In article <firstname.lastname@example.org> WM <email@example.com> writes: > 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.
Nothing more than opinion while you have no idea what has been done in mathematics since 1908. Algebraic number theory is rubbish?
> > 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?
Something that satisfies the axioms of ZF (when you are working within ZF). It is similar to the concepts of group, ring and field. Something that satisfies those axioms is such a thing. But I think you find all those things rubbish.
> > > 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.
Within ZF a sequence is an ordered set. But as you refuse to distinguish beteen an ordered set and a non-ordered set, I think this goes beyond you.
> > > 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 believe in infinite paths, you state they are not in the tree. So we have a direct contradiction to your assertion.
> > > 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. > Everything of that kind is in the tree.
This makes no sense. Every path in the tree (if all paths are finite) is a rational with a power of 2 as the denominator. So 1/3 does not exist as a path. In what way does it exist in the tree?
> > 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.
In what way do numbers like 1/3 exist in your tree? Not as a path, apparently, but as something else. Similar for 'pi' and 'e'. So when you state that the number of paths is countable that does not mean that the number of real numbers is countable because there are apparently real numbers in your tree without being a path. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/