FOM: Large and small

Martin Davis martin at
Fri Apr 30 13:50:25 EDT 1999

At 09:02 AM 4/30/99 -0400, Joe Shipman wrote:
> ... Conway is simply
>reacting against unnecessary insistence on restricting one's language so
>that one has to resort to cumbersome locutions like the Kuratowski
>definition of ordered pair.  (That is, it should be possible in a
>foundational work like ONAG to simply INTRODUCE a concept like ordered
>pair with the appropriate notation and properties, without finding an
>equivalent in the language with only the epsilon relational symbol.)

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
"Sustem of Logistic" to see this.

                           Martin Davis
                   Visiting Scholar UC Berkeley
                     Professor Emeritus, NYU
                         martin at
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