[FOM] The Gold Standard

[Andrew Boucher]

On  23 Feb 2006, at 12:32 AM, Harvey Friedman wrote:

>  In fact, one can already argue, if one
> wants, that the number 0 is Platonistic.
>
> So under this view, you cannot even begin any predicative development
> of mathematics without Platonism.  ...


Here's a way to do arithmetic without assuming the existence of any  
number, not even 0.

Consider the language of Frege Arithmetic, where instead of # one has  
a predicate M, whose first argument is a first-order letter and whose  
second argument is a second-order letter.  (So instead of #P = n one  
has Mn,P.)

Use predicative comprehension.

Use zero(z) to abbreviate

(P)( Mz,P <=> (x) ! Px )

Use these axioms:


(G1) Uniqueness. (P)(n)(m) ( Mn,P & Mm,P => n = m)

(G2) Zero. (P)(n) ( Mn,P & ! zero(n) => (there exists x) Px )

(G3) Successoring: (P)(Q)(a)(n)(m) ( Nn & Sn,m & ! Pa & (x) (Qx <=>  
Px V x = a)
=> (Mm,Q <=> Mn,P) )

(G4) Induction. From:
(z) ( zero(z) => phi(z) ) &
(n)(m) ( Nn & Sn,m & phi(n) => phi(m) )
conclude:
(n) (Nn => phi(n))

This system G develops a large amount of arithmetic - prime number  
theorem, quadratic reciprocity.  Perhaps it proves FLT as well...