In article <email@example.com> WM <firstname.lastname@example.org> writes: > 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.
That is a pretty strange usage of the word "most" in the context of "most newer mathematics". But apparently you think Graham's number 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. > > Why that? Group, ring and field are treated in my lessons.
You think that something that satisfies the ZF axioms being a collection of sets is rubbish, while something that satisfies the ring axioms being a ring is not rubbish?
> > > > > 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.
Right. But there is no finite path that contains them all. I believe in a path that contains them all, and that is an infinite path.
> Every digit that is on the diagonal of Canbtor's list is a > member of a finite initial segment of a real number.
Right, but there is no finite initial segment that contains them all.
> You can only argue about such digits. And all of them (in form of > bits) are present in my binary tree.
Right, but your tree does not contain infinite paths, as you explicitly stated.
> > > 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.
In that case you have a very strange notion of "existing in the tree". Apparently you do *not* mean "existing as a path". So when you say that the number of (finite) paths is countable, I agree, but 1/3 is not included in that, because it is not a path according to your statements.
> > 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?
Apparently not in your tree. In your tree a path is a finite sequence of nodes.
> Everey node of 1/3 (that you > can prove to belong to 1/3) is in the tree.
Right, but there is no path that denotes 1/3.
> > Similar for 'pi' and 'e'. > > Yes. Every digit is available on request.
Right, but there is no path that denotes either 'pi' or '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. > > Wrong. Not only "apparantly" but provably (on request):
There is no proof needed. Apparently there are real numbers in your tree without being a path, because each path is finite (by your own definition).
> Every digit of > every real number that can be shown to exist exists in the tree.
But not every real number is represented in the tree by a path.
> 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?
Yes, by your own admissions. You state (explicitly) that every path is finite and it is easy to prove that every number that is represented by such a path is a rational number with a denominator that is a power of 2. So there are apparently real numbers of which every digit is in the tree that are not represented as a path, like 1/3. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/