On May 3, 12:21 pm, Virgil <vir...@ligriv.com> wrote: > In article > <749e1561-d69b-45af-9001-3be3141ad...@ua8g2000pbb.googlegroups.com>, > Hercules ofZeus <herc.is.h...@gmail.com> wrote: > > > > > On Apr 21, 5:39 pm, Virgil <vir...@ligriv.com> wrote: > > > In article <Me6dnerBAcAL8O7MnZ2dnUVZ_rWdn...@giganews.com>, > > > > fom <fomJ...@nyms.net> wrote: > > > > This is a easy, readable paper of the same > > > > title by Kanamori. A historical analysis > > > > of how infinity entered mathematical discourse. > > > > >http://kurt.scitec.kobe-u.ac.jp/~fuchino/xpapers/infinity.pdf > > > > > OP: Marc Garcia at FOM > > > > > (Virgil -- you will find a familiar proof > > > > at the bottom of page 5) > > > > Yes! A nice version of it, too. > > > > And a nice paper which shows just how far out of any real mathematics WM > > > has put himself. > > > there is no _method_ to any of it though... > > > DEFINE digit1 is different to row1, digit2 is different to row2, and > > so on... > > therefore infinite strings are bigger sets than finite strings... > > > Its merely ONTO, SURJECTIVE definitions thrown directly onto > > a good optical effect of looking down the infinite plane at an angle > > > no new digit sequence is EVER constructed using this 'method' and this > > is provable. > > If you think it provable, then prove it to be provable by proving it. > > Ontherwise, what you merely claim is no evidence. > > > > ---- > > > what would be useful is a procedural system with > > > .. > > > INTERSECTION > > > in( S1, S2 ) <- E(X) XeS1 & XeS2 > > > .. > > > SUBSET > > > ss( S1, S2 ) <- ALL(X) XeS1 -> XeS2 > > > .. > > > EQUALS > > > eq( S1, S2 ) <- ss(S1, S2) & ss(S2, S1) > > > -------------- > > > This checks both ways that all elements of S1 are elements of S2 > > and vice versa! > > > ------------- > > > Then some arithmetic can be added... > > > nat(0) > > nat( s(X) ) <- nat(X) > > > odd( s(0) ) > > odd( s(s(X)) ) <- odd(X) > > > even( 0 ) > > even( s(s(X)) ) <- even(X) > > > --------------- > > > e.g. > > > even(( s(s(s(s(0)))) ) ? > > > > YES > > > -------------- > > > Then sets can be defined using N.S.T. > > > e( A, odds) <- odd( A ) > > e( A, evens) <- even(A) > > e( A, nats ) <- nat( A ) > > > --------------- > > > Now you can use the SET LEVEL OPERANDS. > > > in( nats , odds ) ? > > > > YES > > > .... > > > in( evens , odds ) ? > > > > NO > > > ... > > > ss( odds , nats ) ? > > > > YES > > > ---------------------------- > > > EQUALITY BY EXTENSION > > > fails on infinite sets. > > But inequality by extention succeeds on sets which are no more than > countably infinite. > > > > > What's needed is EQUALITY BY INDUCTION... > > > before you can even HAVE an INFINITE SET THEORY > > > that works! > > While no doubt useful, such a requirement is not at all necessary. >
Induction is necessary if you ever advance from Natural Deduction to Automatic Logic with computable transitive closure.
ss( S1, S2 ) <- ALL(X) XeS1 -> XeS2
There is currently no algorithm to return TRUE or FALSE as to whether
S1 C S2
The End Result of this is that there is no Turing Machine that can compute an anti-diagonal and halt