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A WAY OUT
Harvey M. Friedman
Department of Mathematics
http://www.math.ohiostate.edu/~friedman/
Ohio State University
August 28, 2002
ABSTRACT. We present a way out of Russell’s paradox for
sets in the form of a direct weakening of the usual
inconsistent full comprehension axiom scheme, which, with
no additional axioms, interprets ZFC. In fact, the
resulting axiomatic theory 1) is a subsystem of ZFC +
“there exists arbitrarily large subtle cardinals”, and 2)
is mutually interpretable with ZFC + the scheme of
subtlety.
1. NEWCOMP.
Bertrand Russell [Ru1902] showed that the Fregean scheme of
full comprehension is inconsistent. Given the intuitive
appeal of full comprehension (for sets), this inconsistency
is known as Russell’s Paradox (for sets). The modern view
is to regard full comprehension (for sets) as misguided,
and thereby regard Russell’s Paradox (for sets) as a
refutation of a misguided idea.
We first give an informal presentation of the axiom scheme
investigated in this paper. Informally, the full
comprehension axiom scheme in the language L( ) with only
the binary relation symbol and no equality, is, in the
context of set theory,
Every virtual set forms a set.
We use the term “virtual set” to mean a recipe that is
meant to be a set, but may be a “fake set” in the sense
that it does not form a set. The recipes considered here
are of the form {x:
j
}, where
j
is any formula in L( ).
Other authors prefer to use the term “virtual class”,
reflecting the idea that {x:
j
} always forms a class, with
the understanding that x ranges over sets. Our terminology
reflects the intention to consider only sets, and construct
a powerful set existence axiom.
We say that {x:
j
} forms a set if and only if there is a
set whose elements are exactly the y such that
j
. Here y
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must not be free in
j
(and must be different from x). Thus
{x:
j
} forms a set is expressed by
(
$
y)(
"
x)(x y
j
).
Russell showed that
{x: x x} forms a set
leads to a contradiction in pure logic.
Our way out of Russell’s Paradox is to modify the
inconsistent Fregean scheme in this way:
Every virtual set forms a set, or _____.
We refer to what comes after “or” as the “escape clause”.
The escape clause that we use involves only the extension
of the virtual set and not its presentation.
We are now ready to present the comprehension axiom scheme.
NEWCOMP. Every virtual set forms a set, or, outside any
given set, has two inequivalent elements, where all
elements of the virtual set belonging to the first belong
to the second.
To avoid any possible ambiguity, we make the following
comments (as well as give a formal presentation in section
2).
1. For Newcomp, we use only the language L( ), which does
not have equality.
2. Here “inequivalent” means “not having the same
elements”.
3. The escape clause asserts that for any set y, there are
two unequal sets z,w in the extension of the virtual set,
neither in y, such that every element of z in the extension
of the virtual set is also an element of w.
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
 Fall '08
 JOSHUA
 Math, Sets

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