FOM: Large and small

[Joe Shipman]

Davis:
>Without wanting to argue about any of this, I do think it's worth
>emphasizing the crucial role that the definability of the order pair
made in
>the acceptance of Z, ZF, and ZFC as the most appropraite
axiomatizations of
>mathemtics. You have only to look at the tortured treatment in
Principia in
>which everything has to be done over for relations, or even at Quine's
>"System of Logistic" to see this.

For METAmathematics the Kuratowski definition of the ordered pair and
the Von Neumann definition of ordinals resulted in a big simplification
of the formal system and led to great progress.  For ORDINARY
mathematics, it would have been possible to take integers as urelements
and introduce an ordered pair primitive with the appropriate properties,
and without much more effort formalize the same theorems as were
formalized in the ZFC development.  (That is, using an augmented set
theory rather than the elegant and minimal ZFC formal system.)

In "On Numbers and Games", Conway is trying to found a branch of
ordinary mathematics and is not trying to prove the consistency of his
foundation or any other metamathematical result, so he is quite right to
introduce primitives for pair and Left and Right membership without
showing that the results he arrives at could also have been formalized
in ZFC.  (Foundationalists may require him to remark that such
formalization is possible, and he properly does so at a couple of places
in the book.  This is not in itself a justification of ZFC -- WHATEVER
the most commonly accepted foundation was, authors whose work was not
transparently formalizable in that system would be expected to remark on
whether such formalization was possible, even if they were not expected
to give details.)
-- Joe Shipman