In article <email@example.com> WM <firstname.lastname@example.org> writes: > On 7 Dez., 16:41, "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.
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.
> > > > > 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 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.
> 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.
> > > 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.
What is the inconsistency. Apparently you think that "actual infinity", whatever that may be, apparently at this point a theory, states that parts of 0.000... remain unpainted. As I do not know of any theory that states that I wonder whether you could give me an outline of such a theory. Set theory does not state that.
> 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.
> 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. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/