three quantifiers071402 - 1 THREE QUANTIFIER SENTENCES by...

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THREE QUANTIFIER SENTENCES
by
Harvey M. Friedman*
Ohio State University
July 2, 2002
July 14, 2002
[email protected]
http://www.math.ohio-state.edu/~friedman/
Abstract. We give a complete proof that all 3 quantifier
sentences in the primitive notation of set theory ( ,=), are
decided in ZFC, and in fact in a weak fragment of ZF without
the power set axiom. We obtain information concerning
witnesses of 2 quantifier formulas with one free variable.
There is a 5 quantifier sentence that is not decided in ZFC
(see [Fr02]).
1. Preliminaries.
Daniel Gogol [Go79] presents an argument that all 3
quantifier sentences in the primitive notation of set theory
( ,=) are decided in ZFC. In the author’s words, not all of
the details are presented:
“It is tedious but involves no difficulty to verify that
if...”. p. 5, line 14.
“This can be verified by considering all the possible cases,
but is quite clear if considered carefully. So we omit what
would be a very long verification.” p.8, end.
We give a complete proof that all 3 quantifier sentences in
set theory based on ,=, are decided in ZFC. In fact, we show
that all sentences of somewhat higher complexity are decided
in a weak fragment T of ZF without the power set axiom. We
also give some strong information about witnessing 3
quantifier sentences that begin with an existential
quantifier. Our main results are summarized in Theorem 11.1.
We do not use ideas from [Go79].

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We work entirely within the primitive language of set theory,
which is standard predicate calculus with equality and ,
using the standard quantifiers " , $ , and the standard
connectives ÿ , , , Æ , . We assume the usual complete axioms
and rules of classical logic.
We let T be the following weak set theory.
1. Extensionality.
2. Pairing.
3. Union.
4. Infinity.
5. Foundation.
6. Bounded Separation.
Pairing asserts the existence of {a,b}. Union asserts the
existence of a.
The usual formulation of Infinity in ZF is the existence of a
set containing ∅ and closed under the operation that sends x
to x {x}. Because we are only using Bounded Separation, we
use the stronger version of infinity that asserts the
existence of a set containing ∅ and closed under the
operation that sends x,y to x {y}.
Foundation asserts that every nonempty set has an epsilon
minimal element.
Bounded Separation asserts the existence of {x a: j }, where
j is a formula in which all quantifiers are bounded. I.e.,
all quantifiers are of the forms ( " u v), ( $ u v), where
u,v are distinct variables.
In T, we can prove the existence of a least nonempty
transitive set closed under power set, which we write as
V( w ). In T, we can develop all of the basic facts about V( w )
and its elements, as well as define subsets of V( w ) and
functions on V( w ) by recursion.
Since the focus of the paper is on 3 quantifier sentences, we
strictly follow the simplifying convention that all formulas
will use at most the three distinct variables x,y,z. Letters
such as u,v,w are used as metavariables over the official
variables x,y,z, or variables used to carry out proof
sketches that are to take place in T.

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