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FROM RUSSELL’S PARADOX TO
HIGHER SET THEORY
by
Harvey M. Friedman
Ohio State University
[email protected]
www.math.ohiostate.edu/~friedman/
Philosophy Colloquium
October 10, 1997
ABSTRACT: Russell’s way out of his paradox via the impre
dicative theory of types has roughly the same logical power
as Zermelo set theory  which supplanted it as a far more
flexible and workable axiomatic foundation for mathematics.
We discuss some new formalisms that are conceptually close to
Russell, yet simpler, and have the same logical power as
higher set theory  as represented by the far more powerful
ZermeloFrankel set theory and beyond. END.
The famous Russell’s paradox for sets arises out of the
intuitively appealing naive principle of full comprehension
written
(
$
x)(
"
y)(y x
j
),
where
j
is a logical formula involving
"
,
$
,
, , and
variables in which x is not free. The variables are thought
of as ranging over sets. Here x,y are any distinct variables.
In particular, the simple special case
(
$
x)(
"
y)(y x
y y)
generates the inconsistency by fixing such an x and uni
versally instantiating y by x, thus obtaining
x x
x x.
Russell’s Paradox also makes perfectly good sense in other
contexts besides set theory – e.g., in a theory of
predicates. Here the naive comprehension axiom scheme takes
the notationally similar but distinct form
(
$
P)(
"
Q)(P(Q)
j
),
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where
j
is a logical formula involving
"
,
$
,
,(),
and variables, where P is not free in
j
. This time the
variables are thought of as ranging over predicates. Here P,Q
are any distinct variables.
Why do I draw a distinction between the set version of
Russell’s Paradox and the predicate version? One big
difference between the way sets and predicates are thought of
is in terms of extensionality. We have preferred not to use
equality as primitive, and so extensionality for sets takes
on the form
(
"
z)(z x
z y)
(
"
z)(x z
y z).
Extensionality is in accordance with the usual way of
thinking about sets.
However, the corresponding statement about predicates,
(
"
R)(P(R)
Q(R))
(
"
R)(R(P)
R(Q)),
does not seem appropriate. For instance, one may distinguish
different predicates P,Q such that P,Q fail at every
argument, where P,Q can be distinguished by another predicate
R. Then we have (
"
R)(P(R)
Q(R)) and yet (
"
R)(R(P)
R(Q)).
Yet another important distinction between certain notions of
predication and sets will be relevant later in the
discussion.
Frequently people still use the notation x y even if x,y
are predicates and we are asserting that y holds at x. We
will follow this convention. Then the distinction between
predicates and sets may come at the point at which we decide
to include or exclude EXT.
Now after his discovery of Russell’s Paradox, Russell
embarked on a series of developments to repair the damage. It
is customary to identify two proposals of Russell to get out
from under his Paradox.
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