FOM: conceptions of set in NFU and ZFC

Randall Holmes holmes at
Mon Mar 2 12:32:10 EST 1998

I see that I didn't address this part of Silver's question explicitly.

I think that the concepts of set used in NFU and ZFC are different
in a way which can be articulated informally and perhaps helpfully.

When one looks at a set in NFU, one will only look at its members,
members of members, members of members of members, etc. down to some
concrete finite level.  The stratification criterion is precisely
calibrated to make it difficult (actually impossible) to ask questions
that probe the iterated members of a set to an arbitrary depth.

When one looks at a set in ZFC, one can assume knowledge of its
entire transitive closure, and such knowledge is constantly used
(as, for example, in recognizing ordinals).

This is the reason why I think that NFU is a theory of "sets" in a
narrower sense, while ZFC is "really" a theory of (small, pointed,)
well-founded extensional graphs.  The latter are richer structures
which also support all constructions allowed on (small) sets, so the
choice of ZFC over NFU is not a bad choice for foundations of
mathematics.  But if one really wants to understand the paradoxes, it
is probably a good idea to be aware of the possibility of the NFU
approach (it will at least avert assertions that the universal set or
Frege's natural numbers are inconsistent totalities, which they
patently are not!)

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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