Formulas of L(N,
℘
N,
℘℘
N) are obtained using connectives and
quantifiers in the usual way. Their intended meaning is
obvious.
The
Σ
1
∞
formulas of L(N,
℘
N,
℘℘
N) are the formulas without
any quantifiers over
℘℘
N.
The
Σ
2
1
formulas of L(N,
℘
N,
℘℘
N) are the formulas of
L(N,
℘
N,
℘℘
N) are the formulas that begin with zero or more
existential quantifiers over
℘℘
N, followed by a
Σ
1
∞
formula
of L(N,
℘
N,
℘℘
N).
We use ZC for Zermelo set theory with the axiom of choice,
which we will take to be ZFC with Replacement omitted.
The
Σ
21 sentences of L(N,
℘
N,
℘℘
N) are well known to be very
expressive. For example, the continuum hypothesis is
provably equivalent to a
Σ
2
1
sentence, over ZC.
Throughout the paper, we take the
Σ
2
1
sentences to be the
Σ
2
1
sentences of L(N,
℘
N,
℘℘
N).
Here is the main result of this paper.
THEOREM 7.5. The continuum hypothesis (more generally,
every
Σ
21 sentence) is provably equivalent, over ZC, to a
sentence of the form
(
∃
f:
ℜ
→
ℜ
)(
∀
x,y
∈
ℜ
)(s = t)
where s,t are terms in f,x,y,+,
•
,0,1.
Here are some other results of significance.
THEOREM 6.1. The continuum hypothesis (more generally,
every
Σ
2
1
sentence) is provably equivalent, over ZC, to a
sentence of the form
(
∃
f:
ℜ
2
→
ℜ
)(
∀
x,y
∈
ℜ
)(
ϕ
)
where
ϕ
is quantifier free in f,x,y,<.
THEOREM 5.1. The continuum hypothesis (more generally,
every
Σ
21 sentence) is provably equivalent, over ZC, to a
sentence of the form
(
∃
f:(
℘
N)
2
→
℘
N)(
∀
x,y
∈
℘
N)(
ϕ
)