
Re: mathematical infinite as a matter of method
Posted:
May 3, 2013 3:43 AM


On May 3, 12:21 pm, Virgil <vir...@ligriv.com> wrote: > In article > <749e1561d69b45af90013be3141ad...@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.kobeu.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.
SUBSET
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 antidiagonal and halt
DESPITE YOUR CLAIMS THAT YOU CAN DO EXACTLY THAT
,,,given an infinite List!! X

ODDS { s(0), s(s(s(0))) , s(s(s(s(s(0))))) , ....}
NATS /\ ODDS { s(0), s(s(s(0))) , s(s(s(s(s(0))))) , ....}
Analysis By Extension means testing all parameter values independently
F(a) = G(a) F(b) = G(b) F(c) = G(c)
BY EXTENSION F = G

Its not possible to test Equality by Extension in the inf. case.
Your theory of UR (Uncomputable Reals)
is based on UL (Uncomputable Logic)
Herc  www.BLoCKPROLOG.com

