In article <firstname.lastname@example.org> WM <email@example.com> 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.
> > > 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.
> 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. Can you show how any theory states that? -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/