FOM: NBG and ZFC

[Martin Davis]

At 03:50 PM 1/29/00 -0500, Joseph Shoenfield wrote:
>       One thing is important to understand; each of N,B, and G
>intended a class to be not an arbitrary collection of sets but a
>collection which is definable (from set parameters) in the language
>of set theory.

I don't know how to understand the word "intended", but I doubt very much 
that von Neumann "intended" any such thing.

>     Why did Godel use NBG?   In his original announcement in the
>PNAS, he used the much more familiar ZFC.   To define the construc-
>tible sets in ZFC, he needs to define the operator O, where O(x) is
>the set of subsets of x definable from parameters in the structure
>with universe x and the usual membership relation restricted to x.
>This requires formalizing some syntax and semantics in ZFC.   This
>is no big problem for anyone with a command of elementary logic.
>However, if classes (in the above sense) are available, things are
>more direct; one can simply talk about sets and classes without
>bringing in notions like "formulas" and "validity of formulas".
>I think Godel felt this would make his work easier to understand by
>those unfamiliar with logic.

The finite axiomatization was important too. It enabled Goedel to disguise 
the Skolem hull construction. Instead he could take closure under the 
finite number of operations corresponding to the class existence axioms. 
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, brief as it was, 
that I could say "Aha!"

Martin




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