[FOM] The Gold Standard
Andrew Boucher
Helene.Boucher at wanadoo.fr
Thu Feb 23 01:33:39 EST 2006
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...
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