FOM: Schema N

[Moshe' Machover]

Neil Tennant proposed:

> _______________________________________________________________________
> ADEQUACY CONDITION ON ANY THEORY OF NATURAL NUMBER
> (compare with Tarski's Adequacy Condition on any theory of truth)

> Any adequate theory of number must yield as a theorem each instance
> of the following schema:

>        Schema N:  (E_n x)F(x) if and only if #xF(x)=n*
> ________________________________________________________________________

To which Martin Davis responded:

> I like to place the issue as follows: Frege and Cantor saw that the 
> notion of having the same cardinality is definable without specifying 
> what these cardinalities are. In contemporary terms, they were dealing 
> with an equivalence relation, and Frege specifically proposed to identify 
> the numbers with the equivalence classes. This famously collapsed because 
> the classes are "too large".

> Now an alternative, used for exmaple in old-fashioned number theory texts 
> in dealing with what we'd call residue classes, to using equivalence 
> classes is to choose some suitable representative from each class. That 
> is exactly what von Neumann did for cardinals (and of course ordinals as 
> well) and his treatment has become standard.

> Personally, I find this an entirely satisfactory state of affairs, and am
> dubious about extracting any more philosophical juice from this issue.
.......................................................................

I'd like to see what Neil has to say about this. But as I understand it, 
yes, of course, if you add to ZF the #x operator with the axiom scheme

          #xF(x)= least ordinal having a bijection to {x : F(x)}

then for each n Neil's Schema N is provable. So the [simulation of] 
arithmetic in ZF is adequate in Neil's sense. Nothing very surprising here.

*But* I suppose the point of Neil's Schema N is that each instance of it
may be provable in far more modest first-order theories incorporating some
(first-order) arithmetic. 


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