FOM: expanded comments re NFU

Randall Holmes holmes at
Fri Mar 26 19:21:48 EST 1999

Friedman responding to Holmes:

>A subtle point is that I think there is a fairly good argument that
>the comprehension criterion for NF(U) is a better resolution of the set
>theoretical paradoxes than the comprehension criterion of ZFC --


>but I
>don't think that the foundations of mathematics really depend in any
>essential way on a confrontation with the set theoretical paradoxes
>(the paradoxes were avoidable mistakes).


If the set theoretical paradoxes are not confronted appropriately then we
do not have legitimate foundations of mathematics. There may be various
ways to confront them. E.g., one can propose V(omega+omega) as a picture of
the mathematical universe.

Holmes replies:

The stratification criterion for comprehension is arguably better than
the comprehension requirements for ZFC _as a response to the paradoxes
of set theory_ because it saves more of the set definitions favored by
the school that ran into the paradoxes in the first place (Frege,
Russell et. al.).  It preserves the universe of sets as a Boolean
algebra, for example, and it allows use of the Russell-Whitehead
definitions of cardinal and ordinal numbers.  It can be instructive to
be aware (as many are not!) that these definitions are not in
themselves paradoxical.  

Note that I am not implying here that NFU is a better foundation for
mathematics than ZFC; I do _not_ think that adequate foundations are
obtained as a "fix" from the disastrous "foundations" provided by
"naive set theory".  (Nor do I think this is the only way to motivate
NFU, though it is the historical route taken by Quine to obtain NF).

It is perfectly reasonable to assert that at least one intuitive
approach (leading ultimately to ZFC!) never encounters the paradoxes
of set theory as an issue at all.  If one supposes that sets are
constructed in such a way that one has to have already constructed the
elements of a set S before one can construct S, then one will never
have any reason to consider the universal set (it is obviously not
constructible in this way) nor will one ever be tempted to consider
the genuinely paradoxical Russell class.  Arrival in the Promised Land
of adequate foundations does not require the prior experience of the
disaster of "naive set theory".

One _can_ propose V(omega+omega) (to take your example) as a picture
of the mathematical universe -- without ever addressing the paradoxes
of set theory, because never entertaining the fallacious assumptions
which lead to them.

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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