1 Introduction
End extensions play a fundamental role in the model theory of arithmetic and have been studied intensively. The classical MacDowell–Specker theorem [Reference Mac Dowell and Specker13] showed that every model of
$\mathrm {PA}$
admits a proper elementary end extension. Around two decades later, Paris and Kirby [Reference Paris and Kirby16] studied the hierarchical version of the MacDowell–Specker theorem for fragments of
$\mathrm {PA}$
. In fact, they showed that for countable models, end extendibility with elementarity characterizes the collection strength of the ground model.
Theorem 1.1 (Paris–Kirby).
Let M be a countable model of
$\mathrm {I}\Delta _0$
. For each
$n\in \mathbb {N}$
, M satisfies
$\mathrm {B}\Sigma _{n+2}$
if and only if M has a proper
$\Sigma _{n+2}$
-elementary end extension K.
For the left-to-right direction, the above theorem does not explicitly specify what theory the end extension K can satisfy. The amount of elementarity stated in the theorem already implies
$K\models \mathrm {I} \Sigma _n$
. Paris–Kirby’s proof actually indicates that K cannot always satisfy
$\mathrm {I}\Sigma _{n+1}$
, since this would imply
$M\models \mathrm {B}\Sigma _{n+3}$
. Moreover, for each
$n\in \mathbb {N}$
, Cornaros and Dimitracopoulos [Reference Cornaros and Dimitracopoulos3] constructed a countable model of
$\mathrm {B}\Sigma _{n+2}$
which does not
$\Sigma _{n+1}$
-elementarily end extend to any model of
$\mathrm {I}\Sigma _{n+1}$
. So with
$\Sigma _{n+2}$
-elementarity, the theory that the end extension K can always satisfy lies between
$\mathrm {I}\Sigma _n$
and
$\mathrm {I}\Sigma _{n+1}$
, and the following question arises naturally.
Question 1.2 (Kaufmann–Clote).
For
$n\in \mathbb {N}$
, does every countable model
$M\models \mathrm {B}\Sigma _{n+2}$
have a proper
$\Sigma _{n+2}$
-elementary end extension
$K\models \mathrm {B}\Sigma _{n+1}$
?
The question was included in the list of open problems in [Reference Clote and Krajíček1, p. 12, Problem 33] edited by Clote and Krajíček. It was first raised by Clote in [Reference Clote2], where he noted that the same question in the context of models of set theory had previously been posed by Kaufmann in [Reference Kaufmann8]. In the same paper, Clote [Reference Clote2, Proposition 7] showed that every countable model
$M\models \mathrm {I}\Sigma _{n+2}$
admits a
$\Sigma _{n+2}$
-elementary proper end extension to some
$K\models M\text {-}\mathrm {B}\Sigma _{n+1}$
, which is defined as follows:
$M\text {-}\mathrm {B}\Sigma _{n+1}$
is the class of formulas of the form
$$\begin{align*}\forall{{x}<}\ a \ \exists{y}\ \varphi(x,y)\rightarrow\exists{b}\ \forall{{x}<}\ a \ \exists{{y}<}\ b \ \varphi(x,y), \end{align*}$$
where
$a\in M$
and
$\varphi (x,y)\in \Sigma _{n+1}$
with parameters in K. Cornaros and Dimitracopoulos [Reference Cornaros and Dimitracopoulos3] showed that every countable model
$M\models \mathrm {B}\Sigma _{n+2}$
has a
$\Sigma _{n+2}$
-elementary proper end extension
$K\models \mathrm {B}\Sigma _{n+1}^-$
(the parameter-free
$\Sigma _{n+1}$
-collection). In this paper, we give an affirmative answer to Question 1.2.
The original proof of Theorem 1.1 is based on a first-order restricted ultrapower construction. The ultrapower is generated by a single element when viewed from the ground model. One can show that, by relativizing the proof of the fact that pointwise
$\Sigma _{n+1}$
-definable models do not satisfy
$\mathrm {B}\Sigma _{n+1}$
(e.g.,[Reference Hájek and Pudlák7, Lemma IV.1.41]), such ultrapowers always fail to satisfy
$\mathrm {B}\Sigma _{n+1}$
as required in Question 1.2. To address this issue, we expand our model to a second-order structure that satisfies
$\mathrm {WKL}_0$
; the end extension K will then be a second-order restricted ultrapower with respect to this second-order structure.
For us, one of the motivations for studying this question is to find a model-theoretic characterization of (countable) models of
$\mathrm {I}\Sigma _{n+2}$
analogous to Theorem 1.1. Despite the fact that the end extension in Question 1.2 is insufficient for characterization, a slight generalization of it will suffice. We will show that for any countable model
$M\models \mathrm {I}\Delta _0+\mathrm {exp}$
,
$M\models \mathrm {I}\Sigma _{n+2}$
if and only if M admits a proper
$\Sigma _{n+2}$
-elementary end extension
$K\models M\text {-}\mathrm {I}\Sigma _{n+1}$
, whose definition is similar to that of
$M\text {-}\mathrm {B}\Sigma _{n+1}$
.
The regularity-type principles are the keys to connecting end extensions and the arithmetic theories satisfied by the ground model. Through the end extension, we can employ a “nonstandard analysis” style argument to prove certain types of regularity principles in the ground model. An example of such an argument related to Question 1.2 is provided below.
Proposition 1.3. For each
$n\in \mathbb {N}$
, let
$M\models \mathrm {I}\Delta _0$
. If M admits a proper
$\Sigma _{n+2}$
-elementary end extension
$K\models \mathrm {B}\Sigma _{n+1}$
, then M satisfies the following principle:
$$\begin{align*}\forall{x}\ \exists{{y}<}\ a \ \varphi(x,y)\rightarrow\exists{{y}<}\ a \ \exists^{\mathrm{cf}}{x}\ \varphi(x,y),\end{align*}$$
where
$a\in M $
,
$\varphi (x,y)\in \Pi _{n+1}(M)$
and
$\exists ^{\mathrm {cf}} x$
abbreviates
$\forall {b}\ \exists x>b$
.
Proof. Suppose
$M\models \forall {x}\ \exists {{y}<}\ a \ \varphi (x,y)$
. Because M and K both satisfy
$\mathrm {B}\Sigma _{n+1}$
, the formula
$\forall {x}\ \exists {{y}<}\ a \ \varphi (x,y)$
is equivalent to some
$\Pi _{n+1}$
formula in both M and K. Then by
$\Sigma _{n+2}$
-elementarity,
$K\models \forall {x}\ \exists {{y}<}\ a \ \varphi (x,y)$
. Pick some arbitrary
$d>M$
in K and let
$c<a$
such that
$K\models \varphi (d,c)$
. Now for each
$b\in M$
,
$K\models \exists {{x}>}\ b \ \varphi (x,c)$
and it is witnessed by d. Transferring each of these formulas back to M by elementarity, we have
$M\models \exists {{x}>}\ b \ \varphi (x,c)$
for any
$b\in M$
, which means
$M\models \exists ^{\mathrm {cf}}{x}\ \varphi (x,c)$
.
We call the principle in Proposition 1.3 the weak regularity principle, and denote it by
$\mathrm {WR}\varphi $
. The above proposition, together with the affirmative answer to Question 1.2, implies that
$\mathrm {B}\Sigma _{n+2}\vdash \mathrm {WR}\Pi _{n+1}$
for each
$n\in \mathbb {N}$
.
Similarly to the argument above, we will show that if the end extension K is a model of
$M\text {-}\mathrm {I}\Sigma _{n+1}$
, then the ground model M will satisfy some stronger form of the weak regularity principle that implies
$\mathrm {I}\Sigma _{n+2}$
.
The paper is organized as follows. In Section 2, we present the necessary notations and fundamental facts regarding models of arithmetic. In Section 3, we review the definition of the second-order restricted ultrapower, and state some basic properties. In Section 4, we provide an affirmative answer to Question 1.2 in Theorem 4.3, and present the construction of an end extension that characterizes countable models of
$\mathrm {I}\Sigma _{n+2}$
as mentioned above. In Section 5, we formally introduce the weak regularity principle
$\mathrm {WR}\varphi $
, and calibrate its strength within the I-B hierarchy. Finally, combining the results of Section 4 and Section 5, we establish a model-theoretic characterization of countable models of
$\mathrm {I}\Sigma _{n+2}$
analogous to Theorem 1.1.
2 Preliminaries
We assume the reader is familiar with some basic concepts and facts in the model theory of first- and second-order arithmetic [Reference Hájek and Pudlák7, Reference Kaye9]. We reserve both the symbols
$\mathbb {N}$
and
$\omega $
for the set of standard natural numbers. For each
$n\in \mathbb {N}$
, let
$\Sigma _n$
and
$\Pi _n$
be the usual classes of formulas in the arithmetic hierarchy of first-order arithmetic. Given a model of first-order arithmetic M,
$\Sigma _n(M)$
is the class of
$\Sigma _n$
formulas, potentially including parameters from M that are not explicitly shown.
$\Pi _n(M)$
and other formula classes are defined similarly. A formula is
$\Delta _n$
over M if it is equivalent to both a
$\Sigma _n$
and a
$\Pi _n$
formula in M; if M is clear from context, we simply write
$\Delta _n$
for short.
$\Sigma _n\wedge \Pi _n$
is the class of formulas which is the conjunction of a
$\Sigma _n$
and a
$\Pi _n$
formula, and
$\Sigma _n\vee \Pi _n$
is defined similarly.
$\Sigma _0(\Sigma _n)$
is the closure of
$\Sigma _n$
formulas under Boolean operations and bounded quantification.
$\Sigma _n^0$
,
$\Pi _n^0$
,
$\Delta _n^0$
and
$\Sigma _0(\Sigma _n^0)$
are their second-order variants, respectively. Given a model of second-order arithmetic
$(M,\mathcal {X})$
,
$\Sigma _n^0(M,\mathcal {X})$
is defined similarly to the definition of
$\Sigma _n(M)$
, where the implicit parameters include both first- and second-order parameters in
$(M,\mathcal {X})$
.
$\Pi _n^0(M,\mathcal {X})$
and
$\Delta _n^0(M,\mathcal {X})$
are defined similarly. Finally,
$\exists ^{\mathrm {cf}} x\dots $
is the abbreviation of
$\forall {b}\ \exists {x>b}\dots $
.
For each
$n\in \mathbb {N}$
, let
$\mathrm {B}\Sigma _n$
and
$\mathrm {I}\Sigma _n$
be the collection scheme and the induction scheme for
$\Sigma _n$
formulas respectively. We assume that all the
$\mathrm {B}\Sigma _n$
include
$\mathrm {I}\Delta _0$
, and all the theories considered include
$\mathrm {PA}^-$
, which is the theory of the non-negative parts of discretely ordered rings. Paris and Kirby [Reference Paris and Kirby16, Theorem A] proved that
$\mathrm {I}\Sigma _{n+1}\vdash \mathrm {B}\Sigma _{n+1}\vdash \mathrm {I}\Sigma _n$
and none of the converses holds for each
$n\in \mathbb {N}$
. The hierarchy of theories containing all the
$\mathrm {I}\Sigma _n$
and
$\mathrm {B}\Sigma _n$
is referred to as the I-B hierarchy.
$\mathrm {I}\Sigma _n^0$
and
$\mathrm {B}\Sigma _n^0$
are their second-order counterparts, respectively.
We adopt the standard pairing function, where the code of an ordered pair
$(a,b)$
under this pairing function is denoted by
$\langle a,b\rangle $
. For any element c in some model
$M\models \mathrm {I}\Delta _0+\mathrm {exp}$
, we identify c with a subset of M by defining
$x\in c$
to mean the xth digit in the binary expansion of c is
$1$
. Fixing any proper cut I of M, we say that a set
$A\subseteq I$
is coded in M if there is some
$c\in M$
such that
$A=\{x\in I\mid M\models x\in c\}$
. We define
$$\begin{align*}\mathrm{SSy}_I(M):=\{A\subseteq I\mid A\text{ is coded in }M\}. \end{align*}$$
$\mathrm {RCA}_0$
is the subsystem of second-order arithmetic consisting of
$\mathrm {I}\Sigma _1^0$
and
$\Delta _1^0$
-comprehension. The system
$\mathrm {WKL}_0$
consists of
$\mathrm {RCA}_0$
and a statement asserting that every infinite binary tree has an infinite path. For each
$n\in \mathbb {N}$
, every countable model of
$\mathrm {B}\Sigma _{n+2}^0$
admits a countable
$\omega $
-extension (i.e., an extension only adding second-order objects) to some model satisfying
$\mathrm {WKL}_0+\mathrm {B}\Sigma _{n+2}^0$
[Reference Hájek6].
Considering extensions of models of first-order arithmetic, we say that an extension
$M\subseteq K$
of models of first-order arithmetic is
$\Sigma _n$
-elementary, if all the
$\Sigma _n(M)$
formulas are absolute between M and K, and we write
$M\preccurlyeq _{\Sigma _n} K$
to denote this. For any extension
$M\subseteq K$
and
$d\in K$
, we write
$d>M$
if
$K\models d>c$
for any
$c\in M$
. We say that an extension
$M\subseteq K$
is an end extension, if for every
$d\in K\setminus M$
, we have
$d>M$
. This is denoted by
$M\subseteq _{\mathrm {e}}K$
, or
$M\preccurlyeq _{\mathrm {e},\Sigma _n}K$
if we also have
$M\preccurlyeq _{\Sigma _n}K$
. We say an extension
$M\subseteq K$
is proper if
$M\neq K$
.
Formally, for a model of second-order arithmetic
$(M,\mathcal {X})$
, we view
$(M,\mathcal {X})$
as a two-sorted first-order structure with number sort M and set sort
$\mathcal {X}$
, where elements in
$\mathcal {X}$
are treated as syntactic objects and are not necessarily subsets of M. Under this convention, we write
$(M,\mathcal {X})\subseteq (K,\mathcal {Y})$
if it is an extension of the corresponding two-sorted structure, i.e.,
$M\subseteq K$
,
$\mathcal {X}\subseteq \mathcal {Y}$
, and for any
$x\in M$
and
$A\in \mathcal {X}$
,
$$\begin{align*}(M,\mathcal{X})\models x\in A\iff (K,\mathcal{Y})\models x\in A.\end{align*}$$
Here, on the left-hand side of the equivalence, the second-order object A is interpreted by a subset of M; while it is interpreted by a subset of K on the right-hand side. In this paper, the second-order parts of our extensions usually remain the same, i.e.,
$\mathcal {X}=\mathcal {Y}$
. We write
$(M,\mathcal {X})\preccurlyeq _{\Sigma _n^0}(K,\mathcal {Y})$
if all the
$\Sigma _n^0(M,\mathcal {X})$
formulas are absolute between the two structures. We say that an extension of second-order structures is an end extension if its first-order part is an end extension. We denote this by
$(M,\mathcal {X})\subseteq _{\mathrm {e}}(K,\mathcal {Y})$
, or
$(M,\mathcal {X})\preccurlyeq _{\mathrm {e},\Sigma _n^0}(K,\mathcal {Y})$
if we also have
$(M,\mathcal {X})\preccurlyeq _{\Sigma _n^0}(K,\mathcal {Y})$
.
3 Second-order restricted ultrapowers
The second-order restricted ultrapower resembles the usual ultrapower construction in model theory. Usually, in the model theory of arithmetic, we take the index set for the ultrapower construction to be the model itself, but instead of working on the class of all subsets of the model and all functions from the model to itself, we only consider a restricted class of subsets and functions. For completeness, we review the definition and some basic facts. All the results in this section appear in [Reference Kirby10] except Lemma 3.6, Corollary 3.7, and Theorem 3.9. Throughout this section, we fix some arbitrary second-order structure
$(M,\mathcal {X})\models \mathrm {RCA}_0$
.
Definition 3.1 (Second-order restricted ultrapower).
The second-order part
$\mathcal {X}$
of
$(M,\mathcal {X})$
forms a Boolean algebra under inclusion and Boolean operations. Let
${\mathcal {U}}$
be a non-principal ultrafilter on
$\mathcal {X}$
whose elements are all cofinal in M, and
$\mathcal {F}$
be the class of all the total functions from M to M in
$\mathcal {X}$
. Define an equivalence relation
$\sim $
on
$\mathcal {F}$
by
$$\begin{align*}f\sim g\iff \{i\in M\mid f(i)=g(i)\}\in {\mathcal{U}},\end{align*}$$
where
$f,g\in \mathcal {F}$
. Let
$\mathcal {F}/\mathcal {U}$
be the set of equivalence classes
$[f]$
for
$f\in \mathcal {F}$
modulo
$\sim $
. The interpretations of symbols in the language of first-order arithmetic in
$\mathcal {F}/\mathcal {U}$
are defined by
$$\begin{align*}[f]+[g]&=[f+g],\\ [f]\times[g]&=[f\times g],\end{align*}$$
$$\begin{align*}[f]<[g]\iff \{i\in M\mid f(i)<g(i)\}\in {\mathcal{U}}.\end{align*}$$
Here
$f+g$
and
$f\times g$
are the pointwise addition and multiplication of f and g as functions. M naturally embeds into
$\mathcal {F}/\mathcal {U}$
by identifying elements of M with constant functions. Moreover,
$\mathcal {F}/\mathcal {U}$
is a proper extension of M by considering the equivalence class of the identity function on M.
$\mathcal {F}/\mathcal {U}$
admits a natural second-order expansion inherited from
$\mathcal {X}$
, namely for
$A\in \mathcal {X}$
and
$[f]\in \mathcal {F}/\mathcal {U}$
, we define
$$\begin{align*}[f]\in A\iff \{i\in M\mid f(i)\in A\}\in {\mathcal{U}}.\end{align*}$$
We denote the expanded structure of the ultrapower by
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
. It is easy to show that for
$i\in M$
and
$A\in \mathcal {X}$
,
$$\begin{align*}(M,\mathcal{X})\models i\in A\iff (\mathcal{F}/\mathcal{U},\mathcal{X})\models i\in A.\end{align*}$$
So we may view
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
as an extension of
$(M,\mathcal {X})$
, where the second-order part remains the same. We call
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
the second-order restricted ultrapower of
$(M,\mathcal {X})$
(with respect to
${\mathcal {U}}$
).
The first-order restricted ultrapower is defined similarly, but with
$\mathcal {F}$
and
${\mathcal {U}}$
replaced by the corresponding first-order definable classes. For example, in the construction of an
$\Delta _1$
-ultrapower,
${\mathcal {U}}$
is an ultrafilter on the class of
$\Delta _1$
-definable subsets, and
$\mathcal {F}$
is the class of
$\Delta _1$
-definable total functions.
From now on, we also fix an ultrafilter
${\mathcal {U}}$
on
$\mathcal {X}$
whose elements are all cofinal in M.
Generally, Łoś’s theorem does not hold for restricted ultrapowers, but a restricted version of it does hold:
Theorem 3.2 (Restricted Łoś’s theorem).
Let
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
be the second-order restricted ultrapower of
$(M,\mathcal {X})$
. Then the following holds:
-
(1) If
$\varphi (\overline x)$
is a
$\Sigma _1^0(M,\mathcal {X})$
formula, then
$$\begin{align*}(\mathcal{F}/\mathcal{U},\mathcal{X})\models \varphi(\overline {[f]})\iff \exists {{A}\in{{\mathcal{U}}}}, A\subseteq \{i\in M\mid (M,\mathcal{X})\models \varphi(\overline {f(i)})\}.\end{align*}$$
-
(2) If
$\varphi (\overline x)$
is a
$\Delta _1^0(M,\mathcal {X})$
formula over
$(M,\mathcal {X})$
, then Here the right-hand side makes sense by the
$$\begin{align*}(\mathcal{F}/\mathcal{U},\mathcal{X})\models \varphi(\overline {[f]})\iff \{i\in M\mid (M,\mathcal{X})\models \varphi(\overline {f(i)})\}\in{\mathcal{U}}.\end{align*}$$
$\Delta _1^0$
-comprehension of
$(M,\mathcal {X})$
.
Corollary 3.3.
$(M,\mathcal {X})\preccurlyeq _{\Sigma _2^0}(\mathcal {F}/\mathcal {U},\mathcal {X})$
.
Proof. Let
$(M,\mathcal {X})\models \forall {x}\ \exists {y}\ \psi (x,y)$
for some
$\psi (x,y)\in \Delta _0^0(M,\mathcal {X})$
. By choosing the least witness y of
$\psi (x,y)$
, we may assume
$\psi (x,y)$
defines the graph of a total function
$f\in \mathcal {F}$
, so
$(M,\mathcal {X})\models \psi (x,f(x))$
for all
$x\in M$
. In particular, for each
$g\in \mathcal {F}$
and
$x\in M$
,
$$\begin{align*}(M,\mathcal{X})\models\psi(g(x),f\circ g(x)).\end{align*}$$
Here
$f\circ g$
is the composition of f and g. By Theorem 3.2,
$$\begin{align*}(\mathcal{F}/\mathcal{U},\mathcal{X})\models\psi([g],[f\circ g])\end{align*}$$
for each
$[g]\in \mathcal {F}/\mathcal {U}$
. So
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \forall {x}\ \exists {y}\ \psi (x,y)$
.
We ensure that the ultrapower is an end extension of the ground model by the following definition and lemma.
Definition 3.4. We say that
${\mathcal {U}}$
is additive, if whenever
$f\in \mathcal {F}$
is bounded, then there is a
$c\in M$
such that
$\{i\in M\mid f(i)=c\}\in {\mathcal {U}}$
.
Lemma 3.5. If
${\mathcal {U}}$
is additive, then
$\mathcal {F}/\mathcal {U}$
is an end extension of M.
Proof. If
$\mathcal {F}/\mathcal {U}\models [f]<b$
for some
$b\in M$
, then we may define
$$\begin{align*}g(i)=\begin{cases} 0, & \text{if }f(i)\geqslant b,\\ f(i), & \text{if }f(i)<b. \end{cases}\end{align*}$$
Then g is a bounded function in
$\mathcal {F}$
and
$[g]=[f]$
. The additiveness of
${\mathcal {U}}$
implies that there is some
$c\in M$
such that
$\{i\in M\mid g(i)=c\}\in {\mathcal {U}}$
, that is,
$\mathcal {F}/\mathcal {U}\models [g]= [f]=c$
.
The following lemma and corollary enable us to transfer the comprehension in
$(M,\mathcal {X})$
into the ultrapower via
$\Sigma _2^0$
-elementarity, and reduce the case of
$\mathrm {B}\Sigma _{n+2}$
to
$\mathrm {B}\Sigma _2^0$
uniformly in the construction of our main result.
Lemma 3.6. For each
$n\geqslant 1$
, if
$(M,\mathcal {X})$
satisfies
$\Sigma _n$
-comprehension, then each instance of
$\Sigma _n$
- and
$\Pi _n$
-comprehension in
$(M,\mathcal {X})$
is transferred to
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
. Formally, for any first-order formula
$\varphi (x)$
in
$\Sigma _n(M)$
or
$\Pi _n(M)$
, if there is some
$A\in \mathcal {X}$
such that
$$\begin{align*}(M,\mathcal{X})\models\forall{x}\ (x\in A\leftrightarrow \varphi(x)),\end{align*}$$
then
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \forall {x}\ (x\in A\leftrightarrow \varphi (x))$
as well.
Proof. Fix some
$(M,\mathcal {X})$
that satisfies
$\Sigma _n$
-comprehension. We prove the statement for all the
$\varphi (x)$
in
$\Sigma _k(M)$
and
$\Pi _k(M)$
simultaneously by induction on k.
For
$k=1$
, let
$\varphi (x)$
be any formula in
$\Sigma _1(M)$
or
$\Pi _1(M)$
,
$A\in \mathcal {X}$
and
$(M,\mathcal {X})\models \forall {x}\ (x\in A\leftrightarrow \varphi (x))$
. Since
$\forall {x}\ (x\in A\leftrightarrow \varphi (x))$
is a
$\Pi _2^0(M,\mathcal {X})$
formula,
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \forall {x}\ (x\in A\leftrightarrow \varphi (x))$
by Corollary 3.3.
For the induction step, suppose
$k<n$
and the statement holds for all the formulas in
$\Sigma _k(M)$
and
$\Pi _k(M)$
. Take any
$\varphi (x):=\exists {y}\ \psi (x,y)\in \Sigma _{k+1}(M)$
where
$\psi (x,y)\in \Pi _k(M)$
. Suppose
$$ \begin{align} (M,\mathcal{X})\models\forall{x}\ (x\in A\leftrightarrow \exists{y}\ \psi(x,y)) \end{align} $$
for some
$A\in \mathcal {X}$
. By
$\Sigma _n$
-comprehension in
$(M,\mathcal {X})$
, there exists some
$B\in \mathcal {X}$
such that
$(M,\mathcal {X})$
satisfies
$$ \begin{align} \forall{\langle x,y\rangle}\ (\langle x,y\rangle \in B\leftrightarrow \psi(x,y)). \end{align} $$
(3.6.1) and (3.6.2) in
$(M,\mathcal {X})$
imply that
$(M,\mathcal {X})$
also satisfies
$$ \begin{align} \forall{x}\ (x\in A\leftrightarrow \exists{y}\ \langle x,y\rangle \in B). \end{align} $$
By the induction hypothesis and Corollary 3.3, (3.6.2) and (3.6.3) are transferred to
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
. Combining (3.6.2) and (3.6.3) in
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
, we have
$$\begin{align*}(\mathcal{F}/\mathcal{U},\mathcal{X})\models\forall{x}\ (x\in A\leftrightarrow \exists{y}\ \psi(x,y)).\end{align*}$$
The case for
$\varphi (x)\in \Pi _{k+1}(M)$
is exactly the same. This completes the induction.
Corollary 3.7. If
$(M,\mathcal {X})$
satisfies
$\Sigma _n$
-comprehension for some
$n\in \mathbb {N}$
, then
$M\preccurlyeq _{\Sigma _{n+2}}\mathcal {F}/\mathcal {U}$
when viewed as an extension of models of first-order arithmetic.
Proof. Let
$M\models \forall {x}\ \exists {y}\ \psi (x,y)$
for some
$\psi (x,y)\in \Pi _n(M)$
. By
$\Sigma _n$
-comprehension in
$(M,\mathcal {X})$
, there exists some
$A\in \mathcal {X}$
such that
$(M,\mathcal {X})$
satisfies
$$\begin{align*}\forall{\langle x,y\rangle}\ (\langle x,y\rangle\in A\leftrightarrow \psi(x,y)).\end{align*}$$
Therefore,
$(M,\mathcal {X})$
also satisfies
$$\begin{align*}\forall{x}\ \exists{y}\ \langle x,y\rangle\in A.\end{align*}$$
By Lemma 3.6 and Corollary 3.3, both formulas hold in
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
, which implies
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \forall {x}\ \exists {y}\ \psi (x,y)$
.
For technical reasons, we need another kind of ultrapower construction in the model theory of arithmetic called the coded ultrapower, which appears in [Reference Paris15, Theorem 9] and [Reference Kirby and Paris11, Theorem 3]. We fix some
$M\models \mathrm {I}\Sigma _1$
and a proper end extension
$M\subseteq _{\mathrm {e}} L$
such that
$L\models \mathrm {I}\Delta _0+\mathrm {exp}$
.
Definition 3.8. Let
${\mathcal {U}}$
be an ultrafilter on
$\mathrm {SSy}_M(L)$
whose elements are all cofinal in M, and let
$\mathcal {G}$
be the class of all total functions mapping from M to L that are coded in L. Then
$\mathcal {G}/{\mathcal {U}}$
is defined in the same way as
$\mathcal {F}/\mathcal {U}$
in Definition 3.1, and we call it the M-coded ultrapower of L with respect to
${\mathcal {U}}$
.
For our purpose, only Łoś’s Theorem for
$\Delta _0$
formulas in
$\mathcal {G}/{\mathcal {U}}$
and the fact that
$\mathcal {G}/{\mathcal {U}}\models \mathrm {I}\Delta _0$
are needed. Their proofs essentially appear in [Reference Paris15, Theorem 11], but using a slightly different definition of the coded ultrapowers.
Theorem 3.9. Let
$\mathcal {G}/{\mathcal {U}}$
be a M-coded ultrapower of L. Then the following hold:
-
(1) Let
$\varphi (\overline x)$
be a
$\Delta _0(L)$
formula, then
$$\begin{align*}\mathcal{G}/{\mathcal{U}}\models\varphi(\overline {[f]})\iff \{i\in M\mid L\models\varphi(\overline {f(i)})\}\in {\mathcal{U}}.\end{align*}$$
-
(2)
$\mathcal {G}/{\mathcal {U}}\models \mathrm {I}\Delta _0$
.
4 Constructions of end extensions
In this section, we present the constructions of end extensions by the second-order restricted ultrapower construction. We first answer Question 1.2 affirmatively. In view of Corollary 3.7, it suffices to deal with the case in which
$(M,\mathcal {X})\models \mathrm {B}\Sigma _2^0$
.
Theorem 4.1. For any countable model
$(M,\mathcal {X})\models \mathrm {B}\Sigma _2^0+\mathrm {WKL}_0$
, there is a proper end extension
$(M,\mathcal {X})\preccurlyeq _{\mathrm {e},\Sigma _2^0}(K,\mathcal {X})\models \mathrm {B}\Sigma _1^0$
.
We will give two proofs. In both of our constructions, the end extension is given by a second-order restricted ultrapower construction. The first proof, presented below, was suggested by Tin Lok Wong. We ensure that the ultrapower of
$(M,\mathcal {X})$
satisfies
$\mathrm {B}\Sigma _1^0$
by properly embedding it into a coded ultrapower as an initial segment. The second proof, presented on Page 11, guarantees that the ultrapower satisfies
$\mathrm {B}\Sigma _1^0$
directly by the construction of the ultrafilter.
$\mathrm {WKL}_0$
plays a central role in both constructions.
The first proof of Theorem 4.1.
First by [Reference Enayat and Wong5, Theorem 4.6], there is a countable end extension
$M\subseteq _{\mathrm {e}}L\models \mathrm {I}\Delta _0$
such that
$\mathcal {X}=\mathrm {SSy}_M(L)$
.
Let
$$\begin{align*}\mathcal{G}&:=\{g\mid g\text{ is a total function from } M \text{ to } L \text{ coded in } L\},\\\mathcal{F}&:=\{f\in \mathcal{X}\mid f \text{ is a total function from } M \text{ to } M\}. \end{align*}$$
For any ultrafilter
${\mathcal {U}}$
on
$\mathcal {X}$
whose elements are all cofinal in M, let
$\mathcal {G}/{\mathcal {U}}$
and
$(\mathcal{F}/\mathcal{U},\mathcal{X})$
be the coded ultrapower (see Definition 3.8) and the second-order restricted ultrapower with respect to
${\mathcal {U}}$
respectively. Since
$\mathcal {X}=\mathrm {SSy}_M(L)$
, we may regard
$\mathcal {F}$
as a subset of
$\mathcal {G}$
, and thus
$\mathcal {F}/\mathcal {U}$
naturally embeds into
$\mathcal {G}/{\mathcal {U}}$
.
We want to construct a sufficiently generic ultrafilter
${\mathcal {U}}$
, such that both M and
$\mathcal {F}/\mathcal {U}$
are proper initial segments of
$\mathcal {G}/{\mathcal {U}}$
. We construct
${\mathcal {U}}$
in
$\omega $
many stages. For each
$k\in \mathbb {N}$
, we construct some
$A_k\in \mathcal {X}$
that is cofinal in M and
$A_k\supseteq A_{k+1}$
. We enumerate all the pairs
$\langle f,g\rangle $
such that
$f\in \mathcal {F}$
and
$g\in \mathcal {G}$
as
$\{\langle f_k,g_k\rangle \}_{k\in \mathbb {N}}$
, and all the bounded functions in
$\mathcal {F}$
as
$\{h_k\}_{k\in \mathbb {N}}$
. Here we identify each element
$g\in \mathcal {G}$
with some element in L that codes g.
Stage
$\boldsymbol {0}$
: Set
$A_0=M$
.
Stage
$\boldsymbol {2k+1}$
(
$\mathcal {F}/\mathcal {U}\subseteq _{\mathrm {e}}\mathcal {G}/{\mathcal {U}}$
): Consider
$$\begin{align*}A:=A_{2k}\cap\{x\in M\mid L\models g_k(x)<f_k(x)\}.\end{align*}$$
Since
$L\models \mathrm {I}\Delta _0$
,
$A\in \mathrm {SSy}_M(L)=\mathcal {X}$
. If A is cofinal in M, then let
$A_{2k+1}=A$
. Otherwise let
$A_{2k+1}=A_{2k}$
and proceed to the next stage.
Stage
$\boldsymbol {2k+2}$
(
$M\subseteq _{\mathrm {e}}\mathcal {F}/\mathcal {U}$
): Assume
$h_k$
is bounded by
$b\in M$
. Then
$$\begin{align*}(M,\mathcal{X})\models\exists^{\mathrm{cf}}{x}\ \exists{{y}<}\ b \ (x\in A_{2k+1}\wedge h_k(x)=y).\end{align*}$$
Since
$(M,\mathcal {X})\models \mathrm {B}\Sigma _2^0$
, there is some
$c<b$
such that the set
$$\begin{align*}A_c:=\{x\in M\mid (M,\mathcal{X})\models x\in A_{2k+1}\wedge h_k(x)=c\}\end{align*}$$
is cofinal in M.
$A_c\in \mathcal {X}$
by
$\Delta _1^0$
-comprehension in
$(M,\mathcal {X})$
. Let
$A_{2k+2}=A_c$
and proceed to the next stage.
Finally, let
${\mathcal {U}}:=\{A\in \mathcal {X}\mid \exists {{k}\in } \ {\mathbb {N}} \ A_{k}\subseteq A\}$
. It is not hard to see that
${\mathcal {U}}$
is an ultrafilter and each element of
${\mathcal {U}}$
is cofinal in M. This completes the construction of
${\mathcal {U}}$
.
Verification: First we verify that
$\mathcal {F}/\mathcal {U}\subseteq _{\mathrm {e}}\mathcal {G}/{\mathcal {U}}$
and it is proper. We show that for any
$g\in \mathcal {G}$
and
$f\in \mathcal {F}$
, either
$\mathcal {G}/{\mathcal {U}}\models [g]\geqslant [f]$
or there exists some
$\hat {g}\in \mathcal {F}$
such that
$\mathcal {G}/{\mathcal {U}}\models [g]=[\hat {g}]$
. Take
$k\in \mathbb {N}$
such that
$\langle f_k,g_k\rangle =\langle f,g\rangle $
. At Stage
$2k$
, if
$$\begin{align*}A=A_{2k}\cap\{x\in M\mid L\models g(x)<f(x)\}\end{align*}$$
is cofinal in M, then
$\{x\in M\mid L\models g(x)<f(x)\}\in {\mathcal {U}}$
. Let
$\hat {g}\in \mathcal {G}$
be defined by
$$\begin{align*}\hat{g}(x)=\begin{cases} g(x), & \text{if } g(x)< f(x),\\ 0, & \text{if } g(x)\geqslant f(x). \end{cases}\end{align*}$$
Then
$[\hat {g}]\in \mathcal {F}/\mathcal {U}$
and
$\mathcal {G}/{\mathcal {U}}\models [g]=[\hat {g}]$
. Otherwise, if A is bounded in M, then we are forced to have
$\{x\in M\mid L\models g(x)\geqslant f(x)\}\in {\mathcal {U}}$
. By Theorem 3.9,
$\mathcal {G}/{\mathcal {U}}\models [g]\geqslant [f]$
. The fact that
$\mathcal {F}/\mathcal {U}\subseteq _{\mathrm {e}}\mathcal {G}/{\mathcal {U}}$
is proper follows from considering the constant function
$g(x)\equiv c$
for any
$c>M$
in L.
Next we verify that
$M\subseteq _{\mathrm {e}}\mathcal {F}/\mathcal {U}$
. By Lemma 3.5, it suffices to show that
${\mathcal {U}}$
is additive with respect to
$\mathcal {F}$
. Suppose
$h\in \mathcal {F}$
is bounded and
$h=h_k$
for some
$k\in \mathbb {N}$
. At Stage
$2k+2$
, the choice of
$A_{2k+2}$
forces
$$\begin{align*}\{x\in M\mid (M,\mathcal{X})\models h(x)=c\}\in{\mathcal{U}}\end{align*}$$
for some
$c\in M$
, so
${\mathcal {U}}$
is additive with respect to
$\mathcal {F}$
.
The fact that
$(M,\mathcal {X})\preccurlyeq _{\Sigma _2^0}(\mathcal {F}/\mathcal {U},\mathcal {X})$
follows from Corollary 3.3.
Finally we verify that
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \mathrm {B}\Sigma _1^0$
. By Theorem 3.9,
$\mathcal {G}/{\mathcal {U}}\models \mathrm {I}\Delta _0$
. Since
$\mathcal {G}/{\mathcal {U}}$
is a proper end extension of
$\mathcal {F}/\mathcal {U}$
,
$(\mathcal {F}/\mathcal {U},\mathrm {SSy}_{\mathcal {F}/\mathcal {U}}(\mathcal {G}/{\mathcal {U}}))$
satisfies
$\mathrm {B}\Sigma _1^0$
. It suffices to show that
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
embeds into
$(\mathcal {F}/\mathcal {U},\mathrm {SSy}_{\mathcal {F}/\mathcal {U}}(\mathcal {G}/{\mathcal {U}}))$
. For each
$A\in \mathcal {X}=\mathrm {SSy}_M(L)$
, let
$a\in L$
be the element that codes
$A\subseteq M$
. By Theorem 3.9 and Definition 3.1, it is not hard to prove that for each
$f\in \mathcal {F}$
,
$$\begin{align*}(\mathcal{F}/\mathcal{U},\mathcal{X})\models [f]\in A\iff(\mathcal{G}/{\mathcal{U}},\mathcal{X})\models [f]\in a.\end{align*}$$
So we may embed the second-order part
$\mathcal {X}$
of
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
into
$\mathrm {SSy}_{\mathcal {F}/\mathcal {U}}(\mathcal {G}/{\mathcal {U}})$
by sending A to the subset of
$\mathcal {F}/\mathcal {U}$
coded by a. Since
$(\mathcal {F}/\mathcal {U},\mathrm {SSy}_{\mathcal {F}/\mathcal {U}}(\mathcal {G}/{\mathcal {U}}))\models \mathrm {B}\Sigma _1^0$
, we have
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \mathrm {B}\Sigma _1^0$
.
Even though it is simple, this construction does not reveal a syntactical proof of the fact that
$\mathrm {B}\Sigma _{n+2}\vdash \mathrm {WR}\Pi _{n+1}$
. We ensure that
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \mathrm {B}\Sigma _1^0$
by embedding it in a larger ultrapower
$\mathcal {G}/{\mathcal {U}}$
as an initial segment, and the core argument is wrapped within the construction of
$\mathcal {G}/{\mathcal {U}}$
.
Our second construction directly guarantees each instance of
$\mathrm {B}\Sigma _1^0$
in the ultrapower, and therefore provides more insights. It relies on a simple yet powerful lemma, which states that second-order universes of models of
$\mathrm {WKL}_0$
are closed under the operation that produces choice functions for ranges of
$\Pi _1^0$
-definable multi-valued bounded functions. The lemma also leads to a syntactical proof of the fact that
$\mathrm {B}\Sigma _{n+2}\vdash \mathrm {WR}\Pi _{n+1}$
(see Lemma 5.2).
Lemma 4.2. Fix a model
$(M,\mathcal {X})\models \mathrm {WKL}_0$
. Let
$\theta (x,y,z)\in \Delta _0^0(M,\mathcal {X})$
. If
$(M,\mathcal {X})\models \forall {x}\ \exists {{y}<{f(x)}}\ \forall {z}\ \theta (x,y,z)$
for some total function
$f\in \mathcal {X}$
, then there is a total function
$P\in \mathcal {X}$
such that
$$\begin{align*}(M,\mathcal{X})\models\forall{x}\ (P(x)<f(x)\wedge\forall{z}\ \theta(x,P(x),z)).\end{align*}$$
Proof. Consider the following tree T which is
$\Delta _1^0$
-definable in
$(M,\mathcal {X})$
:
$$\begin{align*}\sigma\in T\iff \forall{{x,z}<{\operatorname{\mathrm{len}}\sigma}}\ (\sigma(x)<f(x)\wedge\theta(x,\sigma(x),z)).\end{align*}$$
Obviously T is bounded by the total function
$f\in \mathcal {X}$
, which means that for any
$\sigma \in T$
and
$x<\operatorname {\mathrm {len}} \sigma $
, we have
$\sigma (x)<f(x)$
. We show that T is infinite. Let
$F(x)=\max _{x'< x}f(x')$
. For any
$x\in M$
, by
$\mathrm {I}\Sigma _1^0$
, let
$\sigma _x\in M$
be a coded sequence of length x such that for any
$x'<x$
and
$y'<F(x)$
,
$$\begin{align*}\sigma_x(x')=y'\iff (M,\mathcal{X})\models\forall{z}\ \theta(x',y',z)\wedge\forall{{w}<}\ {y'} \ \neg\forall{z}\ \theta(x',w,z).\end{align*}$$
Then we have
$$\begin{align*}(M,\mathcal{X})\models\forall{{x'}<{x}}\ \forall{z}\ (\sigma_x(x')<f(x)\wedge \theta(x',\sigma_x(x'),z)).\end{align*}$$
This means
$\sigma _x$
is an element of T of length x, so T is infinite. It is provable in
$\mathrm {WKL}_0$
that every infinite bounded tree has an infinite path (see [Reference Simpson17, Lemma IV.1.4]). Take any such infinite path
$P\in \mathcal {X}$
of T. Clearly P satisfies the requirement as in the statement.
The second proof of Theorem 4.1.
We construct an ultrafilter
${\mathcal {U}}$
on
$\mathcal {X}$
in
$\omega $
many stages. Along the construction, we gradually guarantee that the ultrapower
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \mathrm {B}\Sigma _1^0$
and that
${\mathcal {U}}$
is additive.
Enumerate all the triples
$\{\langle \exists {z}\ \theta _k(x,y,z),f_k, g_k\rangle \}_{k\in \mathbb {N}}$
, where
$\theta _k(x,y,z)\in \Delta _0^0(M,\mathcal {X})$
and
$f_k,g_k$
are total functions in
$\mathcal {X}$
. Enumerate all the bounded total functions in
$\mathcal {X}$
as
$\{h_k\}_{k\in \mathbb {N}}$
. For each
$k\in \mathbb {N}$
, at Stage k we construct a cofinal set
$A_k\in \mathcal {X}$
such that
$A_k\supseteq A_{k+1}$
for all
$k\in \mathbb {N}$
, and the resulting ultrafilter
${\mathcal {U}}:=\{A\in \mathcal {X}\mid \exists {{k}\in {\mathbb {N}}}\ A\supseteq A_k\}$
.
Stage
$\boldsymbol {0}$
: Set
$A_0=M\in \mathcal {X}$
.
Stage
$\boldsymbol {2k+1}$
(
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \mathrm {B}\Sigma _1^0$
): At these stages we want to guarantee the following instances of
$\mathrm {B}\Sigma _1^0$
in
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
:
$$\begin{align*}\forall{{y}<{[g_k]}}\ \exists{z}\ \theta_k([f_k],y,z)\rightarrow\exists{b}\ \forall{{y}<{[g_k]}}\ \exists{{z}<{b}}\ \theta_k([f_k],y,z).\end{align*}$$
The general idea is that we first try to “force” the consequent of the implication above to be true in
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
. If we succeed, then the entire instance is true. Otherwise, we apply Lemma 4.2 to argue that the antecedent is already guaranteed to be false in the ultrapower.
Consider the
$\Sigma _1^0$
-definable set
$$\begin{align*}A:=A_{2k}\cap\{x\in M\mid \exists{b}\ \forall{{y}<{g_k(x)}}\ \exists{{z}<{b}}\ \theta_k(f_k(x),y,z)\}.\end{align*}$$
It is provable in
$\mathrm {RCA}_0$
that every cofinal
$\Sigma _1^0$
-definable set has a cofinal subset in the second-order universe (see [Reference Hájek and Pudlák7, Theorem I.3.22]). If A is cofinal in M, then let
$A_{2k+1}\in \mathcal {X}$
be such a cofinal subset of A and proceed to Stage
$2k+2$
. If A is not cofinal in M, then we let
$A_{2k+1}=A_{2k}$
and proceed directly to Stage
$2k+2$
.
Stage
$\boldsymbol {2k+2}$
(Additiveness of
${\mathcal {U}}$
): This part is exactly the same as the construction of Stage
$2k+2$
in the first proof of Theorem 4.1.
Finally, let
${\mathcal {U}}:=\{A\in \mathcal {X}\mid \exists {{k}\in { }}\ {\mathbb {N}} \ A\supseteq A_k\}$
. This completes the construction of
${\mathcal {U}}$
.
Verification: Let
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
be the corresponding second-order restricted ultrapower. The fact that
$(M,\mathcal {X})\preccurlyeq _{\mathrm {e},\Sigma _2^0}(\mathcal {F}/\mathcal {U},\mathcal {X})$
follows from the exact same reasoning as in the first proof of Theorem 4.1. To show that
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \mathrm {B}\Sigma _1^0$
, consider arbitrary instance of
$\mathrm {B}\Sigma _1^0$
in
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
:
$$\begin{align*}\forall{{y}<{[g]}}\ \exists{z}\ \theta([f],y,z)\rightarrow\exists{b}\ \forall{{y}<{[g]}}\ \exists{{z}<{b}}\ \theta([f],y,z),\end{align*}$$
where
$\theta \in \Delta _0^0(M,\mathcal {X})$
and
$[f],[g]\in \mathcal {F}/\mathcal {U}$
. Here without loss of generality, we assume that
$[f]$
is the only first-order parameter in
$\theta $
. Assume that at Stage
$2k+1$
, we enumerate
$\langle \exists {z}\ \theta _k(x,y,z),f_k, g_k\rangle =\langle \exists {z}\ \theta (x,y,z),f,g\rangle $
, and
$A_{2k}\in \mathcal {X}$
is the cofinal subset of M we obtained from the previous stage. Suppose we are in the first case of the construction at this stage, i.e.,
$$\begin{align*}A=A_{2k}\cap \{x\in M\mid \exists{b}\ \forall{{y}<{g(x)}}\ \exists{{z}<{b}}\ \theta(f(x),y,z)\}\end{align*}$$
is cofinal in M, then by the construction there exists some cofinal subset of A in
${\mathcal {U}}$
. By Theorem 3.2,
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \exists {b}\ \forall {{y}<{[g]}}\ \exists {{z}<{b}}\ \theta ([f],y,z)$
, so the instance of
$\mathrm {B}\Sigma _1^0$
is true. If we are in the second case, assuming that A is bounded by some
$d\in M$
, then
$$\begin{align*}(M,\mathcal{X})\models\forall{{x}>}\ d \ (x\in A_{2k}\rightarrow \forall{b}\ \exists{{y}<}\ {g(x)} \ \forall{{z}<}\ b \ \neg\theta(f(x),y,z)).\end{align*}$$
By
$\mathrm {B}\Sigma _1^0$
in
$(M,\mathcal {X})$
, this is equivalent to:
$$\begin{align*}(M,\mathcal{X})\models\forall{{x}>{d}}\ \exists{{y}<{g(x)}}\ (x\in A_{2k}\rightarrow\forall{z}\ \neg\theta(f(x),y,z)).\end{align*}$$
By Lemma 4.2, there is a total function
$P\in \mathcal {X}$
such that
-
(1)
$(M,\mathcal {X})\models \forall {{x}>{d}}\ P(x)<g(x).$
-
(2)
$(M,\mathcal {X})\models \forall {{x}>{d}}\ (x\in A_{2k}\rightarrow \forall {z}\ \neg \theta (f(x),P(x),z) ).$
Since P is a total function,
$[P]\in \mathcal {F}/\mathcal {U}$
. By Theorem 3.2, (1) implies
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models [P]<[g]$
.
We claim that
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \forall {z}\ \neg \theta ([f],[P],z)$
. Suppose not, then by Theorem 4.1, there is some
${A'}\in {\mathcal {U}}$
such that
${A'}\subseteq \{x\in M\mid \exists {z}\ \theta (f(x),P(x),z)\}$
. But by (2),
$A'\cap A_{2k}$
is bounded by d, which contradicts the fact that
$A'\cap A_{2k}\in {\mathcal {U}}$
. So
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \forall {z}\ \neg \theta ([f],[P],z)$
, and the instance of
$\mathrm {B}\Sigma _1^0$
we considered is vacuously true.
Theorem 4.3. For each
$n\in \mathbb {N}$
and any countable model
$M\models \mathrm {B}\Sigma _{n+2}$
, there is a
$\Sigma _{n+2}$
-elementary proper end extension
$M\preccurlyeq _{\mathrm {e},\Sigma _{n+2}}K\models \mathrm {B}\Sigma _{n+1}$
.
Proof. We first expand M to a second-order structure satisfying
$\mathrm {B}\Sigma _2^0$
by adding all the
$\Sigma _n$
-definable sets into the second-order universe, then we further
$\omega $
-extend it to some countable
$(M,\mathcal {X})\models \mathrm {B}\Sigma _2^0+\mathrm {WKL}_0$
. By Theorem 4.1, there is an ultrapower extension
$(M,\mathcal {X})\preccurlyeq _{\mathrm {e},\Sigma _2^0}(\mathcal {F}/\mathcal {U},\mathcal {X})$
that satisfies
$\mathrm {B}\Sigma _1^0$
. Since all
$\Sigma _n$
-definable subsets of M are in
$\mathcal {X}$
,
$(M,\mathcal {X})$
satisfies
$\Sigma _n$
-comprehension and
$M\preccurlyeq _{\mathrm {e},\Sigma _{n+2}}\mathcal {F}/\mathcal {U}$
by Corollary 3.7.
To show that
$\mathcal {F}/\mathcal {U}\models \mathrm {B}\Sigma _{n+1}$
, suppose that
$\mathcal {F}/\mathcal {U}\models \forall {{x}<}\ {[g]} \ \exists {y}\ \theta (x,y,[f])$
for some
$[g]\in \mathcal {F}/\mathcal {U}$
and
$\theta \in \Pi _n$
, where
$[f]\in \mathcal {F}/\mathcal {U}$
is the only parameter in
$\theta $
. By
$\Pi _n$
-comprehension in M, let
$A\in \mathcal {X}$
be so that
$(M,\mathcal {X})$
satisfies
$$\begin{align*}\forall{\langle x,y,z\rangle}\ (\langle x,y,z\rangle \in A\leftrightarrow\theta(x,y,z)).\end{align*}$$
The structure
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
satisfies the same formula by Lemma 3.6, and thus
$$\begin{align*}(\mathcal{F}/\mathcal{U},\mathcal{X})\models\forall{{x}<}\ {[g]} \ \exists{y}\ \langle x,y,[f]\rangle\in A.\end{align*}$$
By
$\mathrm {B}\Sigma _1^0$
in
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
,
$$\begin{align*}(\mathcal{F}/\mathcal{U},\mathcal{X})\models \exists{b}\ \forall{{x}<}\ {[g]} \ \exists{{y}<}\ b \ \langle x,y,[f]\rangle \in A,\end{align*}$$
which means
$\mathcal {F}/\mathcal {U}\models \exists {b}\ \forall {{x}<}\ {[g]} \ \exists {{y}<}\ b \ \theta (x,y)$
, so
$\mathcal {F}/\mathcal {U}\models \mathrm {B}\Sigma _{n+1}$
.
We now proceed with the construction of end extensions in order to characterize countable models of
$\mathrm {I}\Sigma _{n+2}$
. We first define
$M\text {-}\mathrm {I}\Sigma _{n+1}$
for an end extension
$M\subseteq _{\mathrm {e}} K$
, and introduce some equivalent definitions of it.
Definition 4.4. For each
$n\in \mathbb {N}$
, let
$M,K$
be models of
$\mathrm {I}\Delta _0+\mathrm {exp}$
and
$M\subseteq _{\mathrm {e}} K$
. We say
$K\models M\text {-}\mathrm {I}\Sigma _{n+1}$
if for any
$\varphi (x)\in \Sigma _{n+1}(K)$
and
$a\in M$
,
$$\begin{align*}K\models \varphi(0)\wedge\forall{{x}<}\ a \ (\varphi(x)\rightarrow\varphi(x+1))\rightarrow\forall{{x}<}\ a \ \varphi(x).\end{align*}$$
Notice that we allow parameters from K in
$\varphi $
, while the bound a must be in M.
Lemma 4.5. For each
$n\in \mathbb {N}$
, let M be a model of
$\mathrm {I}\Delta _0+\mathrm {exp}$
,
$K\models \mathrm {I}\Sigma _n$
and
$M\subseteq _{\mathrm {e}} K$
. Then the following are equivalent:
-
(i)
$K\models M\text {-}\mathrm {I}\Sigma _{n+1}$
. -
(ii) For any
$\varphi (x)\in \Sigma _{n+1}(K)$
and
$a\in M$
,
$$\begin{align*}K\models \exists{c}\ \forall{{x}<}\ a \ (\varphi(x)\leftrightarrow x\in c).\end{align*}$$
-
(iii) For any
$\theta (x,y)\in \Pi _{n}(K)$
and
$a\in M$
,
$$\begin{align*}K\models \exists{b}\ \forall{{x}<}\ a \ (\exists{y}\ \theta(x,y)\leftrightarrow\exists{{y}<}\ b \ \theta(x,y)).\end{align*}$$
Proof. We show (i)
$\Leftrightarrow $
(ii) and (ii)
$\Leftrightarrow $
(iii). To show (i)
$\Rightarrow $
(ii), first by modifying a standard argument, one can show that (i) implies the least number principle for
$\Pi _{n+1}(K)$
formulas that are satisfied by some element of M. Then we can pick the least
$c<2^a\in M$
such that
$$\begin{align*}K\models \forall{{x}<}\ a \ (\varphi(x)\rightarrow x\in c).\end{align*}$$
Such c will code
$\varphi (x)$
for
$x<a$
by the minimality of c. To show (ii)
$\Rightarrow $
(i), take some
$c\in M$
that codes
$\{x<a\mid K\models \varphi (x)\}$
. Then, one can prove the instance of
$M\text {-}\mathrm {I}\Sigma _{n+1}$
for
$\varphi (x)$
by replacing
$\varphi (x)$
with
$x\in c$
and applying
$K\models \mathrm {I}\Delta _0$
.
To show (ii)
$\Rightarrow $
(iii), take some
$c\in M$
that codes
$\{x<a\mid K\models \exists {y}\ \theta (x,y)\}$
by (ii). Consider the following
$\Sigma _{n+1}$
formula (over
$\mathrm {I}\Sigma _n$
):
$$\begin{align*}\Phi(v):=\exists{b}\ \forall{{x}<}\ v \ (x\in c\leftrightarrow\exists{{y}<}\ b \ \theta(x,y)).\end{align*}$$
It is not hard to show that
$K\models \Phi (0)\wedge \forall {v}\ (\Phi (v)\rightarrow \Phi (v+1))$
. By
$M\text {-}\mathrm {I}\Sigma _{n+1}$
(from (ii)
$\Rightarrow $
(i)) we have
$K\models \Phi (a)$
, which implies (iii). Finally, to show (iii)
$\Rightarrow $
(ii), let
$\varphi (x):=\exists {y}\ \theta (x,y)$
for some
$\theta \in \Pi _n(K)$
. By (iii), there is some
$b\in K$
such that
$$\begin{align*}K\models\forall{{x}<}\ a \ (\exists{y}\ \theta(x,y)\leftrightarrow\exists{{y}<}\ b \ \theta(x,y)).\end{align*}$$
By
$\mathrm {I}\Sigma _n$
in K, there is some
$c\in K$
that codes
$\{x<a\mid K\models \exists {{y}<}\ b \ \theta (x,y)\}$
. This element c will serve as a witness for (ii).
The condition (ii) in Lemma 4.5 was first studied in [Reference Kossak12] by Kossak. An extension
$M\subseteq K$
is called an
$(n+1)$
-conservative extension if
$M\preccurlyeq _{\Sigma _{n+2}}K$
and it satisfies the condition (ii) in Lemma 4.5. In the same paper, Kossak showed that every countable model of
$\mathrm {I}\Sigma _{n+2}$
has a proper
$(n+1)$
-conservative extension. The construction is also a mild generalization of [Reference Clote2, Proposition 7]. For the sake of completeness, we include a proof of this result here, employing the framework of the second-order restricted ultrapowers.
Theorem 4.6. For any countable model
$(M,\mathcal {X})\models \mathrm {I}\Sigma _2^0$
, there is a proper end extension
$(M,\mathcal {X})\preccurlyeq _{\mathrm {e},\Sigma _2^0}(K,\mathcal {X})$
such that for any
$\Sigma _1^0(K,\mathcal {X})$
formula
$\varphi (z)$
and
$a\in M$
, the set
$\{z<a\mid (K,\mathcal {X})\models \varphi (z)\}$
is coded in K.
Proof. We ensure that the ultrapower
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
satisfies the coding requirement by maximizing each
$\Sigma _1^0$
-definable subset of
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
that is bounded by some element of M.
Enumerate all the pairs
$\{\langle \exists {y}\ \theta _k(x,y,z),a_k\rangle \}_{k\in \mathbb {N}}$
such that
$\theta _k(x,y,z)\in \Delta _0^0(M,\mathcal {X})$
and
$a_k\in M$
. Enumerate all the bounded total functions in
$\mathcal {X}$
as
$\{h_k\}_{k\in \mathbb {N}}$
. At each stage k we construct a cofinal set
$A_k\in \mathcal {X}$
such that
$A_k\supseteq A_{k+1}$
for all
$k\in \mathbb {N}$
, and we define the ultrafilter
${\mathcal {U}}:=\{A\in \mathcal {X}\mid \exists {{k}\in {\mathbb {N}}}\ A\supseteq A_k\}$
.
Stage
$\boldsymbol {0}$
: Set
$A_0=M$
.
Stage
$\boldsymbol {2k+1}$
(Coding
$\Sigma _1^0$
sets): Consider the following
$\Pi _2^0$
formula where
$c<2^{a_k}$
:
$$\begin{align*}\Phi(c):=\exists^{\mathrm{cf}}{x}\ (x\in A_{2k}\wedge \forall{{z}\in{ }}\ c \ \exists{y}\ \theta_k(x,y,z)).\end{align*}$$
Since
$A_{2k}$
is cofinal in M,
$(M,\mathcal {X})\models \Phi (0)$
. By
$\mathrm {I}\Sigma _2^0$
in
$(M,\mathcal {X})$
, there exists a maximal
$c_0<2^{a_k}$
satisfying the formula above. Similar to the second proof of Theorem 4.1, let
$A_{2k+1}\in \mathcal {X}$
be a cofinal subset of the following
$\Sigma _1^0$
-definable subset of M:
$$\begin{align*}\{x\in M\mid (M,\mathcal{X})\models x\in A_{2k}\wedge \forall{{z}\in{ }}\ {c_0} \ \exists{y}\ \theta_k(x,y,z)\}.\end{align*}$$
Stage
$\boldsymbol {2k+2}$
(Additiveness of
${\mathcal {U}}$
): This part is exactly the same as Stage
$2k+2$
in the proof of Theorem 4.1.
Finally, let
${\mathcal {U}}:=\{A\in \mathcal {X}\mid \exists {{k}\in { }}\ {\mathbb {N}} \ A\supseteq A_k\}$
. This completes the construction.
Verification: Let
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
be the corresponding second-order restricted ultrapower. The fact that
$(M,\mathcal {X})\preccurlyeq _{\mathrm {e},\Sigma _2^0}(\mathcal {F}/\mathcal {U},\mathcal {X})$
follows in exactly the same way as in the second proof of Theorem 4.1.
To show the coding requirement of
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
, consider any
$\Sigma _1^0$
formula
$\exists {y}\ \theta ([f],y,z)$
where
$\theta \in \Delta _0^0(\mathcal {F}/\mathcal {U},\mathcal {X})$
and
$a\in M$
. Without loss of generality, we may assume that
$[f]\in \mathcal {F}/\mathcal {U}$
is the only first-order parameter of
$\theta $
.
Let
$k\in \mathbb {N}$
be such that the pair
$\langle \exists {y}\ \theta (f(x),y,z),a\rangle $
was considered at Stage
$2k+1$
of the construction. We claim that the maximal
$c_0\in M$
we obtained in the construction codes
$\{z<a\mid (\mathcal {F}/\mathcal {U},\mathcal {X})\models \exists {y}\ \theta ([f],y,z)\}$
.
On the one hand, for each
$z<a$
such that
$z\in c_0$
, since
$A_{2k+1}\in {\mathcal {U}}$
is a subset of
$\{x\in M\mid (M,\mathcal {X})\models \exists {y}\ \theta (f(x),y,z)\}$
,
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \exists {y}\ \theta ([f],y,z)$
by Theorem 3.2. On the other hand, for each
$z'<a$
such that
$z'\notin c_0$
, if
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \exists {y}\ \theta ([f],y,z')$
, then by Theorem 3.2 again there is some
$B\in {\mathcal {U}}$
such that
$$\begin{align*}B\subseteq \{x\in M\mid (M,\mathcal{X})\models\exists{y}\ \theta(f(x),y,z')\}.\end{align*}$$
Since
$B\cap A_{2k+1}\in {\mathcal {U}}$
,
$B\cap A_{2k+1}$
is cofinal in M. Then we have
$$\begin{align*}(M,\mathcal{X})\models \exists^{\mathrm{cf}}{x} (x\in A_{2k}\wedge \forall{{z}\in{ }}\ {{c_0}\cup\{z'\}}\ \exists{y}\ \theta(f(x),y,z)),\end{align*}$$
which contradicts the maximality of
$c_0$
in the construction. So
$(\mathcal {F}/\mathcal {U},\mathcal {X})\models \neg \exists {y}\ \theta ([f],y,z')$
for
$z'\notin c_0$
.
Theorem 4.7 (Kossak [Reference Kossak12]).
For each
$n\in \mathbb {N}$
and any countable model
$M\models \mathrm {I}\Sigma _{n+2}$
, there is a
$\Sigma _{n+2}$
-elementary proper end extension
$M\preccurlyeq _{\mathrm {e},\Sigma _{n+2}}K\models M\text {-}\mathrm {I}\Sigma _{n+1}$
.
Proof. The proof is mostly the same as that of Theorem 4.3. We expand M to a second-order structure
$(M,\mathcal {X})$
satisfying
$\mathrm {I}\Sigma _2^0+\mathrm {RCA}_0$
by adding all the
$\Delta _{n+1}$
-definable subsets of M into the second-order universe. Such
$(M,\mathcal {X})$
satisfies
$\Sigma _n$
-comprehension. By Theorem 4.6, there exists an ultrapower extension
$(M,\mathcal {X})\preccurlyeq _{\mathrm {e},\Sigma _2^0}(\mathcal {F}/\mathcal {U},\mathcal {X})$
that codes all the
$\Sigma _1^0$
-definable subsets bounded by some element of M. Since all the
$\Sigma _n$
-definable sets of M are in
$\mathcal {X}$
,
$(M,\mathcal {X})$
satisfies
$\Sigma _n$
-comprehension and
$M\preccurlyeq _{\mathrm {e},\Sigma _{n+2}}\mathcal {F}/\mathcal {U}$
by Corollary 3.7.
To show that
$\mathcal {F}/\mathcal {U}\models M \text {-}\mathrm {I} \Sigma _{n+1}$
, we only need to show that
$\mathcal {F}/\mathcal {U}$
satisfies condition (ii) in Lemma 4.5. Let
$\varphi (x):=\exists {y}\ \theta (x,y,[f])$
be any
$\Sigma _{n+1}$
formula, where
$\theta \in \Pi _n$
and
$[f]\in \mathcal {F}/\mathcal {U}$
is the only parameter in
$\theta $
. Since
$(M,\mathcal {X})$
satisfies
$\Sigma _n$
-comprehension, there is some
$A\in \mathcal {X}$
such that
$(M,\mathcal {X})$
satisfies
$$\begin{align*}\forall{\langle x,y,z\rangle}\ (\langle x,y,z\rangle \in A\leftrightarrow \theta(x,y,z)).\end{align*}$$
The same formula holds in
$(\mathcal {F}/\mathcal {U},\mathcal {X})$
by Lemma 3.6, and thus
$$\begin{align*}(\mathcal{F}/\mathcal{U},\mathcal{X})\models\forall{x}\ (\exists{y}\ \langle x,y,[f]\rangle \in A\leftrightarrow\varphi(x)).\end{align*}$$
For any
$a\in M$
, there is some
$c\in K$
that codes
$$\begin{align*}\{x<a\mid (\mathcal{F}/\mathcal{U},\mathcal{X})\models\exists{y}\ \langle x,y,[f]\rangle \in A\}.\end{align*}$$
Such c also codes
$\{x<a\mid \mathcal {F}/\mathcal {U}\models \varphi (x)\}$
.
In Section 5, we will prove the converse of Theorem 4.7 (see Theorem 5.9). We will relate the end extension constructed in Theorem 4.7 to the weak regularity principle, analogous to Proposition 1.3. Notably, the course of the proof will reveal some non-trivial syntactic consequences.
5 The weak regularity principle
In this final section, we introduce the weak regularity principle
$\mathrm {WR}\varphi $
, a variant of the regularity principle, and determine its strength within the I-B hierarchy.
Mills and Paris [Reference Mills and Paris14] introduced the regularity principle
$\mathrm {R} \varphi $
to be the universal closure of the following formula:
$$\begin{align*}\exists^{\mathrm{cf}}{x}\ \exists{{y}<{a}}\ \varphi(x,y)\rightarrow\exists{{y}<}\ a \ \exists^{\mathrm{cf}}{x}\ \varphi(x,y).\end{align*}$$
For any formula class
$\Gamma $
, let
$$\begin{align*}\mathrm{R}\Gamma=\mathrm{I}\Delta_0\cup\{\mathrm{R}\varphi\mid\varphi\in\Gamma\}.\end{align*}$$
It is shown in [Reference Mills and Paris14] that
$\mathrm {R}\Pi _n\Leftrightarrow \mathrm {R}\Sigma _{n+1}\Leftrightarrow \mathrm {B}\Sigma _{n+2}$
for each
$n\in \mathbb {N}$
. The weak regularity principle is defined by replacing the
$\exists ^{\mathrm {cf}} x$
by
$\forall x$
in the antecedent of implication in
$\mathrm {R}\varphi $
.
Definition 5.1. Let
$\varphi (x,y)$
be a formula in first-order arithmetic with possibly hidden variables. The weak regularity principle
$\mathrm {WR}\varphi $
denotes the universal closure of the following formula:
$$\begin{align*}\forall{x}\ \exists{{y}<{a}}\ \varphi(x,y)\rightarrow\exists{{y}<{a}}\ \exists^{\mathrm{cf}}{x}\ \varphi(x,y).\end{align*}$$
For any formula class
$\Gamma $
, define
$$\begin{align*}\mathrm{WR}\Gamma=\mathrm{I}\Delta_0\cup\{\mathrm{WR}\varphi\mid\varphi\in\Gamma\}.\end{align*}$$
The status of the strength of the weak regularity principle among the I-B hierarchy is more complex in comparison with that of the regularity principle. The principles for most natural classes of formulas are equivalent to the collection schemas, whereas induction schemas are only equivalent to the principle for a highly restricted subclass of
$\Sigma _0(\Sigma _{n+1})$
formulas. We will show that for each
$n\in \mathbb {N}$
,
$\mathrm {WR}\Sigma _0(\Sigma _{n})$
and
$\mathrm {WR}(\Sigma _{n+1}\vee \Pi _{n+1})$
are both equivalent to
$\mathrm {B}\Sigma _{n+2}$
, and
$\mathrm {WR}(\Sigma _{n+1}\wedge \Pi _{n+1})$
is equivalent to
$\mathrm {I}\Sigma _{n+2}$
.
The weak regularity principle may also be viewed as an infinitary version of the pigeonhole principle (
$\mathrm {PHP}$
), and similar phenomena arise with the strength of the pigeonhole principle in the I-B hierarchy. Dimitracopoulos and Paris proved in [Reference Dimitracopoulos and Paris4] that
$\mathrm {PHP}\Sigma _{n+1}$
and
$\mathrm {PHP}\Pi _{n+1}$
are equivalent to
$\mathrm {B}\Sigma _{n+1}$
, and that
$\mathrm {PHP}(\Sigma _{n+1}\vee \Pi _{n+1})$
and
$\mathrm {PHP}\Sigma _0(\Sigma _{n+1})$
are equivalent to
$\mathrm {I}\Sigma _{n+1}$
.
Lemma 5.2. For each
$n\in \mathbb {N}$
,
$\mathrm {B}\Sigma _{n+2}\vdash \mathrm {WR}\Pi _{n+1}$
.
Proof. We only show the case of
$n=0$
and the rest can be done by relativizing to the
$\Sigma _n$
-universal set.
Let
$M\models \mathrm {B}\Sigma _{2}$
. We first expand M to a second-order structure satisfying
$\mathrm {B}\Sigma _2^0$
by adding all
$\Delta _1$
-definable subsets of M, then further
$\omega $
-extend it to
$(M,\mathcal {X})$
satisfying
$\mathrm {WKL}_0+\mathrm {B}\Sigma _2^0$
.
Suppose
$M\models{\forall{x}\ } \exists{{y}<{ a}}\ {\forall{z}\ }\theta(x,y,z)$
, where
$\theta(x,y,z)\in\Delta_0(M)$
. Applying Lemma 4.2 for
$\theta $
and
$f(x) \equiv a$
as a constant function, we obtain a total function
$P\in \mathcal {X}$
such that
$$\begin{align*}(M,\mathcal{X})\models \forall{x}\ (P(x)<a\wedge\forall{z}\ \theta(x,P(x),z)).\end{align*}$$
By
$\mathrm {B}\Sigma _2^0$
, there is some
$y_0<a$
such that there are cofinally many x satisfying
$P(x)=y_0$
, which implies
$M\models\exists{{y}<{ a}}\ \exists^{\mathrm{cf}}{x} {\forall{z}\ }\theta(x,y,z)$
. So
$M\models \mathrm {WR}\Pi _1$
.
Corollary 5.3. For each
$n\in \mathbb {N}$
,
$\mathrm {B}\Sigma _{n+2}\vdash \mathrm {WR}{(\Sigma _{n+1}\vee \Pi _{n+1})}$
.
Proof. Fix
$n\in \mathbb {N}$
and let
$M\models \mathrm {B}\Sigma _{n+2}$
,
$\varphi (x,y)\in \Sigma _{n+1}(M)$
and
$\psi (x,y)\in \Pi _{n+1}(M)$
. Suppose
$M\models \forall {x}\ \exists {{y}<}\ a \ (\varphi (x,y)\vee \psi (x,y))$
for some
$a\in M$
. If
$M\models \forall {{x}>}\ b \ \exists {{y}<}\ a \ \psi (x,y)$
for some
$b\in M$
, then by Lemma 5.2,
$M\models \exists {{y}<}\ a \ \exists ^{\mathrm {cf}}{x}\ \psi (x,y)$
and the conclusion holds. Otherwise,
$M\models \exists ^{\mathrm {cf}}{x}\ \exists {{y}<}\ a \ \varphi (x,y)$
, then by
$\mathrm {R}\Sigma _{n+1}$
,
$M\models \exists {{y}<}\ a \ \exists ^{\mathrm {cf}}{x}\ \varphi (x,y)$
and the conclusion holds again.
Remark. There is also a direct model-theoretic proof similar to Proposition 1.3. One only needs to notice that over
$\mathrm {B}\Sigma _{n+1}$
,
$\forall {x}\ \exists {{y}<}\ a \ (\varphi (x,y)\vee \psi (x,y))$
is equivalent to a
$\Pi _{n+2}$
formula.
Lemma 5.4. For each
$n\in \mathbb {N}$
,
$\mathrm {WR}(\Sigma _n\wedge \Pi _n)\vdash \mathrm {I}\Sigma _{n+1}$
.
Proof. We prove for each
$k\leqslant n$
, that
$\mathrm {I}\Sigma _k+\mathrm {WR}(\Sigma _{n}\wedge \Pi _n)\vdash \mathrm {I}\Sigma _{k+1}$
. Then the lemma follows by induction on k. Let
$M\models \mathrm {I}\Sigma _{k}+\mathrm {WR}(\Sigma _{n}\wedge \Pi _n)$
and assume
$M\models \neg \mathrm {I}\Sigma _{k+1}$
. Then there is a proper cut
$I\subseteq M$
defined by some formula
$\varphi (y):=\exists {x}\ \theta (x,y)$
, where
$\theta (x,y)\in \Pi _k(M)$
. Let
$$\begin{align*}\mu(x,y):=\forall{{y'}<}\ y \ \exists{{x'}<}\ x \ \theta(x',y').\end{align*}$$
Then
$\mu (x,y)\in \Pi _k(M)$
over
$\mathrm {I}\Sigma _k$
. Define
$$\begin{align*}J=\{y\in M\mid M\models \exists{x}\ \mu(x,y)\}.\end{align*}$$
It is not hard to show that J is closed downward, closed under successors, and
$J\subseteq I$
by its definition, that is, J is a proper cut of M contained in I.
For any
$x\in M$
, if
$M\models \mu (x,y)$
for all
$y\in J$
, then
$y\in J$
is defined by
$\mu (x,y)$
in M, which contradicts our assumption that
$M\models \mathrm {I}\Sigma _k$
. So for each
$x\in M$
, we may take the largest
$y\in J$
satisfying
$\mu (x,y)$
by
$\mathrm {I}\Sigma _k$
. Fixing some arbitrary
$a>J$
, we have
$$\begin{align*}M\models \forall{x}\ \exists{{y}<}\ a \ (\mu(x,y)\wedge\neg\mu(x,y+1)).\end{align*}$$
Applying
$\mathrm {WR}(\Sigma _n\wedge \Pi _n)$
, there is some
$y_0<a$
such that
$$\begin{align*}M\models \exists^{\mathrm{cf}}{x}\ (\mu(x,y_0)\wedge\neg\mu(x,y_0+1)).\end{align*}$$
By the definition of
$\mu (x,y)$
, this implies
$y_0\in J$
and
$y_0+1\notin J$
, which contradicts the fact that J is closed under successors. So
$M\models \mathrm {I}\Sigma _{k+1}$
.
Lemma 5.5 (Independently by Leszek A. Kołodziejczyk).
For each
$n\in \mathbb {N}$
,
$\mathrm {WR}{\Sigma _0(\Sigma _n)}\vdash \mathrm {B}\Sigma _{n+2}$
.
Proof. Let
$M\models \mathrm {WR}\Sigma _0(\Sigma _{n})$
. We show
$M\models \mathrm {R}\Pi _n$
, which is equivalent to
$\mathrm {B}\Sigma _{n+2}$
. By Lemma 5.4,
$M\models \mathrm {I}\Sigma _{n+1}$
. Suppose
$M\models \exists ^{\mathrm {cf}}{x}\ \exists {{y}<}\ a \ \varphi (x,y)$
for some
$\varphi \in \Pi _n(M)$
, and without loss of generality, we assume
$M\models \exists {{y}<}\ a \ \varphi (0,y)$
. For each
$z\in M$
, we find the largest
$x<z$
such that
$M\models \exists {{y}<}\ a \ \varphi (x,y)$
, and associate z with all the witnesses
$y<a$
such that
$\varphi (x,y)$
. Formally,
$$\begin{align*}M\models\forall{z}\ \exists{{y}<}\ a \ \exists{{x}<}\ z\ (\varphi(x,y)\wedge\forall{{x'}\in{(x,z)}}\ \neg\exists{{y}<}\ a \ \varphi(x',y)),\end{align*}$$
where
$(x,z)$
refers to the open interval between x and z. By
$\mathrm {WR}\Sigma _0(\Sigma _n)$
,
$$\begin{align*}M\models\exists{{y}<}\ a \ \exists^{\mathrm{cf}}{z}\ \exists{{x}<}\ z \ (\varphi(x,y)\wedge\forall{{x'}\in{(x,z)}}\ \neg\exists{{y}<}\ a \ \varphi(x',y)),\end{align*}$$
which implies
$M\models \exists {{y}<}\ a \ \exists ^{\mathrm {cf}}{x}\ \varphi (x,y)$
.
Theorem 5.6. For each
$n\in \mathbb {N}$
,
$\mathrm {WR}\Sigma _0(\Sigma _n)\Leftrightarrow \mathrm {WR}(\Sigma _{n+1}\vee \Pi _{n+1})\Leftrightarrow \mathrm {B}\Sigma _{n+2}$
.
Proof. The fact that
$\mathrm {WR}\Sigma _0(\Sigma _n)\vdash \mathrm {B}\Sigma _{n+2}$
follows from Lemma 5.5. The fact that
$\mathrm {B}\Sigma _{n+2}\vdash \mathrm {WR}(\Sigma _{n+1}\vee \Pi _{n+1})$
follows from Corollary 5.3. For
$\mathrm {WR}(\Sigma _{n+1}\vee \Pi _{n+1})\vdash \mathrm {WR}\Sigma _0(\Sigma _n)$
, note that every
$\Sigma _0(\Sigma _n)$
formula is equivalent to some
$\Delta _{n+1}$
formula over
$\mathrm {I}\Sigma _{n}$
(see [Reference Hájek and Pudlák7, Lemma I.2.50]), and
$\mathrm {WR}(\Sigma _{n+1}\vee \Pi _{n+1})\vdash \mathrm {I}\Sigma _n$
by Lemma 5.4.
The following proposition is an analog of Proposition 1.3 for
$M\preccurlyeq _{\mathrm {e},\Sigma _{n+2}}K\models M\text {-}\mathrm {I}\Sigma _{n+1}$
and
$\mathrm {WR}(\Sigma _{n+1}\wedge \Pi _{n+1})$
. It also leads to a model-theoretic proof of the fact that
$\mathrm {I}\Sigma _{n+2}\vdash \mathrm {WR}(\Sigma _{n+1}\wedge \Pi _{n+1})$
.
Proposition 5.7. Let
$M\models \mathrm {I}\Delta _0+\mathrm {exp}$
. For each
$n\in \mathbb {N}$
, if there is a proper
$\Sigma _{n+2}$
-elementary end extension
$M\preccurlyeq _{\mathrm {e},\Sigma _{n+2}}K\models M\text {-}\mathrm {I}\Sigma _{n+1}$
, then
$M\models \mathrm {WR}(\Sigma _{n+1}\wedge \Pi _{n+1})$
.
Proof. Let
$\theta (x,y,z)\in \Sigma _{n}(M)$
,
$\sigma (x,y,w)\in \Pi _{n}(M)$
and
$$\begin{align*}\varphi(x,y):=\forall{z}\ \theta(x,y,z)\wedge \exists{w}\ \sigma(x,y,w).\end{align*}$$
Suppose
$M\models \forall {x}\ \exists {{y}<}\ a \ \varphi (x,y)$
for some
$a\in M$
. Over
$\mathrm {I}\Sigma _{n+1}$
, this is equivalent to the following
$\Pi _{n+2}$
formula:
$$\begin{align*}\forall{x}\ \forall{b}\ \exists{{y}<}\ a \ (\forall{{z}<}\ b \ \theta(x,y,z)\wedge\exists{w}\ \sigma(x,y,w)).\end{align*}$$
$M\models \mathrm {I}\Sigma _{n+1}$
by Theorem 1.1, so both M and K satisfy the above formula by elementarity. Pick some arbitrary
$d>M$
in K, then
$$\begin{align*}K\models \forall{b}\ \exists{{y}<}\ a \ (\forall{{z}<}\ b \ \theta(d,y,z)\wedge \exists{w}\ \sigma(d,y,w)).\end{align*}$$
By Lemma 4.5(iii), there is some
$b\in K$
such that
$$\begin{align*}K\models \forall{{y}<}\ a \ (\forall{z}\ \theta(d,y,z)\leftrightarrow\forall{{z}<}\ b \ \theta(d,y,z)),\end{align*}$$
which implies
$$\begin{align*}K\models \exists{{y}<}\ a \ (\forall{z}\ \theta(d,y,z)\wedge\exists{w}\ \sigma(d,y,w)).\end{align*}$$
Pick a witness
$c<a$
in M such that
$K\models \forall {z}\ \theta (d,c,z)\wedge \exists {w}\ \sigma (d,c,w)$
, i.e.,
$K\models \varphi (d,c)$
. Now for each
$b\in M$
,
$K\models \exists {{x}>}\ b \ \varphi (x,c)$
, which is witnessed by d. Transferring each of these formulas to M, we have
$M\models \exists {{x}>}\ b \ \varphi (x,c)$
for any
$b\in M$
. So
$M\models \exists ^{\mathrm {cf}}{x}\ \varphi (x,c)$
.
Theorem 5.8. For each
$n\in \mathbb {N}$
,
$\mathrm {WR}(\Sigma _{n+1}\wedge \Pi _{n+1})\Leftrightarrow \mathrm {I}\Sigma _{n+2}$
.
Proof. The fact that
$\mathrm {WR}(\Sigma _{n+1}\wedge \Pi _{n+1})\vdash \mathrm {I}\Sigma _{n+2}$
follows from Lemma 5.4. For the other direction, given any countable model
$M\models \mathrm {I}\Sigma _{n+2}$
, there is a proper end extension
$M\preccurlyeq _{\mathrm {e},\Sigma _{n+2}}K\models M\text {-}\mathrm {I}\Sigma _{n+1}$
by Theorem 4.7, and then
$M\models \mathrm {WR}(\Sigma _{n+1}\wedge \Pi _{n+1})$
by Proposition 5.7.
Remark. In Hájek–Pudlák [Reference Hájek and Pudlák7, Lemma I.2.49], it was shown that for each
$n\in \mathbb {N}$
, every
$\Sigma _0(\Sigma _{n+1})$
formula is equivalent to the following normal form:
$$\begin{align*}Q_{1}{u_1}< {v_1} \ \dots Q_{k}{u_k}< {v_k} \ \Psi(u_1\dots u_k,v_1\dots v_k,w_1\dots w_l),\end{align*}$$
where
$k,l\in \mathbb {N}$
, each
$Q_i$
for
$i\leq k$
is either
$\forall$
or
$\exists$
.
$\Psi $
is a Boolean combination of
$\Sigma _{n+1}$
formulas and the variable sets
$\{u_i\}_{i\leqslant k}$
,
$\{v_i\}_{i\leqslant k}$
and
$\{w_i\}_{i\leqslant l}$
are pairwise disjoint.
Our proof of Proposition 5.7 can be refined to show that
$\mathrm {I}\Sigma _{n+2}\vdash \mathrm {WR}\varphi $
, where
$\varphi (x,y)\in \Sigma _0(\Sigma _{n+1})$
, and if written in the normal form above, x does not appear in
$\{v_i\}_{i\leqslant k}$
, i.e., x is not permitted to appear as the bound of any bounded quantifiers in front of a Boolean combination of
$\Sigma _{n+1}$
formulas. In contrast, the instance
$\varphi (z,y)$
used to prove
$\mathrm {WR}\Sigma _0(\Sigma _n)\vdash \mathrm {B}\Sigma _{n+2}$
in Lemma 5.5 starts with
$\exists x<z$
explicitly.
Finally, we prove the converse of Theorem 4.7, and establish the characterization of countable models of
$\mathrm {I}\Sigma _{n+2}$
as promised.
Theorem 5.9. Let M be a countable model of
$\mathrm {I}\Delta _0$
. For each
$n\in \mathbb {N}$
,
$M\models \mathrm {I}\Sigma _{n+2}$
if and only if M admits a proper
$\Sigma _{n+2}$
-elementary end extension
$K\models M\text {-}\mathrm {I}\Sigma _{n+1}$
.
Proof. The direction from left to right follows by Theorem 4.7. For the other direction, if
$M\preccurlyeq _{\mathrm {e},\Sigma _{n+2}}K\models M\text {-}\mathrm {I}\Sigma _{n+1}$
, then
$M\models \mathrm {WR}(\Sigma _{n+1}\wedge \Pi _{n+1})$
by Proposition 5.7, and thus
$M\models \mathrm {I}\Sigma _{n+2}$
by Theorem 5.8.
The main remaining problem now is to find a purely syntactic proof of Theorem 5.8. We conjecture that a more refined tree construction similar to the approach in Lemma 4.2 would solve the problem.
Problem 5.10. Give a direct proof of the fact that
$\mathrm {I}\Sigma _{n+2}\vdash \mathrm {WR}(\Sigma _{n+1}\wedge \Pi _{n+1})$
(and also
$\mathrm {I}\Sigma _{n+2}\vdash \mathrm {WR}\varphi $
where
$\varphi (x,y)\in \Sigma _0(\Sigma _{n+1})$
as described in the remark after Theorem 5.8) without using end extensions.
Acknowledgments
This work is contained in the author’s Ph.D. thesis. I would like to thank my two supervisors Tin Lok Wong and Yue Yang for their guidance, encouragement, and helpful discussions. I am also indebted to Leszek A. Kołodziejczyk and Linus Richter for carefully reading drafts of the paper and providing many helpful suggestions. Finally, I would like to thank the anonymous referee for valuable suggestions concerning this paper.
Funding
The author’s research was partially supported by the Singapore Ministry of Education grant MOE-000538-01 as well as by the NUS grant WBS A-0008494-00-00 and A-0008454-00-00.
Open access
