2
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.