In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 7 Dez., 16:41, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > In article > > <5f8c7e7f-ec83-45ba-a584-b61475969...@j4g2000yqe.googlegroups.com> WM > > <mueck...@rz.fh-augsburg.de> writes: > > > On 3 Dez., 16:27, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > > > > > > > > No. I asked you for a mathematical definition of "actual > > infinity" > > > > > > and you told me that it was "completed infinity". Next I asked > > you > > > > > > for a mathematical definition of "completed infinity" but you > > have > > > > > > not given an answer. So I still do not know what either "actual > > > > > > infinity" or "completed infinity" are. > > > > > > > > > > Both are nonsense. But both are asumed to make sense in set theory. > > > > > > > > No, set theory does not contain a definition of either of them. > > > > So in what way do they make sense in set theory? As I have not seen a > > definition of them at all, I can not see what that means. > > 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.
As yet, no one has shown that the axiom of infinity causes any problems with mathematics, at least in ZF or NBG, though it does cause some in those who are trying to be mathematicians. > > > > > > > The axiom of infinity is a definition of actual infinity. > > > > > "There *exists* a set such that ..." > > > > > Without that axiom there is only potential infinity, namely Peano > > > > > arithmetic. > > > > > > > > I see neither a definition of the words "actual infinity" neither > > > > a definition of "potential infinity". Or do you mean that "potential > > > > infinity" is Peano arithmetic (your words seem to imply that)? > > > > > > > > So we can say that in "potential infinity" consists of a set of > > axioms? > > > > > > Here is, to my knowledge, the simplest possible explanation. > > > > Darn, I ask for a definition, not for an explanation. > > The definition of an actually infinite set is given in set theory by > the axiom of infinity. You should know it or know where to find it. > (You can look it up in my book.) > > The definition of a potentially infinite set is given by > 1 in N > n in N then n+1 in N.
If N is to be a set, then it must be actually infinite by your own definition. And if it is not to be a set, then it is improper to even speak of "all naturals". > > > > > Consider > > > the infinite binary tree: > > > > > > 0 > > > /\ > > > 0 1 > > > /\ /\ > > > 0 1 0 1 > > > ... > > > > > > Paint all paths of the form > > > 0.111... > > > 0.0111... > > > 0.00111... > > > 0.000111... > > > and so on. > > > Potential infinity then says that every node and every edge on the > > > outmost left part of the tree gets painted. > > > Actual infinity says that there is a path 0.000... parts of which > > > remain unpainted. And that is wrong. > > > > So "potential infinity" and "actual infinity" are theories? But I do not > > know of any theory that states that there is any part of the path 0.000... > > that is unpainted. > > That is the inconsistency of set theory.
Since, as far as I am aware, no set theory says that, the only 'inconsistency' is in WM's understanding.
Which is why his math in Wolkenmuekenheim is such a mess. > > 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.
WM allows that he set of nodes can be so formed even in WM's world , but then in WM's world, he says the set of paths cannot be formed. > > Nevertheless set theory says that there is something missing in a tree > thus constructed. What do you think is missing?
The set of paths. Note that, as a set of nodes, each path must be an infinite set such that for each pair of nodes one is an initial string of the other.
So that, among other things, in WM's world the set of nodes is not allowed to have a power set.
> (If nothing is > missing, there are only countably many paths.)
According to WM's rules there can be no set of paths to count, so it can't be countable.