Date: May 2, 2013 8:53 PM
Author: Hercules ofZeus
Subject: Re: mathematical infinite as a matter of method

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.

----


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..