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. > > > The axiom of infinity is adefinition 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. 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.