In article <53058a2b-9675-469e-b3b2-095661656ef3@e5g2000yqn.googlegroups.com>, Newberry <newberryxy@gmail.com> wrote:
> On Jun 23, 9:09 pm, Virgil <Vir...@home.esc> wrote: > > In article > > <b0258a2f-1679-492d-99e9-3397093bb...@t10g2000yqg.googlegroups.com>, > > > > Newberry <newberr...@gmail.com> wrote: > > > On Jun 23, 1:49 pm, Virgil <Vir...@home.esc> wrote: > > > > In article > > > > <8cd5db0d-46a5-4deb-8fe5-9a28acad5...@k39g2000yqb.googlegroups.com>, > > > > > > Newberry <newberr...@gmail.com> wrote: > > > > > Cantor's proof starts with the assumption that a bijection EXISTS, not > > > > > that it is effective. > > > > > > Actually, a careful reading shows Cantor's proof merely assumes an > > > > arbitrary INJECTION from N to R (originally from N to the set of all > > > > binary sequences, B) which is NOT presumed initially to be surjective, > > > > and then directly proves it not to be surjective by constructing > > > > > OK, so construct it assuming injection. > > > > It is simplest to do for an injection from N to B. > > A member of B, a binary sequence, is itself a function from N to an > > arbitrary two element set which, without loss of generalization, we may > > take to be S = {0,1}. > > A list of such functions is then equivalent to a single function from > > NxN, the Cartesian product to {0,1}, say F(-,-), so that for each m in > > N, the function F(m,-): N -> {0,1}: m |--> F(n,m) is one of the binary > > functions in the list. > > For any such list of binary functions, define a new function > > g(-):N -> {0,1}:n |--> 1 - F(n,n), or more briefly, g(n) = 1-F(n,n). > > > > This new function is the desired "antidiagonal" for the list of lists > > F(-,-) and g(-) is not in the original list since g(-) differers from > > each F(n,-) at n. > > Now costruct it when F is not effectvely computable.
I don't have to construct it, merely define it. > > > > > > > > > > > > > > > the > > > > "antidiagonal"as a member of the codomain not in the image. > > > > > > Thus proving that ANY injection from N to R (or B) fails to be > > > > surjective. > > > > > > For some unknown reason, the DIRECT "anti-diagonal" proofs given by > > > > Cantor, and his followers, are often misrepresented as being proofs by > > > > contradiction, but they never were.
