In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 20 Jun., 16:33, Newberry <newberr...@gmail.com> wrote: > > On Jun 19, 6:06 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote: > > > > > Newberry <newberr...@gmail.com> writes: > > > >> Because every infinite sequence of digits represents a real number? > > > >> And > > > >> the antidiagonal is one such sequence? > > > > > > If it does not exist then it does not represent anything let alone a > > > > number. > > > > > > Now it is clear that it does not exist. Since all the reals are on the > > > > list and the anti-diagonal would differ from any of them. This > > > > violates the assumption. Hence the anti-diagonal does not exist. > > > > > Wow. Are you saying that the *sequence of digits* specified by the > > > anti-diagonal does not exist? > > > > That's right. There is no formula or algorithm to construct the list. > > It means that you would have to flip each and every digit one by one. > > And that is impossible. > > Well, there are some lists defined by finite definitions like > 0.1 > 0.11 > 0.111 > ... > > But the number of these lists and the number of their diagonals is > countable. > > > > > Anyway, in a sense, you're right. If we assume that every real is > > > represented by a sequence on the list, then we can prove that every > > > sequence occurs on the list (ignoring the issue of multiple > > > representations). And yet, we can also show that a particular sequence > > > is not on the list. > > > > Which sequence is that? > > O, don't be unfair. Matheolgicians "prove" their claims. They are not > used to give examples. Remember Zermelo who "proved" that every set > can be well ordered. Although every sober mind today knows that > hisproof is wrong, set theory is built upon this lie.
Given the axiom of choice, every set is in principle well orderable.
But without the axiom of choice, it is not always possible.
So WM must be rejecting the axiom of choice for himself.
But he does not have either the right, or, more importantly, the power to impose that rejection on everyone else.