In article <bfcd6301-e7f0-4ced-906d-c111d378429e@j32g2000yqh.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 11 Jun., 15:46, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > In article > > <8d9c63c5-b605-4a98-81f2-815875aae...@21g2000vbk.googlegroups.com> WM > > <mueck...@rz.fh-augsburg.de> writes: > > > On 4 Jun., 04:04, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > > > > That is the logic of potential infinity, i.e., of incomplete sets. > > > > > > > > Eh? As far as I know logic is *not* about sets. It is about axioms, > > > > statements and inference rules. > > > > > > Logic has been obtained from the bahviour of parts of reality that can > > > be called sets. > > > > O. Still, as far as I know logic is *not* about sets. > > > > > > > In Cantor's diagonal argument you can use the same logic : There is > > no > > > > > last line, therefore there is always a line beyond the checked > > lines, > > > > > But there you don't. > > > > > > > > Because you do not check the lines in order. It is always your basic > > > > assumption that you first check the first line and after that the next > > > > line. That is wrong. > > > > > > That is necessary because you cannot find the n-th line unless you > > > know the line number n - 1 or some equivalent mark. > > > > You are wrong. A list is a mapping from N to the elements of the list. > > Through that list, given a number n, you find the n-th element of the list > > without referring to any previous elements of the list. > > But you cannot find number n without referring to the numbers less > than n.
Nor without equally implicitly referring to the numbers greater than it. > > > To give an example, > > let's have a list of positive rational numbers through the mapping given a > > lnong time ago by David Tribble. To get the 51st element of the list, we > > calculate the mapping: write 51 in binary, create from it a continued > > fraction > > as described on > > <http://homepages.cwi.nl/~dik/english/mathematics/mueck/mapping.html> > > which is [0, 1, 2, 3], calculate it and we find 7/9. So we know the 51st > > element without knowing the 50th element. Where did I come at the 47th > > rational in that list (which is 7/11)? > > How can you write 51 without knowing what it is? Of course you must > count.
I can write lots of numbers by suitably randomized processes without even knowing the number I write, much less what numbers might precede it.
> > > And in my binary tree I give a definition of an end of a path that > > > contains aleph_0 nodes. > > > > I have not seen such a definition. What is the definition of "end of a > > path"? > > Virgil calls it tail. That is a good name. > I begin the construction with one path p_0. Then I construct another > path p_1. All nodes that p_1 does not have in common with p_0, is the > tail of p_1. All nodes of the tail are mapped on p_1. Then p_2 is > constructed. All nodes of p_2 that differ from p_0 and p_1 are mapped > on p_2. And so on. You see the ratio of paths and nodes per path at > any stage during the construction is 0.
Actually, WM is still suffering from a misunderstanding of the definition of a path. That definition does snot require any sort of "construction" such as Wm envisions.
The "parent of" relation between nodes generates an "ancestor of" partial order on the nodes of a tree in the obvious way. A subset of the set of nodes of that tree is a path if and only if (1) it is totally ordered by the ancestor relation., so that for any two of its nodes one must be an ancestor of the other, and (2) it is maximal among such totally ordered sets, so that there is no node that can be to it added without destroying the total property.
Every set of nodes in every tree satisfying both properties is a path, and every set of nodes failing either is not a path.
Those properties, and only those, confer pathhood.
For WM to claim that any set of nodes in a tree which satisfies those properties is not a path is both false and stupid, but he persists in doing so anyway.
> > I don't see a difference. But I can assuer you, there is no decimal > expansion for irrational numbers.
WM's assurances have to often proved false to be trusted.
Besides which, who says that the existence of a decimal expansion is required for a number to exist? For a particular form of numeral, but there is no requirement that any number have any particular form of name before it can exist as as number. At least not in serious mathematics.
> > If one rejects actual infinity, then there is not "a continuous > function nowhere differentiable".
Then there are also no continuous functions at all, and nothing to replace them with.
