On 11 Dez., 03:50, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > In article <fec95b83-39c5-4537-8cf7-b426b1779...@k17g2000yqh.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> 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?
The answer is an explicit no. "Most" here concerns the magnitude of numbers involved. There was much ado about inaccessible cardinals. > > > > 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.
Why that? Group, ring and field are treated in my lessons.
> > > > > 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.
You believe in infinite paths. But you cannot name any digit that underpins your belief. Every digit that you name belongs to a finite path. Every digit that is on the diagonal of Canbtor's list is a member of a finite initial segment of a real number.
You can only argue about such digits. And all of them (in form of bits) are present in my binary tree. > > > > > 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?
It exists in that fundamentally arithmetical way: You can find every bit of it in my binary tree constructed from finite paths only. You will fail to point to a digit of 1/3 that is missing in my tree. Therefore I claim that every number that exists is 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.
Isn't a path a sequence of nodes, is it? Everey node of 1/3 (that you can prove to belong to 1/3) is in the tree.
> Similar for 'pi' and 'e'.
Yes. Every digit is available on request.
> 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.
Wrong. Not only "apparantly" but provably (on request): Every digit of every real number that can be shown to exist exists in the tree.
Or would you say that a number, every existing digit of which can be shown to exist in the tree too, is not in the tree as a path?