FOM: the ZFC notion of set

Randall Holmes holmes at
Fri Feb 27 16:37:37 EST 1998

(In response to Neil Tennant, Fri, 27 Feb 1998 13:58:26 -0500 (EST))

I should first point out a formal error in what I said.

The notion "set of ZFC" is equivalent (if one's domain of objects
is of inaccessible size) to "isomorphism type of (pointed) well-
founded extensional relation with domain smaller than the universe".
Without the size qualification, one gets ZFC - Power Set every time!
This is a third-order concept of higher-order logic.

Tennant asked what this would prove to someone who thought second-order
logic was just set theory in disguise.  My answer to this is that the
notion of set in second-order logic (or nth order logic for any finite n)
is rather different from the notion of set in ZFC.  It is actually a typed
notion, and it does not admit unlimited iteration as the set notion of
ZFC does.

I think that second-order logic is a kind of set theory (not at all in
disguise); so is nth order logic.  I don't necessarily exclude it from
logic by calling it set theory.  It is a different kind of set theory
from ZFC, though, so it is worth pointing out that ZFC can be
interpreted in it (under suitable conditions).

The definition of sets of ZFC as isomorphism types of small
well-founded extensional relations comes up in NFU, where this is the
only way to simulate Zermelo-style set theory.  Induction/recursion on
the membership relation, which is essential in ZFC, is forbidden in
NFU, since it requires one to consider unstratified predicates.  The
idea of using isomorphism types of relations to represent sets also
comes up in the modelling of anti-foundation axioms.

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at
not glimpse the wonders therein. |

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