
Re: mathematical infinite as a matter of method
Posted:
May 2, 2013 8:53 PM


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.

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

Then an INFINITE SET THEORY would be possible:
eq( odds , in( nats , odds ) ) ? #
> YES
This is comparing (VIA AXIOM OF EXTENSIONALITY)
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))))) , ....}

The PROBLEM IS no computer can do this...
eq( odds , in( nats , odds ) ) ?
automatically...
because by default EQUALITY BY EXTENSION
fails on infinite sets.

EQ makes 2 calls to SS (subset)
SUBSET
ss( S1, S2 ) < ALL(X) XeS1 > XeS2
...
which is easy to program when S1 is finite..

What's needed is EQUALITY BY INDUCTION...
before you can even HAVE an INFINITE SET THEORY
that works!
Herc  www.BLoCKPROLOG.com
# /\ and in() are not interchangable..

