenmuekenheim, and are of no importance anywhere else.
> > > 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?
If WM chooses to work in a mathematics without sets, he is quite free to do so, but has no power to impose such limits on anyone else, except possibly his poor captive students. > > > > > 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!
Then one does not have a COMPLETE infinite binary tree, but only a non-set of (finite) nodes.
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?
In your model I do not see any infinite paths, since every infinite path requires an infinite set of nodes and your model disallows all infinite sets. > > 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)?
Unless one is allowed to have a SET OF INFINITELY MANY nodes, one can not have even the fractional part of pi.
Similarly for any proper fraction whose denominator is not a power of 2.
A sequence which is not (actually) infinite (an image of the SET N), must have a last term, and thus be incapable of converging.