In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 16 Aug., 19:17, Dick <DBatche...@aol.com> wrote: > > > Situation 1. You want transfinite sets to exist. > > Simple. Adopt the usual axioms of set theory. Then:_ > > you get a contradiction:
Maybe so in WM's matheology, but not in mathematics. > > Supertask 1: > Fill in 1 to 10, remove 1, > fill in 11 to 20, remove 2, > fill in 21 to 30, remove 3, > and so on. > After all, all numbers are removed. > > > Supertask 2: > Fill in 1 to 10, remove 10, > fill in 11 to 20, remove 20, > fill in 21 to 30, remove 30, > and so on. > After all, not all numbers are removed. > > But the quantities are identical in both cases for every finite step.
That differing sequences should have differing results in not a problem in mathematics however much it may be in WM's matheology. > > Or: > > Definition: Two infinite paths, A and B, of the Binary Tree can be > distinguished at level n > <==> There are two different nodes a and b at level n, such that a is > in A and b is in B. > > Definition: Two infinite paths A and B are different <==> There exists > a level n such that A and B can be distinguished at level n. > > Therorem: There is no level n where uncountably many path can be > distinguished. > > Corollary: In the complete infinite Binary Tree at most countably many > paths can be distinguished.
Does not follow unless there is a level at which all paths can be distinguished, which is not the case.
So that WM's matheology leads him, as usual, to false claims.