FOM: Reply to Davis

[Joseph Shoenfield]

     In my communication on NBG, I stated that N, B, and G all intended
a class to be a collection of sets definable from set parameters in the
language of set theory.   Martin Davis questions this, I think quite
justly.   I think it is true, but I can't produce any quotes to justify
it.   I should have been content to say that the above notion of
"class" is the one now universally used by main line set theorists
(whether they operate in ZFC or NBG); that it fits very well with the
class existence axioms of NBG; and that the notion of a class as an
arbitrary collection of sets raises many problems which are not of much
interest to set theorists.
     Martin also points out that the finite axiomatizability of NBG
was important for Godel.    This is true in the sense that this finite
set of axioms gives rise to Godel's "fundamental operations" which
are the key to defining the constructible sets.  Martin says:
>     For me, when I was learning the subject, this device served to
>obscure what was happening.   It was only when I saw the PNAS article
>... that I could say "Aha!".
     I had the same experience, and so did other logicians interested
in set theory (e.g., Dana Scott).   But I do not think it was due to
the use of NBG, but to the failure of the book to explain what was
going on.   In 1957 when Kreisel and I were at the Institute, I per-
suaded him to ask Godel about this.   Godel said that he made this
clear in his lectures but that this failed to appear in the book.
Clearly there was a lack of communication between Godel and George
Brown; I will not try to guess whose fault it was.
     I must confess that I have never read through von Neumann's
article.   I learned about NBG first from Bernays' article.    I
once asked Mostowski (who had gone through the von Neumann article
when it first appeared) if I should read it.   He said he was quite
disappointed in it, because, after choosing functions rather classes
as the fundamental notion, von Neumann did not look for axioms which
were clearly about functions, but merely translated Zermelo's axioms
into his language (using characteristic functions).   I have always
felt that it would be interesting to find a set of axioms for set
theory which really used functions as the basic notion.   We all know
that in developing recursion theory, it is best to use functions as
the basic notions, and define recursive sets in terms of recursive 
functions.   Can this be done in set theory and would the resulting
axiom system be useful?