1 Introduction
In [Reference Boney, Grossberg, Lieberman, Rosický and Vasey4], the authors introduced the notion of a
$\mu $
-abstract elementary class (
$\mu $
-AEC for short, see Definition 1.4 below).
$\mu $
-AECs are a framework for the model theory of
$\mathbb {L}_{\infty , \mu }$
just as abstract elementary classes (AECs, which are
$\aleph _0$
-AECs) are a framework for the model theory of
$\mathbb {L}_{\infty , \omega }$
.
A natural question in both cases is to find an example where the relevant logic is weaker than the new framework. For AECs, the quantifier
$Q_{\aleph _1}$
(‘there exists uncountably many’) and its larger generalizations provide natural examples: there are
$\mathbb {L}(Q_{\aleph _1})$
-axiomatizable classes that are not axiomatizable in
$\mathbb {L}_{\infty , \omega }$
but that are AECs when equipped with the natural notion of morphisms.Footnote
1
Indeed, similar examples exist at higher cardinalities: fixing
$\mu <\lambda $
, the quantifier
$Q_\lambda $
‘there exists at least
$\lambda $
-many’ can axiomatize classes not axiomatizable by
$\mathbb {L}_{\infty , \mu }$
, but nonetheless these classes form a
$\mu $
-AEC under the natural choice of morphism (see [Reference Boney, Grossberg, Lieberman, Rosický and Vasey4, Example 8, p. 3051]). However, once ‘
$\mu $
’ becomes a parameter, these counter examples become less satisfying:
$Q_\lambda $
is expressible in
$\mathbb {L}_{\infty , \lambda }$
, so a change of parameter (that is already changing) solves the issue. This motivates the main question of this article, left implicit in [Reference Boney, Grossberg, Lieberman, Rosický and Vasey4].
Question 1.1. Is there a
$\mu $
-AEC that is not axiomatizable by
$\mathbb {L}_{\infty , \infty }$
?
Recent work of Shelah and Villaveces [Reference Shelah and Villaveces20] shows that the answer is ‘no’ for
$\mu = \aleph _0$
. We use stationary logic (introduced by Shelah [Reference Shelah18]) to give an affirmative answer for
$\mu =\aleph _1$
(Theorems 2.2 and 3.2). This shows that the framework of
$\mu $
-AECs goes beyond
$\mathbb {L}_{\infty , \infty }$
, just as AECs go beyond
$\mathbb {L}_{\infty , \omega }$
. We use an example of Barwise et al. [Reference Barwise, Kaufmann and Makkai1, Section 5] to give an explicit class of structures that witness this.
Stationary logic is a fragment of second-order logic that allows for mondaic quantifiers that state that there is a club of realizations in
$\mathcal {P}_{\omega _1}M$
satisfying some formula (or dually that there is a stationary set of realizations), but no other second-order quantifiers (see Section 1.1 for details). Importantly, the work of Fuchino et al. [Reference Fuchino, Rodrigues and Sakai7–Reference Fuchino, Rodrigues and Sakai9] has connected the downward Löwenheim–Skolem–Tarski behavior of this logic to certain diagonal reflection properties that are independent of
$\text {ZFC}$
. Recently, Cox [Reference Cox6] has shown that this downward Löwenheim–Skolem–Tarski property restricted to just instances of the
$\text {stat\, }$
quantifier (called there the
$\Pi ^1_1$
fragment) is already equivalent to such a principle. In Section 2, we work with the
$\Sigma _1^1$
fragment (using just positive instances of the
$aa$
quantifier) to avoid axioms independent of
$\text {ZFC}$
.
The proof follows recent results [Reference Boney2] on making AECs from classes of models axiomatized by the cofinality quantifier. With that quantifier (as with the ‘almost all’ quantifier
$aa$
), we wanted to allow only positive instances of the quantifier. There was an additional issue of ‘accidental’ instances of the quantifier occurring: a countable union of end-extending linear orders will have cofinally
$\omega $
regardless of the cofinality of the individual orders. Thus, we defined a notion of ‘deliberate’ use of quantifiers by deciding for each collection of parameters when want to enforce the quantifier holds for them; this is laid out in Definition 2.1.
Category theory offers another perspective on Question 1.1. The class of
$\mu $
-AECs over all
$\mu $
coincides with the class of accessible categories with monos, up to equivalence of categories (see [Reference Boney, Grossberg, Lieberman, Rosický and Vasey4, Theorems 4.3 and 4.10]). In turn, every accessible category is equivalent to a category of models axiomatized by an
$\mathbb {L}_{\infty , \infty }$
-theoryFootnote
2
by classic results [Reference Makkai and Paré15, Theorem 3.2.1 and Corollary 4.3.3]. This seems to preclude an affirmative answer to Question 1.1 if not for the key word ‘equivalent’. As we previously discussed [Reference Boney and Vasey5, Section 4], part of this categorical equivalence changes the underlying set of the models to be unrecognizable from their original presentation presentation. Responding to a related question of Makkai and Rosický, Henry [Reference Henry10] answered a related version of this question by showing that, for each
$\mu $
, the
$\mu $
-AEC of ‘sets of size at least
$\mu ^+$
’ is not even equivalent to an
$\mathbb {L}_{\infty , \mu }$
-axiomatizable class. However, it is axiomatizable by
$\mathbb {L}(Q_\mu )$
and, thus, axiomatizable by
$\mathbb {L}_{\infty , \mu ^+}$
. For more on this, see Section 2 and Question 2.4.
1.1 Definitions and notation
We consider several fragments of second-order logic,
$\mathbb {L}^2$
. In this logic, we add second-order variables, quantifiers for these variables,Footnote
3
and the atomic formula
$`x \in X$
’ between a first-order variable and a second-order variable.
Notation 1.2. We use the following conventions:
-
(1) First-order variables are denoted by lowercase letters at the end of the alphabet ( $x, y, z, \dots $
), and second-order variables are represented similarly by uppercase letters (
$X, Y, Z, \dots $
). -
(2) First-order parameters are denoted by lowercase letters, typically starting at the beginning of the alphabet ( $a, b, \dots $
), and second-order parameters are denoted by uppercase letters starting at the beginning of the alphabet (
$A, B, \dots $
). -
(3) In all cases, a bold version of the letter is used to denote a string of the appropriate variable length.
-
(4) In formulas with first- and second-order variables, we use a semicolon to separate them, e.g., $\phi (\textbf {x}; \textbf {Z}, Y)$
means that
$\phi $
is a formula with first-order free variables a subset of the tuple
$\textbf {x}$
and the second-order free variables are a subset of the tuple
$\textbf {Z}$
with the singleton Y.
Following [Reference Fuchino, Rodrigues and Sakai7], we define the logic
$\mathbb {L}^{2_{\omega _1}} \subset \mathbb {L}^2$
. This logic contains first-order logic, allows for no quantification over second-order variables, and restricts the second-order parameters to be countable (or finite) subsets of the universe. Although this logic has no meaningfully second-order sentences, it is useful to define the desired strong substructure relation in Definition 2.1. Crucially, since we restrict to countable sets, this logic is expressible in
$\mathbb {L}_{\omega _1, \omega _1}$
, so it inherits many of that logic’s desirable qualities (namely, downward Löwenheim–Skolem–Tarski and closure under
$\omega _1$
-directed colimits). We will use (Definition 2.1(4)) elementary substructure with respect to
$\mathbb {L}^{2_{\omega _1}}$
as considered by [Reference Fuchino, Rodrigues and Sakai7]. Note that [Reference Fuchino, Rodrigues and Sakai7, Theorem 1.1(2)] states that the Downward Löwenheim–Skolem–Tarski property for
$\mathbb {L}^{2_{\omega _1}}$
(simply called
$\mathcal {L}^{\aleph _0}$
there) to
$\aleph _1$
is equivalent to the Continuum Hypothesis since a model could be forced to contain all countable subsets of a countable set. However, the Downward Löwenheim–Skolem–Tarski property for this logic down to
$2^{\aleph _0}$
is a theorem of
$\text {ZFC}$
.
The main logic we consider is stationary logic,
$\mathbb {L}(aa)$
(under the standard
$\omega _1$
interpretation). This is a fragment of second-order logic that interacts with the club filter
$\mathcal {C}(M)$
on
$\mathcal {P}_{\omega _1} M$
. Recall
Alternatively, Kueker [Reference Kueker13, Section 1.1c] gives a game-theoretic definitions. Given a logic
$\mathcal {L}$
,
$\mathcal {L}(aa)$
augments it by adding a supply of second-order variables, the atomic formula ‘
$x\in X$
’, and the quantifier
$aa\, s$
over these variables where
Following [Reference Cox6, Definition 2.1], we define some fragments of
$\mathcal {L}(aa)$
(which are small parts of a larger hierarchy):
-
(1) the $\Sigma _1$
-fragment of stationary logic
$\mathcal {L}(aa)$
, denoted
$\mathcal {L}^{\Sigma _1}(aa)$
, consists of all formulas of the form $$ \begin{align*}aa\, Y_1, \dots, aa\,Y_n \psi(\textbf{x}; \textbf{X}, Y_1, \dots, Y_n),\end{align*} $$where $\psi (\textbf {x}; \textbf {X}, \textbf {Y}) \in \mathcal {L}(aa)$
contains no second-order quantifiers;
-
(2) the $\Pi _1$
-fragment of stationary logic
$\mathcal {L}(aa)$
, denoted
$\mathcal {L}^{\Pi _1}(aa)$
, consists of all formulas of the form $$ \begin{align*}stat\, Y_1, \dots, stat\,Y_n \psi(\textbf{x}; \textbf{X}, Y_1, \dots, Y_n),\end{align*} $$where $\psi (\textbf {x}; \textbf {X}, \textbf {Y}) \in \mathcal {L}(aa)$
contains no second-order quantifiers.
Any reference to a class-sized logic should be interpreted as referring to some set-sized fragment of it where we do not bother to reference the particular fragment. For instance, ‘a theory of
$\mathbb {L}_{\infty , \infty }(\tau )$
’ really means ‘a theory of
$\mathbb {L}_{\lambda , \lambda }(\tau )$
for some
$\lambda $
’ (although see Rosický [Reference Rosický17] for some work with proper class sized theories).
We typically use
$\mathcal {L}$
to refer to an abstract logic, and
$\mathbb {L}$
to refer to particular logics. Thus, ‘
$\mathbb {L}(aa)$
’ is the first-order logic
$\mathbb {L}$
augmented by the ‘almost all’ quantifier
$aa$
, while
$`\mathcal {L}(aa)$
’ treats
$\mathcal {L}$
as some arbitrary logic (to be plugged in later) that is further augmented by the ‘almost all’ quantifier.
The above describes the ‘standard
$\omega _1$
interpretation’ of
$aa$
, where the club filter is on
$\mathcal {P}_{\omega _1} M$
. We can also vary the
$\omega _1$
parameter to define
$aa_\lambda $
, which restricts second-order variables to subsets of size
$<\lambda $
and the semantics of
$aa_\lambda $
ask if the definable set is in the club filter on
$\mathcal {P}_\lambda M$
.
In our attempts to ‘Skolemize’
$\mathbb {L}(aa)$
, we will make heavy use of the following result of Menas that provides a basis for the various club filters.
Fact 1.3 [Reference Menas16], see also [Reference Kanamori11, Proposition 25.3].
Fix
$\mu $
and an infinite set X. For each
$\mathcal {C} \subset \mathcal {P}_\mu X$
,
$\mathcal {C}$
contains a club iff there is a function
$F:[X]^2 \to \mathcal {P}_\mu X$
such that
Finally, we define the main model-theoretic framework we use:
$\mu $
-AECs.
$\mu $
-AECs were introduced in [Reference Boney, Grossberg, Lieberman, Rosický and Vasey4] as a generalizations of Shelah’s AECs [Reference Shelah and Baldwin19];
$\mu $
-AECs capture and extend
$\mathbb {L}_{\infty , \mu }$
-axiomatizable classes in the same way AECs capture and extend
$\mathbb {L}_{\infty , \omega }$
-axiomatizable classes.
Definition 1.4 [Reference Boney, Grossberg, Lieberman, Rosický and Vasey4, Definition 2.2].
Fix a
$<\mu $
-ary language
$\tau $
. Given a class of
$\tau $
-structures
$\mathbb {K}$
and a binary relation
$\prec _{\mathbb {K}}$
on
$\mathbb {K}$
, we say that
$(\mathbb {K}, \prec _{\mathbb {K}})$
is a
$\mu $
-AEC iff
-
(1) $\prec _{\mathbb {K}}$
refines
$\tau $
-substructure; -
(2) $\mathbb {K}$
and
$\prec _{\mathbb {K}}$
are both closed under isomorphism; -
(3) (Coherence) if $M_0 \subset M_1$
and
$M_0 \prec _{\mathbb {K}} M_2$
and
$M_1 \prec _{\mathbb {K}} M_2$
, then
$M_0 \prec _{\mathbb {K}} M_1$
; -
(4) (Tarski–Vaught axioms) $\mathbb {K}$
is closed under
$\mu $
-directed colimits of
$\prec _{\mathbb {K}}$
systems and such a colimit is the standard colimit of the structures; -
(5) (Downward Löwensheim–Skolem–Tarski) there is a minimal cardinal $\text {LS}(\mathbb {K})$
such that if
$M \in \mathbb {K}$
and
$X \subset M$
, then there is
$N \prec _{\mathbb {K}} M$
containing X of size
$(X+\text {LS}(\mathbb {K}))^{<\mu }$
.
We refer to
$\prec _{\mathbb {K}}$
as the ‘strong substructure relation’.
$\mu $
-AECs have a nice characterization in terms of accessible categories where all morphisms are mono.
Fact 1.5 [Reference Boney, Grossberg, Lieberman, Rosický and Vasey4, Theorems 4.3 and 4.10].
-
(1) Every $\mu $
-AEC is an
$(\mu +\text {LS}(\mathbb {K}))^+$
-accessible category where all morphisms are mono. -
(2) Every $\mu $
-accessible category where all morphisms are monos is equivalent of a
$\mu $
-AEC with Löwenheim–Skolem number
$(\mu +\nu )^{<\mu }$
, where
$\nu $
is the number of morphisms between
$\mu $
-presentable objects of the category.
Note that [Reference Lieberman, Rosický and Vasey14, Definition 3.6] gives a more parameterized version of accessibility that better lines up with model-theoretic definitions like
$\mu $
-AECs.
2 Positive, deliberate axiomatizations
Following [Reference Boney2], we give a formal framework for positive, deliberate uses of the
$aa$
quantifier. The framework there was specific to cofinality quantifiers and did not contemplate second order quantifiers, so we must repeat ourselves.
Definition 2.1. Fix a (first-order) language
$\tau $
.
-
(1) $\tau _* = \tau _*^{\mathbb {L}^{\Sigma _1}(aa)} = \tau \cup \{R_\phi (\textbf {x}; \textbf {Z}) \mid \phi (\textbf {x}; \textbf {Z}, Y) \in \mathbb {L}^{\Sigma _1}(aa)\}.$
-
(2) The base theory
$$ \begin{align*}T^{\mathbb{L}^{\Sigma_1}(aa)}_{\tau} = \{ \forall \textbf{Z}\forall \textbf{x} \left( R_\phi(\textbf{x}; \textbf{Z}) \to aa\, Y \phi(\textbf{x}; \textbf{Z}, Y) \right) \mid \phi(\textbf{x}; \textbf{Z}, Y) \in \mathbb{L}^{\Sigma_1}(aa)\}.\end{align*} $$
-
(3) Given a theory $T \subset \mathbb {L}^{\Sigma _1}(aa)(\tau )$
, define $$ \begin{align*} T^* :=\ & \text{"the result of replacing each use of } `aaY\phi(\textbf{x}; \textbf{Z}, Y)' \text{with}\\ & `R_\phi(\textbf{x}; \textbf{Z})' \text{ in the inductive construction of each } \psi \in T"\\ T^+ :=\ & T^* \cup T^{\mathbb{L}^{\Sigma_1}(aa)}_\tau. \end{align*} $$
-
(4) Given $\tau _*$
structures
$M \subset N$
, we define $$ \begin{align*}M \prec_{(+)} N\end{align*} $$to mean that M is an elementary substructure of N for all formulas in $\mathbb {L}^{2_{\aleph _1}}(\tau _*).$
Note that the second-order variables—including the universal quantifier in
$T^{\mathbb {L}^{\Sigma _1}(aa)}_\tau $
—are restricted to countable subsets. Thus,
$T^+$
is essentially an
$\mathbb {L}_{\omega _1, \omega _1}(\tau _*)$
-theory; this will be important in showing closure under
$\omega _1$
-directed colimits.
We can now show that we have built an
$\omega _1$
-AEC.
Theorem 2.2. Fix
$T \subset \mathbb {L}^{\Sigma _1}(aa)(\tau )$
and fix
Then
$\mathbb {K}_T^+$
is an
$\omega _1$
-AEC with
$LS(\mathbb {K}_T^+) =|T| + \aleph _1$
.
Note that the
$\mu $
-AEC axioms only guarantee models of size
$\kappa ^{<\mu }$
for
$\kappa \geq LS(\mathbb {K})$
, so there is a built-in dependence on the value of the continuum.
Proof. The proof of the various axioms is standard except for the Tarski–Vaught axiom (
$\omega _1$
-directed colimits) and the downward Löwenheim–Skolem–Tarski axiom.
-
• Tarski–Vaught: Suppose that $\{M_i \in \mathbb {K}^+_T \mid i \in I\}$
is an
$\omega _1$
-directed system so
$i < j$
implies that
$M_i \prec _{(+)} M_j$
. We can build the colimit of this system of
$\tau $
-structures $$ \begin{align*}M_I := \bigcup_{i \in I} M_i.\end{align*} $$Since $T^*$
is essentially,
$\mathbb {L}_{\omega _1, \omega _1}$
, this all transfers nicely and we are left to show that
$M_I$
satisfies
$T^{\mathbb {L}^{\Sigma _1}(aa)}_\tau $
. Fix
$\textbf {a} \in M_I$
;
$\textbf {A} \in [M_I]^{\aleph _0}$
such that $$ \begin{align*}M_I \vDash R_\phi(\textbf{a}; \textbf{A})\end{align*} $$and we want to show that
$$ \begin{align*}\mathcal{C}:= \{B \in [M_I]^{\aleph_0} \mid M_I \vDash \phi(\textbf{a}; \textbf{A}, B)\}\end{align*} $$contains a club. By $\omega _1$
-directedness, there is some
$i \in I$
so
$M_i$
contains the parameters
$\textbf {a}$
and
$\textbf {A}$
.
-
– Unbounded: Let $B_0 \in [M_I]^{\aleph _0}$
. By
$\omega _1$
-directedness, find
$j> i$
so
$B_0 \subset M_j$
. Then
$M_j \vDash `R_\phi (\textbf {a}; \textbf {A})$
’ and, thus, $$ \begin{align*}M_j \vDash aa Y \phi(\textbf{a}; \textbf{A}, Y).\end{align*} $$So by unboundedness in $M_j$
, there is
$B_1 \supset B_0$
in
$[M_j]^{\aleph _0}$
such that $$ \begin{align*}M_j \vDash \phi(\textbf{a}; \textbf{A}, B_1).\end{align*} $$Then $B_1 \in [M_I]^{\aleph _0}$
and $$ \begin{align*}M_I \vDash \phi(\textbf{a}; \textbf{A}, B_1)\end{align*} $$as desired.
-
– Closed: Let $B_n \in [M_I]^{\aleph _0}$
be increasing so $$ \begin{align*}M_I \vDash \phi(\textbf{a}; \textbf{A}, B_n).\end{align*} $$Find $i< j \in I$
so
$\cup B_n \subset M_j$
. By elementarity, we have, for each
$n <\omega ,$
$$ \begin{align*}M_j \vDash \phi(\textbf{a}; \textbf{A}, B_n).\end{align*} $$The realizations in $[M_j]^{\aleph _0}$
are club. Then we have that
$M_j \vDash `R_\phi (\textbf {a}; \textbf {A}, \cup B_n)$
’, so $$ \begin{align*}M_j \vDash \phi(\textbf{a}; \textbf{A}, \cup B_n).\end{align*} $$By $\mathbb {L}^{2_{\aleph _1}}(\tau _*)$
-elementarity,
$M_I$
thinks this as well.
Note that since the structure $M_I$
is computed as the colimit in
$\text {Set}$
and
$\prec _{(+)}$
is defined nicely, this structure satisfies the second part of the Tarski–Vaught axioms. -
-
• Downward Löwenheim–Skolem–Tarski: Let $M \vDash T^+$
. First, we take advantage of the fact that
$T^*$
is essentially
$\mathbb {L}_{\omega _1, \omega _1}$
to (classically) Skolemize it: there is
$\tau ^{sk}$
and an expansion
$M^{sk}$
of M such that, for any
$Y \subset M$
, we have $$ \begin{align*}\langle Y \rangle_{M^{sk}} \prec_{\mathbb{L}_{\omega_1, \omega_1}} M,\end{align*} $$where $\langle \cdot \rangle _{M^{sk}}$
is the closure under the functions of
$M^{sk}$
. Notice that the functions of
$M^{sk}$
can be
$<\omega _1$
-ary.
However, we are missing from this conclusion that $\langle Y \rangle _{M^{sk}}$
satisfies
$T^{\mathbb {L}^{\Sigma _1}(aa)}_\tau $
; that is, that we have forced the appropriate subsets to contain a club. For this, we use Menas’ result Fact 1.3 to further Skolemize to $$ \begin{align*}\tau^{sk+} := \tau^{sk} \cup \left\{F_\phi:[M]^{2+\ell(\textbf{x}) + \omega \cdot \ell(\textbf{Z})} \to [M]^{\aleph_0} \mid \phi(\textbf{x}; \textbf{Z}, Y) \in \mathbb{L}^{\Sigma_1}(aa) \right\}\end{align*} $$and find a $\tau ^{sk+}$
-expansion
$M^{sk+}$
such that, for all
$\textbf {a} \in M$
and all
$\textbf {A} \in [M]^{\aleph _0}$
, we have $$ \begin{align*}\left(M\vDash R_\phi(\textbf{a}; \textbf{A})\right) \implies \left( \mathcal{C}\left(F^{M^{sk+}}_\phi(\cdot, \cdot, \textbf{a}, \textbf{A})\right) \subset \left\{t \in [M]^{\aleph_0} \mid M \vDash `\phi(\textbf{a}; \textbf{A}, t)\text{'} \right\}\right).\end{align*} $$Above, the interpretation $F_\phi ^{M^{sk+}}$
is chosen to witness that
$\phi (\textbf {a};\textbf {A}, M)$
contains a club.
Now fix $X \subset M$
, and we want to find
$M_0 \prec _{(+)} M$
such that
$X \subset M_0$
and
$\|M_0\| \leq (|T| + \aleph _1 + |X|)^{\aleph _0}$
. Working in the expansion
$M^{sk+}$
defined above, set $$ \begin{align*}M_0 := \langle X \rangle_{M^{sk+}}.\end{align*} $$From this $X \subset M_0$
and is of the desired size. From above, $$ \begin{align*}M_0 \prec_{(+)} M\end{align*} $$and $M_0 \vDash T^*$
. So it remains to show that
$M_0 \vDash T^{\mathbb {L}^{\Sigma _1}(aa)}_\tau $
.
Fix a formula $\phi (\textbf {x}; \textbf {Z}, Y)$
and parameters
$\textbf {a} \in M_0$
;
$\textbf {A} \in [M_0]^{\aleph _0}$
, and suppose that $$ \begin{align*}M_0 \vDash R_\phi(\textbf{a}; \textbf{A}).\end{align*} $$Write $G:[M]^2 \to [M]^{\aleph _0}$
for
$F_\phi ^{M^{sk+}}(\cdot , \cdot , \textbf {a}; \textbf {A})$
. By elementarity,
$M \vDash R_\phi (\textbf {a}; \textbf {A})$
, so this means that $$ \begin{align*}\mathcal{C}(G) \subset \left\{B \in [M]^{\aleph_0} \mid M \vDash `\phi(\textbf{a}; \textbf{A}, B)\text{'} \right\}.\end{align*} $$Since $M_0$
is closed under G, we have that
$G \upharpoonright [M_0]^2: [M_0]^2 \to [M_0]^{\aleph _0}$
. Let
$B \in \mathcal {C}(G\upharpoonright [M_0]^2) \subset [M_0]^{\aleph _0}$
. Then
$B \in \mathcal {C}(G)$
, so $$ \begin{align*}M \vDash \phi(\textbf{a}; \textbf{A}, B).\end{align*} $$By elementarity, this means that $M_0 \vDash \phi (\textbf {a}; \textbf {A}, B)$
. Thus, $$ \begin{align*}\mathcal{C}(G\upharpoonright [M_0]^2) \subset \{B \in [M_0]^{\aleph_0} \mid M_0 \vDash \phi(\textbf{a}; \textbf{A}, B)\}\end{align*} $$so the latter contains a club, as desired.So we have verified each instance of
$$ \begin{align*}M_0 \vDash \forall \textbf{Z} \forall \textbf{x}\left(R_\phi(\textbf{x};\textbf{Z}) \to aa Y \phi(\textbf{x}; \textbf{Z}, Y)\right).\end{align*} $$Thus, $M_0 \in \mathbb {K}^+_T$
, as desired.
Putting these together, we have shown that
$\mathbb {K}^+_T$
is an
$\omega _1$
-AEC.
A version of the Downward LST result appears as [Reference Cox6, Lemma 4.1] by using internal approachability of countable sets. Our more model-theoretic version using Skolem functions is morally equivalent.Footnote
4
Additionally, our version does not restrict the size of the language. There is also a more refined Skolemization possible: all second-order variables are quantified by
$aa$
, so instead of the Skolemization of
$aa Y \phi (\textbf {x}; \textbf {Z}, Y)$
being finitary functions depending on the infinite parameters
$\textbf {x}; \textbf {Z}$
, we could incorporate the (future) Skolemization of the
$\textbf {Z}$
to make the functions entirely finitary. However, this much more technical formulation provides no advantage in the end result, so we avoid it.
We can also generalize this to infinitary logics and other interpretations of
$aa$
; the proof is a straightforward generalization of the proof of Theorem 2.2 so we omit it.
Theorem 2.3. Fix
$T \subset \mathbb {L}^{\Sigma _1}_{\kappa , \lambda }(aa_\mu )(\tau )$
and set
Then
$\mathbb {K}^+_T$
is a
$(\lambda +\mu )$
-AEC with
$LS(\mathbb {K}^+_T) = |T| + \mu +\kappa $
.
In particular, if
$T \subset \mathbb {L}^{\Sigma _1}_{\kappa , \omega _1}(aa)(\tau )$
, then
$\mathbb {K}^+_T$
is an
$\omega _1$
-AEC.
From the set-theoretic work of [Reference Cox6–Reference Fuchino, Rodrigues and Sakai9], we know that some on the theories in
$\mathbb {L}(aa)$
that form a
$\mu $
-AEC is necessary (for a ZFC result at least). In particular, [Reference Cox6, Theorem 3.1] shows that extending Theorem 2.3 to
$\mathbb {L}^{\Pi _1}(aa)$
(which allows for the stationary quantifier) already implies
$DRP_{internal}$
, a reflection property beyond
$\text {ZFC}$
. To prove
$DRP_{internal}$
, Cox uses this downward Löwenheim–Skolem property to
$\omega _1$
on the following
$\Pi _1$
-formulas (in the language of set theory):
-
•
$$ \begin{align*}\phi(x) = "stat\, Z \left( \exists p \left( p = Z \cap \bigcup x \wedge p \in x\right) \right)\text{"}\end{align*} $$
-
•
$$ \begin{align*}\psi(x) = "stat\, Z \left( \exists p \exists \alpha \left(p = Z \cap x \wedge \alpha < x \wedge \alpha = \sup p\right) \right).\text{"}\end{align*} $$
However, the literature is concerned with making the Löwenheim–Skolem–Tarski property hold down to cardinals below the continuum. The general question of forming a
$\mu $
-AEC would allow any threshold size, which the current literature doesn’t seem to explore. Thus, the following question is still open.
Question 2.4. Is there a theory T from
$\mathbb {L}(aa)$
or
$\mathbb {L}(aa) \cup \mathbb {L}_{\infty , \infty }$
such that
$\textrm {Mod }(T)$
cannot be made into an accessible category for any choice of morphism?
Or is there some set-theoretic hypothesis that implies every
$\mathbb {L}_{\infty , \infty }(aa)$
-axiomatizable class can be made into an accessible category via the right choice of morphism?
Sean Cox points out that the existence of a supercompact cardinal
$\kappa $
implies a downward Löwenheim–Skolem–Tarski property for
$\mathbb {L}(aa)$
or even
$\mathbb {L}_{\kappa , \kappa }(aa_\lambda )$
for
$\lambda < \kappa $
. However, the closure of these classes under directed colimits is unknown. In particular, while the reflection of stationary sets is well-studied, the closure of stationary sets under sufficiently directed colimits seems unexplored.
3 The example of Barwise–Kaufmann–Makkai
To make explicit the claim that there is an
$\omega _1$
-AEC that is not axiomatizable by
$\mathbb {L}_{\infty , \infty }$
, we use an example of Barwise et al. [Reference Barwise, Kaufmann and Makkai1]. Throughout, we will use the following notation: given an ordering
$(A, <)$
and subset
$X \subset A$
, we use
to denote the obvious ordered set with the inherited linear ordering, perhaps more properly and awkwardly written as
$\left (X, <\upharpoonright (X\times X)\right )$
.
Fix the language
where
$E, <,$
and R are binary relations and the rest are unary relations.
The key idea of the example is to code two isomorphic well orders
$(U,<)$
and
$(V,<)$
without exhibiting the actual isomorphism between them. Instead,
$\mathcal {P}_{\omega _1} U$
and
$\mathcal {P}_{\omega _1} V$
are connected with the
$aa$
quantifier to imply that the isomorphism must exist.
First, define
$\psi \in \mathbb {L}_{\omega _1, \omega _1}(\tau )$
that says the following:
-
(1) $(U, <)$
and
$(V, <)$
are disjoint well orders; -
(2) E is an extensional binary relation on $U\times P_U$
and
$V \times P_V$
; -
(3) using E, we can identify $P_U$
as
$\mathcal {P}_{\omega _1}U$
and
$P_V$
as
$\mathcal {P}_{\omega _1}V$
; -
(4) R relates isomorphic pieces of $P_U$
and
$P_V$
: $$ \begin{align*}\forall x,y \left[ R(x,y) \leftrightarrow \left(P_U(x) \wedge P_V(y) \wedge \left[(x, <) \cong (y, <)\right]\right)\right].\end{align*} $$
Second, define
$\phi \in \mathbb {L}(aa)$
as
Two key facts:
Fact 3.1 [Reference Barwise, Kaufmann and Makkai1, Section 5].
-
(1) Given $M \vDash \phi $
, we have
$M \vDash \psi $
iff
$(U^M, <^M)$
and
$(V^M, <^M)$
are isomorphic. -
(2) There is no $\chi \in \mathbb {L}_{\infty , \infty }(\tau )$
so $$ \begin{align*}\textrm{Mod }(\chi) = \textrm{Mod }(\phi \wedge \psi).\end{align*} $$
Note that Fact 3.1(2) says that
$\textrm {Mod }(\phi \wedge \psi )$
cannot be the set of models of an
$\mathbb {L}_{\kappa , \lambda }$
-elementary class for any
$\kappa $
and
$\lambda $
. This gives us a proof of the following theorem that motivates this article.
Theorem 3.2.
$\mathbb {K}_{\phi \wedge \psi }^+$
is an
$\omega _1$
-AEC (and thus forms an accessible category) where the models of
$\mathbb {K}^+_{\phi \wedge \psi }$
are not axiomatizable by an
$\mathbb {L}_{\infty , \infty }$
-sentence. Moreover, it is closed under
$\omega _1$
-directed colimits which are created in
$\text {Set}$
and it’s
$LS$
number is
$\omega _1$
.
Proof. From its definition,
$\phi \in \mathbb {L}^{\Sigma _1}(aa)(\tau )$
, so
$\phi \wedge \psi \in \mathbb {L}^{\Sigma _1}_{\omega _1, \omega _1}(\tau )$
. Then we can use Theorem 2.3 for the positive part and Fact 3.1(2) for the negative part.
The abstract theory of accessible category theory implies that
$\mathbb {K}^+_{\phi \wedge \psi }$
is equivalent to an
$\mathbb {L}_{\infty , \omega _1}$
-axiomatizable category. This abstract equivalence is built by ‘changing the universe’, so that a model
$M \in \mathbb {K}^+_{\phi \wedge \psi }$
is represented by the collection of all embeddings from
$\omega _1$
-sized models of
$\mathbb {K}^+_{\phi \wedge \psi }$
into M. The ability of such a ‘universe changing’ functor to greatly impact the axiomatizability of a class of structures has been observed before (see the discussion in Section 1).
However, the Barwise–Kaufmann–Makkai example illustrates something much more subtle. By being more intentional about turning
$\textrm {Mod }(\phi \wedge \psi )$
into an
$\mathbb {L}_{\infty ,\infty }$
-axiomatizable class, we can give a very explicit description of the class and an equivalence which shows that the change in axiomatization comes not from changing the universe, but from a failure of Beth definability. This is more akin to the functorial expansions that were used to axiomatize AECs in [Reference Boney, Baldwin and Iovino3, Theorem 3.2.3].
We know that isomorphisms between well-orders are unique. Thus, the result of [Reference Barwise, Kaufmann and Makkai1] implying existence of an isomorphism in models of
$\phi \wedge \psi $
actually implies the uniqueness of the isomorphism; in other words, this isomorphism is implicitly definable in
$\mathbb {L}_{\omega _1, \omega _1}(aa)(\tau )$
.
On the other hand, Fact 3.1(2) shows that the existence of an isomorphism
$U \to V$
is not implicitly definable in
$\mathbb {L}_{\infty , \infty }(\tau )$
.
We define our strong substructure relation by taking advantage of this mismatch. When trying to define a strong substructure
$M \prec ^+ N$
, there are two issues to consider:
-
(1) We want it to interact well with $\psi \in \mathbb {L}_{\omega _1, \omega _1}$
. To do so, let
$\mathcal {F} \subset \mathbb {L}_{\omega _1, \omega _1}(\tau )$
be the smallest elementary fragment containing the sentence
$\phi $
; recall this the smallest set of formulas containing
$\phi $
and closed under subformulas and all first-order operations.Footnote
5
Crucially, if
$M \prec _{\mathcal {F}} N$
, then, for any
$a \in P^M_U \cup P^M_V$
, we have $$ \begin{align*}\{x \in M : M \vDash `xEa\text{'}\} = \{x \in N : N \vDash `xEa\text{'}\}.\end{align*} $$This means that they agree about the isomorphism from $P^M_U$
to
$\mathcal {P}_{\omega _1}(U^M)$
and from
$P^M_V$
to
$\mathcal {P}_{\omega _1}(U^V)$
.
-
(2) We want them to agree on the implicit isomorphism defined by $\phi $
(see Example 3.3 gives an example of structures that agree on the explicit structure, but disagree on the implicit isomorphism). To do this, given
$M \vDash \phi \wedge \psi $
, set
$F_M:U^M \to V^M$
to be the unique isomorphism. We want M being a strong substructure of N to mean that $$ \begin{align*}F_M = F_N \upharpoonright U^N.\end{align*} $$
Example 3.3. We illustrate with a non-example. Consider the structure N that comes from setting
$U^N = V^N = \omega +\omega $
(or use your favorite ordinal in place of
$\omega $
). We will build
$M \subset N$
by setting
$U^M = V^M = \omega $
, but we use different parts of
$\omega +\omega $
:
$U^M$
is an initial segment of
$U^N$
, while
$V^M$
is a final segment of
$V^N$
. Note that
$N \vDash \phi \wedge \psi $
since
$U^N \cong V^N$
, but that we don’t have
$M \prec ^* N$
since they build incompatible isomorphisms.
The outer box represents
$U^N \cup V^N$
and the inner box represents
$U^M \cup V^M$
.
Thus,
$M\prec _{\mathcal {F}} N$
but
$M \not \prec ^+ N$
.
To streamline our definition, we define the auxiliary class of models where the implicitly definable isomorphism is explicit.
Definition 3.4.
-
(1) Set $\tau ^+ = \tau \cup \{F\}$
for a unary function symbol and
$\psi _F$
to be the first-order statement that
$F: U \to V$
is a bijection. -
(2) Set $\mathbb {K}^+ = \left (\textrm {Mod } (\phi \wedge \psi _F), \prec _{\mathcal {F}}\right )$
and define the map $$ \begin{align*} \textrm{Mod } (\phi \wedge \psi) \to&\ \textrm{Mod } (\phi \wedge \psi_F)\\ M \mapsto&\ M^+ = (M, F_M), \end{align*} $$where $F_M$
is the unique isomorphism
$U^M \to V^M$
.
Definition 3.5. Define the class
$\mathbb {K} = \left ( \textrm {Mod }(\phi \wedge \psi ), \prec ^+\right )$
, where
$\prec ^+$
is defined by, for any
$M, N \vDash \phi \wedge \psi $
,
We now collect several straightforward results.
Proposition 3.6.
-
(1) $\mathbb {K}^+$
is
$\mathbb {L}_{\omega _1, \omega _1}$
-axiomatizable and, therefore, an
$\omega _1$
-AEC. -
(2) The map $M \mapsto M^+$
induces an isomorphism (and thus an equivalence) between
$\mathbb {K}$
and
$\mathbb {K}^+$
that preserves the underlying sets (a concrete isomorphism). -
(3) $\mathbb {K}$
is an
$\omega _1$
-AEC, but models of
$\mathbb {K}$
are not axiomatizable by any
$\mathbb {L}_{\infty , \infty }$
sentence.
Proof. (1) follows because
$\mathcal {F}$
is a fragment of
$\mathbb {L}_{\omega _1, \omega _1}$
containing the relevant sentences (see item (3) in [Reference Boney, Grossberg, Lieberman, Rosický and Vasey4, Section 2]). (2) is the content of saying the bijection
$F_M$
is implicitly definable from
$\phi \wedge \psi $
. (3) follows from (1) and (2).
Work in progress with Mick Walker reveals that there is a deep connection between interpolation (seen as a property that implies Beth definability) and the axiomatization of
$\mu $
-AECs.
Acknowledgments
We thank Sean Cox for helpful discussions around these ideas and the referee for their thorough report.
Funding
The author was supported by the National Science Foundation under grants DMS-2137465 and DMS-2339018.
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