In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> In unary or in von Neumann's representation there is the whole > counting process incorporated in each number. In decimal, you need > only count the powers of 10. Anyhow addressing a number means > counting.
I can address sqrt(2) other than by counting.
> > Of course you can check the n-th element before checking any other. > But in order to find the n-th element, you have to count from 1 to n, > either in unary or at least in the powers of 10. Therefore it is not > correct to talk about simultaneity.
The problem of accessing a number is quite different from the problem of its existence. There are even numbers which exist in certain contexts none of which are individually accessible.
> > > > Assuming countability again. > > Countability of nodes only. I am only interested to cover every node, > i.e., to exhaust the tree. I am a follower of Eudoxos.
Countability of members of set of nodes does not imply countability of subsets of that set. > > > Suppose we have the set of paths where each > > path goes to the right after some specific (for that path) node. > > That would imply all paths with tails 111... No problem. > > > Is there > > a node in your tree that is not occupied by any of the paths in that set? > > No, it isn't. Every path node is occupied by at least the head of a > path. > > > Are there possible other paths that are *not* in that set of paths (like > > a path that alternates going left and right)? > > It is said so. But there is no possibility, after having completed the > construction, to introduce such a path. They have sneaked in.
However sneaky, they are there, in the sense that there are sets of nodes that satisfy the definition of being paths that were not built in by WM's construction but which appeared despite he attempts to eliminate them.
So to keep those unwanted paths out, WM will have to redefine paths so that only sets of nodes explicitly constructed can be counted as paths.
> > True is: There are only rational sums of decimal or binary expansions.
True is: any sequence of decimal digits following "0." which can be unambiguously described defines a real number in [0,1]. And that includes lots of irrationals.
> > No. The limit is not rational. But the limit cannot be represented by > a sequence. Therefore the limit is not in any Cantor-list. That is the > big error of set theory: to believe that irrational numbers have > rational representations. Every binary or decimal sequence is a > rational representation.
There are real numbers which have precise definitions but no known decimal expansions.
WM conflates a particular form of number name with the number itself. A name is not the thing named.
And there are, in general, lots of different names for any number, any one of which suffices to establish its existence. > > I skip the rest because the main point now has become fairly clear: > The paths in the binary tree. We should concentrate on that problem.
As soon as one has the set of all nodes of members of an infinite sequence of nested finite binary trees, one has a countable, partially ordered by the transitive closure of the 'parent of' relation set of nodes in which every maximal totally ordered subset is, by definition, a path.
So there are uncountably many such paths in that tree.