FOM: naive or brainwashed?

Charles Parsons parsons2 at
Thu Mar 12 17:53:40 EST 1998

I applaud most of what Martin says in his posting. I hope he'll write up
the talk so that those of us who won't be at the meeting can read it.

There's one statement I would quarrel with, that Zermelo's axiomatization
of set theory was "based on" cumulative types. There's no hint in his 1908
paper or others of that time that that was the way he understood his
system. That it could be interpreted that way was a subsequent discovery,
requiring the work of a few people on the set-theoretic side (Mirimanoff,
von Neumann, Zermelo again (1930), Goedel) and a change in the way type
theories were looked at, so that one could abstract from the feature that
different types of entity were represented by variables of different
syntactic categories. Possibly Goedel in his 1933 MAA lecture was the first
to put the latter clearly. I think it likely that Tarski saw the point in
the 1930's, but it's not clear in his monograph on truth. Quine deserves
some credit here, since his work on set theory was based on the aim of
starting with the ideas of the theory of types and developing a first-order
theory, although it first led him into the detour (interesting in its own
right) of NF and ML. His address, "Unification of universes in set theory"
(JSL 21, 1956) describes how one gets from the simple theory of types to
conventional set theory. None of that would have been news to Goedel, but I
don't think it was conventional wisdom at the time.

Quine's way of thinking about these things in the 1930s is described in his
"The inception of 'New Foundations'" in the enlarged edition of _Selected
Logic Papers_ (Harvard 1995).

Martin no doubt knows most of this and just expressed himself in a
misleading way.

Charles Parsons

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