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Topic: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Replies: 39   Last Post: Oct 14, 2012 11:56 PM

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 Graham Cooper Posts: 4,495 Registered: 5/20/10
Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Posted: Oct 13, 2012 8:20 PM

On Oct 14, 10:06 am, George Greene <gree...@email.unc.edu> wrote:
> On Oct 13, 4:30 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>

> > These are just INFERENCES mind you.
>
> > You still owe me some BASE THEOREMS, otherwise all your "proofs" are
> > Oracular.

>
> NO, I DON'T.
>
> In the NON-logical context, in the SET THEORY or NUMBER THEORY (Peano
> Arithmetic)
> or group theory OR ANY OTHER AXIOMATIC THEORY context, in the context
> of anything
> for which you might actually be USING first-order logic, any
> investigation TO WHICH you might
> be APPLYING first-order logic, what YOU are calling a "base theorem"
> is called AN AXIOM.
> That's what axioms ARE FOR.  They are true DESPITE not having any
> proofs (other than "it's an axiom".)
>
> In the pure-logical context, there is no difference between a logical
> axiom and the conclusion of an inference
> rule that infers/derives that conclusion FROM AN EMPTY set of
> premises.
> I'm NOT being ORACULAR when I insist that P V ~ P
> is true, withOUT any proof in a FIRST-order system.  P V ~P
> First-order PRESUMES all that IN ONE handwave!

George, I don't care!

*I* use MODUS PONENS for inference and ANY OLE AXIOM and ANY THEOREM
SO FAR.

a & (a->b) --> b

*I* use double arrow --> for NEW THEOREM

If you want to use a list of inferences distinct from axioms, may I
suggest

(P ^ Q) --> Q

so we know it's not a THEOREM/AXIOM

The only lesson here is that there is SOME / ONE rule that operates ON
the theory by creating --> new formula which is distinct from the
theorems.

*************

In MY TERMINOLOGY, since you use "model" to mean "deduction sequence"
is meaningless.

A MODEL is CREATED in the theory like so:

a & (a ->[p->q]) -> [p->q]

NOW you have a 2ND LEVEL OF INFERENCE RULES [p->q]

You could JUST USE MODUS PONENS TO DO THIS and a set of tautology
formula

but you like to have a list

1 P --> P
2 !P --> !P
...

which forces the prover into a single model.

Herc

Date Subject Author
10/5/12 Graham Cooper
10/5/12 Frederick Williams
10/7/12 Charlie-Boo
10/5/12 Graham Cooper
10/5/12 Frederick Williams
10/5/12 Graham Cooper
10/7/12 Graham Cooper
10/8/12 Graham Cooper
10/9/12 Graham Cooper
10/11/12 Graham Cooper
10/12/12 Graham Cooper
10/12/12 Graham Cooper
10/12/12 camgirls@hush.com
10/12/12 Richard Tobin
10/12/12 camgirls@hush.com
10/13/12 george
10/13/12 Graham Cooper
10/14/12 george
10/13/12 Graham Cooper
10/13/12 george
10/13/12 george
10/13/12 Graham Cooper
10/14/12 Graham Cooper
10/14/12 Graham Cooper
10/14/12 Graham Cooper
10/5/12 Scott Berg
10/5/12 Curt Welch
10/6/12 Mike Terry
10/6/12 Graham Cooper