Am Samstag, 7. Oktober 2017 01:48:42 UTC+2 schrieb Shobe, Martin: > On 10/6/2017 3:22 PM, WM wrote:
> > There are aleph_0 steps completing all paths, all are distinct, each one is completed by its own step. If so, then there are not more than aleph_0 paths. > > None of the paths are completed at any step.
Of course not. But set theory predicts completion. "We get to them simply by counting beyond the normal countable infinite, i.e., in a very natural and uniquely defined consistent continuation of the normal counting in the finite." [D. Hilbert: "Über das Unendliche", Math. Annalen 95 (1925) p. 169]
Set theorists can construct the complete Binary Tree as far as its nodes and edges are concerned. Unfortunetely this includes the completion of all paths.
Alas even "in the infinite" a path cannot branch into two paths without creating a node. Because a node is defined as a branching point, no increase in paths is possible without the same increase in nodes.
Not necessary to mention, at every level the cross-section of the Binary Tree, i.e., the number of nodes at that level, is finite. And, as an upper estimate, even lining up all nodes of the Binary Tree on a single level would limit the set of paths to aleph_0.