Bulletin of the Section of Logic
Volume 42:1/2 (2013), pp. 11–20
Morteza Moniri1
S. Hosein Sajjadi
REGULAR CUTS IN MODELS OF BOUNDED
ARITHMETIC
Abstract
We study some model theoretic properties of cuts in models of bounded arithmetic
theories like S2i and T2i . We consider suitable versions of the notion of regular
cut in models of these theories. We study the relation between this notion and
some versions of the collection axiom. As a consequence, we construct certain
extensions of models of theories of bounded arithmetic. Each of these extensions
contains a nonstandard initial segment which is a model of S2 .
2010 Mathematics Subject Classification: 03F30, 03H15
Key words: Bounded arithmetic, regular cut, collection axiom
1.
Introduction
Let M be a nonstandard model of Peano arithmetic. A non-empty proper
subset I of M is a cut of M if I is closed under successor and also is
downward closed.
A cut I is called regular if for any definable (coded) function f which
is bounded on I by an element of I there is an unbounded subset of I
such that f is constant on it. This property generalizes the same obvious
property of the standard cut of natural numbers in non-standard models
1 This
research was in part supported by a grant from IPM (No. 89030121).
12
Morteza Moniri and S. Hosein Sajjadi
of bounded arithmetic. Regular cuts of models of Peano arithmetic have
been extensively studied, see e.g. [3] and [5] for book-length descriptions.
In this paper we study parallel results for models of Buss’ theories
of bounded arithmetic. In particular, we prove that for any model M of
bounded arithmetic and any regular cut I of M of certain degree there is
a partial elementary extension K of M such that I is also a cut in K and
this extension adds a new element between I and M r I.In addition, K
contains a nonstandard cut which is a model of S2 . In this direction, we
mostly follow [7] and [5].
2.
Some Backgrounds
In this paper we study some model theoretic properties of Buss’ theories
S2i and T2i of bounded arithmetic. Here, we give some general information
about these theories (see [1] for the details).
The language of the theories S2i and T2i is obtained by adding the
function symbols x x2 y (= x2 rounded down to the nearest integer), |x| (=
the length of binary representation for x) and # (x#y = 2|x||y| ) to the
usual language of first-order arithmetic.
The base theory BASIC is a finite set of quantifier-free formulas expressing basic properties of the relation and function symbols. Σb0 = Πb0 is
the class of all sharply bounded formulas. A sharply bounded formula is a
bounded formula in which all quantifiers are sharply bounded, i.e. of the
form ∃x 6 |t| or ∀x 6 |t| where t is a term which does not contain x.
The syntactic classes Σbi+1 , Πbi+1 of bounded formulas are defined by
counting alternations of bounded quantifiers ignoring sharply bounded quantifiers. A formula ϕ is in ∆bi with respect to a model (resp., theory), if there
is a Σbi -formula and a Πbi -formula such that ϕ is equivalent to each of them
in the model (resp., theory). In this paper, by a class of formulas we mean
one of these classes.
The theory S2i is axiomatized by BASIC plus the Σbi − P IN D axioms,
i.e.
[ϕ(0) ∧ ∀x (ϕ(bx/2c) → ϕ(x))] → ∀xϕ(x),
where ϕ(x) is a Σbi -formula that can have more free variables besides
x. The classical theory T2i is BASIC plus the Σbi − IN D axiom.
Throughout this paper, i denotes a natural number greater than or
equal 1, unless it is explicitly stated that it can also be 0.
Regular Cuts in Models of Bounded Arithmetic
13
Let M be a nonstandard countable model of BASIC. A non-empty
proper subset I of M is said to be a cut of M if I is closed under successor
and whenever a ∈ I and M |= b < a, then b ∈ I. If I is a cut of M we
write I ⊂e M . By c > I we mean c is greater than all elements of I. I
is small if there is an a ∈ M such that I < |a| in the obvious sense. By
a definable function (or coded function) of M we mean a function that its
graph is definable by a formula ϕ(x, y) in M (maybe with parameters) and
for each x ∈ M there is a unique y ∈ M such that M satisfies ϕ(x, y). By
a definable function f : I −→ M , we mean a definable function over M
restricted to I. Also, when we say that a subset X of I is definable we
mean X is intersection of I with a definable subset of M .
In [6], we defined a strong version of the notion of cut in models of
bounded arithmetic and proved some basic (overspill) properties for it.
Here we recall the ones that we need in this paper.
Definition 2.1 Let M |= BASIC and I ⊂e M . We say that I is a p-cut
if bx/2c ∈ I implies x ∈ I and whenever a ∈ I and M |= b ≤ a then b ∈ I.
Note that p-cuts are those cuts that are closed under addition. For
an example of a cut which is not a p-cut, let a ∈ M |= BASIC be a
non-standard element. Then
Ia = {x ∈ M : ∃n ∈ N x ≤ a + n}
is a cut but not a p-cut (a ∈ I and 2a 6∈ I). If I is a p-cut then
|I| = {|x| : x ∈ I}
is a cut, because from BASIC we have S(|x|) = |2x| and if |b| ≤ |a| then
b < 2a. Moreover, if I is proper, then |I| is a small cut.
Fact 2.2 (Overspill). Let M |= T2i and I ⊂e M be a proper cut and
ϕ(x) ∈ Πbi such that for all x ∈ I, M |= ϕ(x). Then there is c > I such
that M |= ∀x ≤ c ϕ(x).
Proof. If not, then I would be definable by the Πbi -formula ∀x ≤ y ϕ(x)
which is impossible. Fact 2.3 (Sharp
Let M |= S2i and I ⊂e M be a small cut
S overspill).
b
b
and ϕ be a (Σi Πi )-formula such that for all x ∈ I, M |= ϕ(x). Then
there is c ∈ M such that |c| > I and M |= ∀x ≤ |c| ϕ(x).
14
Morteza Moniri and S. Hosein Sajjadi
Proof. If not, then
{z : M |= ∀x ≤ |z| ϕ(x)} = {z : |z| ∈ I}
S
would be a proper (Σbi Πbi )-definable p-cut which is impossible. 3.
Regular cuts
The notion of regular cut is an important notion in the model theory of
P A. In this section we study a suitable version of the definition for models
of the theories of bounded arithmetic.
Definition 3.1. Let M |= BASIC and I ⊂e M . I is said to be Γ-regular
if for every a ∈ I and any Γ-definable function f : I −→< a in M there is
an unbounded A ⊂ I such that f is constant on A.
Note that if I is a Γ-regular cut in a model M S21 , then for any
a ∈ M , a > I implies |a| > I. To see this, note that for the function
|, | : I −→< |a| where |a| ∈ I, the inverse image of any b < |a| is bounded
in I by 2b0 where b = |b0 |. So proper regular cuts are small.
Proposition 3.2. If M |= S2i , then Σbi -regularity, Πbi -regularity and ∆bi regularity of cuts of M are all equivalent.
Proof. We just show that any proper Σbi -regular cut I in a model M |= S2i
is Πbi -regular. Let f : I −→< a be definable by a Πbi -formula ϕ(x, y). Now
consider the formula
∃y ≤ |c| (ϕ(x, y) → y < a).
The bound |c| in the above formula comes from the fact that I is regular
and so small. Every i ∈ I satisfies this formula. So, by overspill (Fact 2.3),
there is b ∈ M such that |b| > I and
M |= ∀x < |b| ∃y ≤ |c| (ϕ(x, y) → y < a).
Now consider the formula
(x < |b| ∧ (ϕ(x, y) → y < a)) ∨ (x ≥ |b| ∧ y = a).
This formula is Σbi and obviously defines a function that its restriction to
I is equal to f . Regular Cuts in Models of Bounded Arithmetic
15
Before presenting our main results, we recall some definitions from
model theory of PA.
Let M ⊂ K and I ⊂e M . We say that K is an I-end elementary
extension of M , denoted by M I K, if M K, I ⊂e K and there is
c ∈ K such that
K |= I < c < M r I.
If I is a p-cut in M and M ⊂ K, K is said to be a weak I-end elementary
extension of M , denoted by M |I| K, if M K, |I| ⊂e K, and there is
c ∈ K such that
K |= |I| < |c| < M r |I|.
For every set Γ of formulas, M ≺Γ,I K and M ≺Γ,|I| K are defined
naturally, elementary with respect to the class Γ.
The following axioms are suitable versions of the collection axioms.
Definition 3.3. Let M |= BASIC and I ⊂e M . We say that I |= B ∗ Πbi
if whenever θ(x, y, z) ∈ Πbi is a formula with parameters from M , a ∈ I
and
∀x < a ∃y ∈ I ∀z ∈ I M |= θ(x, y, z),
then there is e ∈ M , such that |e| ∈ I and
∀x < a ∃y < |e| ∀z ∈ I M |= θ(x, y, z).
Note that in the above definition, if I has the mentioned property, then
it can not contain any large element. To see this choose the formula x < y
as θ.
Definition 3.4. Let M |= BASIC and I is a p-cut of M . We say that
I |= B ∗ LΣbi if whenever for θ(x, y, z) ∈ Σbi and a ∈ I we have
∀x < |a| ∃y ∈ I ∀z ∈ |I| M |= θ(x, y, z),
then there is b ∈ I such that
∀x < |a| ∃y < b ∀z ∈ |I| M |= θ(x, y, z).
The proof of the following theorems, Theorem 4.6-4.9, are straightforward modifications of the proof of [7], Theorem 4.
Theorem 3.5. Let M, K |= T2i+1 , I ⊂e M contains no large elements and
M ≺Πbi ,I K. Then I |= B ∗ Πbi .
16
Morteza Moniri and S. Hosein Sajjadi
Proof. Let ϕ(x, y, z) be a Πbi -formula with possible parameters from M ,
a ∈ I, and
∀x < a ∃y ∈ I ∀z ∈ I M |= ϕ(x, y, z).
For each x < a, let yx be such that ∀z ∈ I M |= ϕ(x, yx , z). By overspill
there is zx ∈ M − I such that M |= ∀z < zx ϕ(x, yx , z). So K |= ∀z <
zx ϕ(x, yx , z). Choose c ∈ K such that I < c < M − I. For each x < a we
have, K |= ∀z < c ϕ(x, yx , z). Thus we have
∀x < a ∃y ∈ I K |= ∀z < c ϕ(x, y, z).
Pick the least e such that
K |= ∀x < a ∃y < |e| ∀z < c ϕ(x, y, z)
Such an e exists because K |= T2i+1 . It is clear that |e| ∈ I and we have
∀x < a ∃y < |e| ∀z ∈ I M |= ϕ(x, y, z).
Let M |= BASIC, I ⊂e M and I |= B ∗ Πbi . Then I is
Theorem 3.6.
Σbi -regular.
Proof. Let f : I −→< a be Σbi -definable in M and for each c < a, f −1 (c)
is bounded in I. So
∀x < a ∃y ∈ I ∀z ∈ I M |= z > y −→ f (z) 6= x.
Notice that f (z) 6= x is a Πbi - formula and since I |= B ∗ Πbi there is e, such
that |e| ∈ I and
∀x < a ∃y < |e| ∀z ∈ I M |= z > y −→ f (z) 6= x.
This implies that
∀x < a ∀z ∈ I M |= z > |e| −→ f (z) 6= x
which contradicts the fact that f : I −→< a. Note that in Theorem 3.5, if M contains a large element a, the condition
M ≺Πbi ,I K does not hold for I = Log(M ). Since otherwise, by Theorem
3.6, Log(M ) = I would be Σbi -regular. This is impossible since considering
the function f : Log(M ) −→< |a| defined by f (x) = |x|, for each b such
that |b| < |a|, the inverse image of |b| in Log(M ) would be bounded above
Regular Cuts in Models of Bounded Arithmetic
17
by 2b. This means that there is no weak end extension of a model of
T2i+1 which adds an element between Log(M ) and M \ LogM . By a weak
end extension of M we mean an extension that has Log(M ) as an initial
segment.
Theorem 3.7. Let M, K |= S2i , I be a p-cut in M and M ≺Σbi , |I| K.
Then I |= B ∗ LΣbi .
Proof. Let ϕ(x, y, z) be a Σbi -formula and a ∈ I such that
∀x < |a| ∃y ∈ I ∀z ∈ |I| M |= ϕ(x, y, z).
For each x < |a|, let yx ∈ I be such that ∀z ∈ |I| M |= ϕ(x, yx , z). By Sharp
overspill (Fact 2.3), there is zx ∈ M \I such that M |= ∀z < |zx | ϕ(x, yx , z).
So K |= ∀z < |zx | ϕ(x, yx , z). Choose c ∈ K such that |I| < |c| < M \ |I|.
Hence, for all x < |a|, K |= ∀z < |c| ϕ(x, yx , z). Thus we have
∀x < |a| ∃y ∈ I K |= ∀z < |c| ϕ(x, y, z).
Pick the least-length e such that
K |= ∀x < |a| ∃y < e ∀z < |c| ϕ(x, y, z).
Because be/2c ∈ I and I is p-cut, we have e ∈ I and
∀x < |a| ∃y < e ∀z ∈ |I| M |= ϕ(x, y, z).
Theorem 3.8. Let M |= BASIC, I be a p-cut in M, and I |= B ∗ LΣbi .
Then |I| is Πbi -regular.
Proof. Let a ∈ I and f : |I| −→< |a| be a Πbi -definable function. Assume
that, for each c < |a|, f −1 (c) is bounded on |I|. So
∀x < |a| ∃y ∈ I ∀z ∈ |I| M |= z > |y| −→ f (z) 6= x.
Since the formula z > |y| −→ f (z) 6= x is equivalent to a Σbi -formula and
I |= B ∗ LΣbi , there is e ∈ I such that
∀x < |a| ∃y < e ∀z ∈ |I| M |= z > |y| −→ f (z) 6= x.
So we have
∀x < |a| ∀z ∈ |I| M |= z > |e| −→ f (z) 6= x.
This contradicts the fact that f : |I| −→< |a|. 18
Morteza Moniri and S. Hosein Sajjadi
Lemma 3.9 Let M |= T2i and I ⊂e M be ∆bi -regular. Assume that X ⊆ I is
b
∆bi -definable and unbounded in I. Let f : I −→< a, where
T a ∈ I, is a ∆i −1
definable function. Then there is c < a such that f (c) X is unbounded
in I.
Proof. Note that I is small as it is regular. So there is e such that |e| > I.
Therefore, considering the function |, | : I −→< ||e||, we conclude that
||e|| > I. Now let ψ(x) defines the set X. Define a function g : I −→< a
as follows.
If x < ||e|| ∧ ψ(x), then g(x) = f (x). If
x < ||e|| ∧ ¬ψ(x) ∧ ∃z < |e| (|z| > x ∧ ψ(|z|)) ∧ ∀z 0 < |z| ¬(z 0 > x ∧ ψ(z 0 )),
then g(x) = f (|z|). Since I is ∆bi -regular and g is definable of the same
complexity, then there is c < a such that g −1 (c) is unbounded in I and so
f −1 (c) ∩ X is unbounded in I. Theorem 3.10. Let M |= T2i+1 and I ⊂e M be ∆bi+1 -regular. Then there
is a structure K such that M ≺I,∆bi K.
Proof. We give a modification of the proof of [5], Theorem 7.2.4, which
is a similar result for models of P A.
Let X0 = I. Let f0 , f1 , · · · be an enumeration of all ∆bi+1 -definable
functions which are bounded on I. Let a ∈ I. The function f0 : I −→< a
has the properties of Lemma 3.9, so there is c < a in M such that f0−1 (c)∩X0
is unbounded. Put f0−1 (c) ∩ X0 = X1 . Proceed inductively and define
the ∆bi+1 -definable sets X0 , X1 , · · · . Let us denote this family of sets by
E. E has the finite intersection property and so there is an ultrafilter D
over I such that E ⊆ D, see [2], Proposition 4.1.3, for basic properties
of ultrafilters. Let D0 be the set of all ∆bi+1 -definable members of D. By
construction, for each ∆bi+1 -definable function f : I −→< a, there is c < a
such that f −1 (c) ∈ D0 .
Q
Consider
modulo D0 , i.e. D0 M . Let K be the
Q the ultrapower of M
subset of D0 M including all Σbi -definable functions of S2i restricted to I,
more exactly, all equivalent classes of such functions under the equivalence
relation
f ∼ g ⇔ {i ∈ I : f (i) = g(i)} ∈ D0 .
One can show that Loś’s theorem, restricted to ∆bi -formulas, holds for K.
For the atomic case, note that all Σbi -definable functions are closed under
Regular Cuts in Models of Bounded Arithmetic
19
+, ·, S, |.|, x x2 y and #. We just check the induction case for ∃. So let ϕ(x)
be a ∆bi -formula of the form ∃y < t ψ(x, y). Let
{x ∈ I : M |= ∃y < t ψ(f1 (x), y)} ∈ D0 .
where f1 ∈ K.
Since f1 is a Σbi -definable function of S2i , there is a term t1 such that
S2i ` ∀x∃y[∃z < t1 ((z = f1 (x)) ∧ (ϕ(z) → ψ(z, y)) ∧ (¬ϕ(z) → y = x))].
By Buss’ witnessing theorem ([1], p. 86, Theorem 1), there is a Σbi -definable
function f ∗ such that
{x ∈ I : M |= ψ(f1 (x), f ∗ (x))} ∈ D0 .
So, by induction hypothesis, we get K |= ψ([f1 ], [f ∗ ]).
Using Loś’s theorem we show that M ≺∆bi K as usual. Below we prove
that M ≺I K.
First of all, we show that I ⊂e K. We just check that I is downwardclosed. Let f be a constant function with value a ∈ I. Let g ∈ K and
g < f . This means that {i ∈ I : g(i) < f (i)} ∈ D0 . Consider the set
A = {i ∈ I : g(i) < a}. A is defined by a Σbi -formula, say ψ. Define the
function h : I −→< a + 1 by
h(x) = y ↔ [(ψ(x) → y = g(x)) ∧ (¬ψ(x) → y = a)].
By construction of D0 , there exists c < a + 1 such that h−1 (c) ∈ D0 . But
h−1 (a) ∈
/ D0 , and for c < a, h−1 (c) = g −1 (c). So there is c < a in I such
that g ∼ c. This means that g = c in D0 and so g ∈ I. To see M ≺I K, it
is easy to show that I < idI < M r I.
Corollary 3.11. Let M |= T2i+2 . Then there is an extension K of M
such that K |= S2i and K contains a nonstandard initial segment which is
a model of S2 .
Proof. Consider the standard initial segment in M and consider the
extension K of M as in Theorem 3.10. Let
J = {x ∈ K : ∀y ∈ M \ N, x < y}.
Now for any a ∈ K we have a > J implies |a| > J. So J is closed under
exponentiation and any bounded formula in J is equivalent to a sharply
bounded formula.
20
Morteza Moniri and S. Hosein Sajjadi
References
[1] S. R. Buss, Bounded Arithmetic, Bibliopolis, Naples, 1989.
[2] C. C. Chang and J. J. Keisler, Model Theory, North-Holland, 1990.
[3] P. Hájek and P. Pudlák, Metamathematics of First Order Arithmetic,
Springer-Verlag, 1993.
[4] J. Krajı́cek, Bounded Arithmetic, Propositional Logic and Complexity Theory, Cambridge University Press, Cambridge, 1995.
[5] R. Kossak and J. H. Schmerl, The Structure of Models of Peano Arithmetic, Clarendon Press, Oxford, 2006.
[6] M. Moniri and S. H. Sajjadi, Cuts and overspill properties in models of
bounded arithmetic, to appear.
[7] A. Pillay, Cuts in models of arithmetic, Lecture Notes in Mathematic,
no. 890, pp. 13–20.
Department of Mathematics
Institute for Research in Fundamental Sciences (IPM)
P.O. Box 19395-5746, Tehran, Iran
e-mail: ezmoniri@gmail.com
Department of Mathematics
Shahid Beheshti University, G. C., Evin
Tehran, Iran
e-mail: s.hoseinsajadi@gmail.com