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Topic: mathematical infinite as a matter of method
Replies: 25   Last Post: May 4, 2013 11:24 PM

 Messages: [ Previous | Next ]
 Graham Cooper Posts: 4,495 Registered: 5/20/10
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
>  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.

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 anti-diagonal 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

Date Subject Author
4/21/13 fom
4/21/13 Virgil
5/2/13 Hercules ofZeus
5/2/13 fom
5/2/13 Virgil
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 Brian Q. Hutchings
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 fom
5/4/13 Graham Cooper
5/3/13 Graham Cooper
5/3/13 fom
5/3/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/4/13 fom
5/4/13 Graham Cooper
5/3/13 Graham Cooper