In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 10 Dez., 16:35, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > > > > Before 1908 there was quite a lot of mathematics possible. > > > > Yes, and since than quite a lot of newer mathematics has been made > > available. > > Most of it being rubbish.
WM can only speak for the quality of his own mathematics. > > > Moreover, before 1908 mathematicians did use concepts without actually > > defining them, which is not so very good in my opinion. > > Cantor gave a definition of set. What is the present definition?
There are a number of them, but in the better set theories, sets are only "defined" by asserting their properties as axioms.
Thus anything that satisfies all the axioms in one of those systems of axioms is a set in that system. > > > > > > 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? > > > > Ok, so N is not a set. What is it? > > N is a sequence of natural numbers.
There are no sequences which are not also sets. All it takes is a way to distinguish the elements of a sequence form non-elements to make it set. At least in many set theories. What is WM's definition of a sequence that prohibits its members from being all the members of some set? > > > > There is not even one single infinite path! > > > > Eh? So there are no infinite paths in that tree? > > In fact no, but every path that you believe in is also in the tree, > i.e., you will not be able to miss a path in the tree.
I, and everyone sane, will "miss" absolutely every path in WM's perversion of a "complete infinite binary tree" since in such a tree no finite set of nodes, which is all WM allows, covers a *complete* path. > > > > > 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? > > > > If there are no infinite paths in that tree, 1/3 is not in that tree. > > 1/3 does not exist as a path. But everything you can ask for will be > found in the tree.
I ask for 1/3 as a path in MY complete infinite binary trees. Along with all other rationals and reals in [0,1].
> Everything of that kind is in the tree.
What is in my tree is mostly missing in yours, which includes only those binary rationals in [0.1]. > > > Otherwise 1/3 would be a rational with a denominator that is a power of > > 2 (each finite path defines such a number). > > > > > 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)? > > > > By the axiom of infinity there *are* infinite paths in that tree. So your > > statement that there are none is a direct contradiction of the axiom of > > infinity. > > Try to find something that exists in your opinion but that does not > exist in the tree that I constructed.
Infinite paths as sets of nodes exist in my opinion, and that of most mathematicians, and they all require a set, or sequence, containing infinitely many nodes, which your sort of binary tree does not contain but everyone else's does. Note that outside of Wolkenmuekenheim, all infinite sequences are just special types of infinite sets (functions having N as domain).