An exposition of the compactness of $L(Q^\mathrm{cf})$

TL;DR

This paper provides an exposition on the compactness properties of the logic $L(Q^{cf})$, which involves the class of regular cardinals, highlighting its foundational aspects.

Contribution

It offers a detailed explanation of the compactness theorem for $L(Q^{cf})$ logic across any set of regular cardinals, clarifying its theoretical framework.

Findings

Establishes the compactness of $L(Q^{cf})$ for all sets of regular cardinals.

Clarifies the foundational aspects of $L(Q^{cf})$ logic.

Provides a comprehensive exposition of the logic's properties.

Abstract

We give an exposition of the compactness of $L(Q^\mathrm{cf})$, for any set $C$ of regular cardinals.

Equations

$${\exists }^{\mathrm{cf}}xA(x),\text{and}{\forall }^{\mathrm{cf}}xA(x)$$

$${\exists }^{\mathrm{cf}}x{\forall }^{\mathrm{cf}}yG(x,y)$$

$${\exists }^{\mathrm{cf}}y{\forall }^{\mathrm{cf}}x\neg G(x,y).$$

$$K=\Bigl{\{}(x,z)\,\Bigm{|}\,\exists y^{\prime}\;\bigl{(}\forall y\;(y^{\prime}\leq y\to G(x,y))\;\land\;H(y^{\prime},z)\bigr{)}\Bigr{\}}$$

$$G^{\prime}=\Bigl{\{}(x,y)\Bigm{|}\exists x^{\prime}\;\bigl{(}x\leq x^{\prime}\land\forall y^{\prime}\;(y\leq y^{\prime}\to G(x^{\prime},y^{\prime}))\bigr{)}\Bigr{\}}.$$

$$G^{\prime }=(\neg ((\neg G^{-1}{)}^{\mathrm{anti}}{)}^{-1}{)}^{\mathrm{anti}}.$$

$$G=\{(x,y)\mid \exists \alpha (x\le x_{\alpha }\wedge y_{\alpha }\le y)\}$$

$${\exists }^{\mathrm{cf}}x\exists yG(x,y)\text{and}$$

$$\displaystyle\forall y^{\prime}\,\exists x^{\prime}\,\forall xy\;\bigl{(}(x^{\prime}\leq x\land y\leq y^{\prime})\to\neg G(x,y)\bigr{)}.$$

$$T^{\prime }=\mathrm{Th}(M,m{)}_{m\in M}\cup \{m{<}_{\phi }c\mid m\in M\}$$

$$M{\models }_C{Q}^{\mathrm{cf}}xy\phi (x,y,\bar{c})\iff \mathrm{cf}\{(a,b)\mid M{\models }_C\phi (a,b,\bar{c})\}\in C.$$

$$M^{\ast }\models Q^{\mathrm{cf}}xy\phi (x,y,\bar{c})\iff M^{\ast }\models R_{\phi }(\bar{c}).$$

$$M^{\ast }\models R_{\phi }(\bar{c})\iff M{\models }_C{Q}^{\mathrm{cf}}xy\phi (x,y,\bar{c}).$$

$$M^{\ast }\models Q^{\mathrm{cf}}xy\phi (x,y,\bar{c})\iff \mathrm{cf}\{(a,b)\mid M^{\ast }\models \phi (a,b,\bar{c})\}\in C$$

$$Q^{\mathrm{cf}}xy\phi (x,y,{\bar{c}}_1)\leftrightarrow Q^{\mathrm{cf}}xy\phi (x,y,{\bar{c}}_2),$$

Full text

1 Introduction

We present here a new and short exposition of the proof of the compactness of the logic $L(Q^{\operatorname{cf}}_{C})$, first-order logic extended by the cofinality quantifier $Q^{\operatorname{cf}}_{C}$, where $C$ is a class of regular cardinals. The logic and the proof of compactness are due to S. Shelah. The Compactness Theorem was stated and proved in [8], but this article is not self-contained and some fundamental steps of the proof must be found in the earlier article [7]. The interested reader consulting these two articles will soon realise that the structure of the proof is not completely transparent and that to fully understand the details requires a lot of work.

The most popular case of the cofinality quantifier is the logic $L(Q^{\operatorname{cf}}_{\omega})$ of the quantifier of cofinality $\omega$, that is, $C=\{\omega\}$. Our motivation comes from the application of $L(Q^{\operatorname{cf}}_{\omega})$ in [1] to an old problem on expandability of models. An anonymous referee of a preliminary version of [1] did not accept the validity (in ZFC) of the compactness proof presented in [8], apparently confused by the assumption of the existence of a weakly compact cardinal made at the beginning of the article. The assumption only applies to a previous result on a logic stronger than first-order logic even for countable models.

Our proof of compactness of $L(Q^{\operatorname{cf}}_{C})$ uses some ideas of [8], but it is more in the spirit of Keisler’s proof in [5] of countable compactness of the logic $L(Q_{1})$ with the quantifier of uncountable cardinality. However we use a simpler notion of weak model. J. Väänänen in the last chapter of [10] offers also a proof of compactness of $L(Q^{\operatorname{cf}}_{\omega})$ in Keisler’s style, but it is incomplete and only gives countable compactness (see I. Hodkinson’s review in [4]).

There are some other proofs in the literature, but also unsatisfactory. The proof by H-D. Ebbinghaus in [3], based on a set-theoretical translation, is just an sketch and the proof of J.A. Makowsky and S. Shelah in [6] only replaces part of Shelah’s argument in [8] by a different reasoning and does not include all details.

2 Connections

For a linear ordering $(X,<)$ we use the expressions

[TABLE]

for $\forall x^{\prime}\,\exists x\;(x^{\prime}\leq x\land A(x))$, and $\exists x^{\prime}\,\forall x\;(x^{\prime}\leq x\to A(x))$, respectively. The variables $x,x^{\prime}$ will range over the set $X$, $y,y^{\prime}$ over $Y$ and $z,z^{\prime}$ over $Z$.

Definition**.**

Let $X$ and $Y$ be two linear orderings. A connection between $X$ and $Y$ is a relation $G\subset X\times Y$ with satisfies

[TABLE]

Note that $X$ and $Y$ cannot be connected if $X$ or $Y$ has a last element.

Remark 2.1**.**

If $X$ has no last element, the relation $x\leq y$ connects $X$ with itself. 2. 2.

If $G$ connects $X$ and $Y$, then $\neg G^{-1}=\{(y,x)\mid\neg G(x,y)\}$ connects $Y$ and $X$. 3. 3.

If $G$ connects $X$ and $Y$, and $H$ connects $Y$ and $Z$, then

[TABLE]

connects $X$ and $Z$.

Proof.

1. and 2. are easy to see. We will not use 3. and leave the proof to the reader. ∎

Remark 2.2**.**

If $X$ and $Y$ are connected by $G$, then they are also connected by

[TABLE]

$G^{\prime}$* is antitone in $x$ and monotone in $y$.*

Proof.

It is easy to see that G^{\mathrm{anti}}=\bigl{\{}(x,y)\bigm{|}\exists x^{\prime}\;(x\leq x^{\prime}\land G(x^{\prime},y))\bigr{\}} connects $X$ and $Y$ and is antitone in $x$. Now it can be seen that

[TABLE]

∎

Lemma 2.3**.**

Two linear orders without last element are connected if and only if they have the same cofinality.

Proof.

If $\operatorname{cf}(X)=\operatorname{cf}(Y)=\kappa$, choose two increasing cofinal sequences $(x_{\alpha}\mid\alpha<\kappa)$ and $(y_{\alpha}\mid\alpha<\kappa)$ in $X$ and $Y$. Then

[TABLE]

connects $X$ and $Y$.111It suffices to assume that the $y_{\alpha}$ are increasing. Also one can use $G=\{(x_{\alpha},y)\mid y_{\alpha}\leq y)\}$.

For the converse assume that $\operatorname{cf}(X)=\kappa$, and that $G$ connects $X$ and $Y$. Choose a cofinal sequence $(x_{\alpha}\mid\alpha<\kappa)$ in $X$ and elements $y_{\alpha}$ in $Y$ such that $y_{\alpha}\leq y\to G(x_{\alpha},y)$ for all $y$. Then the $y_{\alpha}$ are cofinal in $Y$. To see this we use that there are cofinally many $y$ such that $\neg G(x,y)$ for sufficiently large $x$, which implies that $\neg G(x_{\alpha},y)$ for some $\alpha$. This implies $y<y_{\alpha}$. ∎

Lemma 2.4**.**

Assume that $G\subset X\times Y$ satisfies

[TABLE]

Then $G^{\prime}=\{(x,y)\mid\exists y^{\prime}\;(y^{\prime}\leq y\land G(x,y^{\prime}))\}$ connects $X$ and $Y$.

Note that a connecting $G$ which is monotone in $y$ satisfies (3) and (4).

Proof.

This is a straightforward verification. ∎

3 The Main Lemma

Consider a $L$-structure $M$ with two (parametrically) definable linear orderings, $<_{\varphi}$ and $<_{\psi}$ of its universe, both without last element. We say that $\varphi$ and $\psi$ are definably connected if there is a definable connection between $(M,<_{\varphi})$ and $(M,<_{\psi})$.

Recall that a formula $\varphi(x)$ isolates a partial type $\Sigma(x)$ in a theory $T$ if it is consistent with $T$ and implies $\Sigma(x)$ in $T$ (see Definition 4.1.1 in [9] or the definition of locally realizing a type in [2]). $T$ isolates $\Sigma(x)$ if some formula $\varphi(x)$ does it in $T$.

Lemma 3.1**.**

If $\varphi$ and $\psi$ are not definably connected, and $c$ is a new constant, the theory

[TABLE]

does not isolate the partial type $\Sigma(y)=\{n<_{\psi}y\mid n\in M\}$.

Proof.

Assume that $\gamma(c,y)$, for some $L(M)$-formula $\gamma(x,y)$, isolates $\Sigma(y)$ in $T^{\prime}$. This means that

$T^{\prime}\cup\{\gamma(c,y)\}$ is consistent. 2. 2.

$T^{\prime}\vdash\gamma(c,y)\to n<_{\psi}y$ for all $n\in M$.

We show that the relation $G$ defined by $\gamma(x,y)$ has properties (3) and (4) of Lemma 2.4, where $X=(M,<_{\varphi})$ and $Y=(M,<_{\psi})$. This will contradict the hypothesis of our Lemma.

That $T^{\prime}\cup\{\gamma(c,y)\}$ is consistent means that for all $m\in M$ the theory $\operatorname{Th}(M,m)_{m\in M}$ does not prove $m\leq_{\varphi}c\to\neg\exists y\;\gamma(c,y)$, which means that $M\models\exists x(m\leq_{\psi}x\land\exists y\;\gamma(x,y))$. This is exactly condition (3) of 2.4.

That $T^{\prime}\vdash\gamma(c,y)\to n<_{\psi}y$ means that there is an $m\in M$ such that $\operatorname{Th}(M,m)_{m\in M}$ proves $(m\leq_{\varphi}c\land\gamma(c,y))\to n<_{\psi}y$, which means M\models\forall xy\;\bigl{(}(m\leq_{\varphi}x\land y\leq_{\psi}n)\to\neg\gamma(x,y)\bigr{)}. The existence of such $m$ for all $n$ is exactly condition (4) of 2.4. ∎

Corollary 3.2**.**

Assume $\kappa$ is regular, $|M|,|L|\leq\kappa$, and $<_{\varphi}$ is a definable linear ordering of $M$ without last element. Then there is an elementary extension $N$ of $M$ such that:

$M$* is not $<_{\varphi}$-cofinal in $N$.* 2. 2.

If $<_{\psi}$ is a definable linear ordering of $M$ of cofinality $\kappa$, and $\psi$ and $\varphi$ are not definably connected, then $M$ is $<_{\psi}$-cofinal in $N$.

Proof.

Let $c$ be a new constant and let $T^{\prime}=\operatorname{Th}(M,m)_{m\in M}\cup\{m<_{\varphi}c\mid m\in M\}$. By Lemma 3.1, $T^{\prime}$ does not isolate any of the types $\Sigma_{\psi}(y)=\{n<_{\psi}y\mid n\in M\}$. Each $\Sigma_{\psi}(y)$ consists of a $<_{\psi}$-ordered chain of formulas increasing in strength. So by regularity of $\kappa$, for any $<_{\psi}$ of cofinality $\kappa$ the type $\Sigma_{\psi}(y)$ cannot be isolated neither by means of a set of $<\kappa$ formulas. By the $\kappa$-Omitting Types Theorem (see Theorem 2.2.19 in [2]), there is a model of $T^{\prime}$ omitting all types $\Sigma_{\psi}(y)$ for any $<_{\psi}$ of cofinality $\kappa$. This gives the elementary extension $N$. ∎

This corollary applies in particular to the case $\kappa=\omega$. Here the assumption on the cofinality of $<_{\psi}$ is not needed since it is the only possible cofinality in a countable model, and the Omitting Types Theorem used in the proof is the ordinary one for countable languages and countably many non-isolated types.

4 Completeness

For a language $L$ let $L(Q^{\operatorname{cf}})$ be the set of formulas which are built like first-order formulas but using an additional two-place quantifier $Q^{\operatorname{cf}}xy\;\varphi$, for different variables $x$ and $y$. Let $C$ be class a of regular cardinals and $M$ an $L$-structure. For a binary relation $R$ on $M$, we write “$\operatorname{cf}R\in C$ ” for “$R$ is a linear ordering of $M$, without last element and cofinality in $C$ ”.

The satisfaction relation $\models_{C}$ for $L$-structures $M$, $L(Q^{\operatorname{cf}})$-formulas $\psi(\bar{z})$, and tuples $\bar{c}$ of elements of $M$ is defined inductively, where the $Q^{\operatorname{cf}}$-step is

[TABLE]

We say that $M$ is a $C$-model of $T$, a set of $L(Q^{\operatorname{cf}})$-sentences, if $M\models_{C}\psi$ for all $\psi\in T$.

A weak structure $M^{\ast}=(M,\ldots)$ is an $L^{\ast}$-structure, where $L^{*}$ is an extension of $L$ by an $n$-ary relation $R_{\varphi}$ for every $L(Q^{\operatorname{cf}})$-formula $\varphi(x,y,z_{1},\dotsc,z_{n})$. Satisfaction is defined using the rule

[TABLE]

In weak structures every $L(Q^{\operatorname{cf}})$-formula is equivalent to a first-order $L^{\ast}$-formula, and conversely. So the $L(Q^{\operatorname{cf}})$-model theory of weak structures is the same as their first-order model theory.

Note that the $C$-semantics of $M$ is given by the semantics of the weak structure $M^{\ast}$ if one sets

[TABLE]

The following lemma is clear:

Lemma 4.1**.**

The $C$-semantics of $M$ is given by the weak structure $M^{\ast}$ if and only if

[TABLE]

for all $\varphi$ and $\bar{c}$.

The following property of weak structures $M^{\ast}$ can be expressed by a set $\operatorname{SA}(L)$ of $L(Q^{\operatorname{cf}})$ sentences (the Shelah Axioms):

If the $L(Q^{\operatorname{cf}})(M)$-formula $\varphi(x,y)$ satisfies $M^{\ast}\models Q^{\operatorname{cf}}xy\;\varphi(x,y)$ then $\varphi$ defines a linear ordering $<_{\varphi}$ without last element. Furthermore, if $\psi(x,y)$ defines a linear ordering $<_{\psi}$ and $M^{\ast}\models\neg Q^{\operatorname{cf}}xy\;\psi(x,y)$, there is no definable connection between $(M,<_{\varphi})$ and $(M,<_{\psi})$.

Lemma 4.2**.**

$L$-structures with the $C$-semantics are models of $\operatorname{SA}(L)$.

Proof.

This follows from Lemma 2.3. ∎

Theorem 4.3**.**

Let $C$ be a non-empty class of regular cardinals, different from the class of all regular cardinals. An $L(Q^{\operatorname{cf}})$-theory $T$ has a $C$-model if and only if $T\cup\operatorname{SA}(L)$ has a weak model.

Proof.

One direction follows from Lemma 4.2. For the other direction assume that $T\cup\operatorname{SA}(L)$ has a weak model.

Claim 1: If $L$ is countable, $T$ has a an $\{\omega\}$-model of cardinality $\omega_{1}$.

Proof. Let $M_{0}^{\ast}$ be a countable weak model of $T\cup\operatorname{SA}(L)$. Consider a linear ordering $<_{\varphi}$ without last element and $M_{0}^{\ast}\models\neg\,Q^{\operatorname{cf}}xy\,\varphi$. Then by Corollary 3.2 for $\kappa=\omega$ and the axioms $\operatorname{SA}(L)$, there is an elementary extension $M_{1}^{\ast}$ such that $M_{0}$ is not $<_{\varphi}$-cofinal in $M_{1}$, but $<_{\psi}$-cofinal is in $M_{1}$ for every $\psi$ with $M_{0}^{\ast}\models Q^{\operatorname{cf}}xy\,\psi$. We may assume that $M^{\ast}_{1}$ is countable. Continuing in this manner, taking unions at limit stages, one constructs an elementary chain of countable weak models $M^{\ast}_{0}\prec M^{\ast}_{1}\dotsb$ of length $\omega_{1}$ with union $M^{\ast}$, such that

If $<_{\varphi}$ is a linear ordering of $M^{\ast}$ without last element and $M^{\ast}\models\neg\,Q^{\operatorname{cf}}xy\,\varphi$, and if the parameters of $\varphi$ are in $M_{\alpha}$, then for uncountably many $\beta\geq\alpha$, $M_{\beta}$ is not $<_{\varphi}$-cofinal in $M_{\beta+1}$. 2. 2.

If $M^{\ast}\models Q^{\operatorname{cf}}xy\,\psi$, and the parameters of $\psi$ are in $M_{\alpha}$, then $M_{\alpha}$ is $<_{\psi}$-cofinal in $M$.

It follows that, if $M^{\ast}\models\neg\,Q^{\operatorname{cf}}xy\,\varphi$, then either $\varphi$ does not define a linear ordering without last element, or $<_{\varphi}$ has cofinality $\omega_{1}$. And, if $M^{\ast}\models\,Q^{\operatorname{cf}}xy\,\psi$, then $<_{\psi}$ has cofinality $\omega$. By Lemma 4.1 $M$ is an $\{\omega\}$-model of the $L(Q^{\operatorname{cf}})$-theory of $M^{\ast}$, and whence an $\{\omega\}$-model of $T$. This proves Claim 1.

Let $L^{\prime}$ be the extension of $L$ which has for every $L(Q^{\operatorname{cf}})$-formula $\varphi(x,y,\bar{z})$ a new relation symbol $V_{\varphi}$ of arity $2+2\cdot|\bar{z}|$. Let $\operatorname{SK}$ be the set of axioms which state that if $\varphi(x,y,\bar{c}_{1})$ and $\varphi(x,y,\bar{c}_{2})$ define linear orderings without last elements, and

[TABLE]

then $V_{\varphi}(x,y,\bar{c}_{1},\bar{c}_{2})$ defines a connection between the two orderings.

Claim 2: $T\cup\operatorname{SA}(L^{\prime})\cup\operatorname{SK}$ has a weak model.

Proof: By compactness we may assume that $L$ is countable. Then $T$ has an $\{\omega\}$-model $M$ of cardinality $\omega_{1}$, by Claim 1. If the $L(Q^{\operatorname{cf}})$-formula $\varphi(x,y,\bar{c}_{1})$ and $\varphi(x,y,\bar{c}_{2})$ define linear orderings without last element, and $M\models_{C}Q^{\operatorname{cf}}xy\,\varphi(x,y,\bar{c}_{1})\,\leftrightarrow\,Q^{\operatorname{cf}}xy\,\varphi(x,y,\bar{c}_{2})$, then the two orderings have the same cofinality, namely $\omega$ or $\omega_{1}$, and there is a connection between them by Lemma 2.3. We interpret $V_{\varphi}(x,y,\bar{c}_{1},\bar{c}_{2})$ by any such connection. This yields an expansion of $M$, which is an $\{\omega\}$-model of $T\cup\operatorname{SA}(L^{\prime})\cup\operatorname{SK}$. This proves Claim 2.

To prove the theorem, we choose two regular cardinals $\lambda<\kappa$ such that $|L|\leq\kappa$ and either $\lambda\not\in C$ and $\kappa\in C$ or conversely. Let $M_{0}^{\ast}$ be a weak model of $T\cup\operatorname{SA}(L^{\prime})\cup\operatorname{SK}$. It $M_{0}^{\ast}$ is finite, it is a $C$-model of $T$ for trivial reasons222$\operatorname{SA}(L)$ is used here.. Otherwise we may assume that $M_{0}^{\ast}$ has cardinality $\kappa$ and all $L(Q^{\operatorname{cf}})$-definable linear orderings without last element have cofinality $\kappa$. Let us first assume that $\lambda\not\in C$ and $\kappa\in C$.

Consider an $L(Q^{\operatorname{cf}})$-definable linear ordering $<_{\varphi}$ without last element and $M_{0}^{\ast}\models\neg\,Q^{\operatorname{cf}}xy\,\varphi$. Then by Corollary 3.2 and the axioms $\operatorname{SA}(L^{\prime})$, there is an elementary extension $M_{1}^{\ast}$ such that $M_{0}$ is not $<_{\varphi}$-cofinal in $M_{1}$, but $<_{\psi}$-cofinal in $M_{1}$ for every $L(Q^{\operatorname{cf}})$-definable ordering with $M_{0}^{\ast}\models Q^{\operatorname{cf}}xy\,\psi$. So $(M_{1},<_{\psi})$ has still cofinality $\kappa$. We may assume that $M^{\ast}_{1}$ has cardinality $\kappa$. Then by the axioms $\operatorname{SK}$, every $L(Q^{\operatorname{cf}})$-definable ordering $<_{\psi}$ of $M_{1}$ with $M_{1}^{\ast}\models Q^{\operatorname{cf}}xy\,\psi$ is connected to an $M_{0}$-definable ordering $<_{\psi_{0}}$, and so has cofinality $\kappa$.

Continuing in this manner, taking unions at limit stages, one constructs an elementary chain of weak models $M^{\ast}_{0}\prec M^{\ast}_{1}\dotsb$ of length $\lambda$ with union $M^{\ast}$, such that

If $<_{\varphi}$ is an $L(Q^{\operatorname{cf}})$-definable linear ordering $<_{\varphi}$ of $M^{\ast}$ without last element and $M^{\ast}\models\neg\,Q^{\operatorname{cf}}xy\,\varphi$, and if the parameters of $\varphi$ are in $M_{\alpha}$, then for $\lambda$-many $\beta\geq\alpha$, $M_{\beta}$ is not $<_{\varphi}$-cofinal in $M_{\beta+1}$. 2. 2.

If $M^{\ast}\models Q^{\operatorname{cf}}xy\,\psi$, and the parameters of $\psi$ are in $M_{\alpha}$, then $M_{\alpha}$ is $<_{\psi}$-cofinal in $M$.

It follows that, if $M^{\ast}\models\neg\,Q^{\operatorname{cf}}xy\,\varphi$, then either $\varphi$ does not define a linear ordering without last element, or $<_{\varphi}$ has cofinality $\lambda$. And, if $M^{\ast}\models\,Q^{\operatorname{cf}}xy\,\psi$, then $<_{\psi}$ has cofinality $\kappa$. By Lemma 4.1 $M\restriction L$ is an $C$-model of the $L(Q^{\operatorname{cf}})$-theory of $M^{\ast}$, and whence a $C$-model of $T$.

The proof in the case $\lambda\in C$ and $\kappa\not\in C$ is, mutatis mutandis, the same. ∎

Corollary 4.4**.**

For every class $C$ of regular cardinals, the logic $L(Q^{\operatorname{cf}}_{C})$ is compact.

We have always assumed that whenever $Q^{\operatorname{cf}}xy\varphi(x,y,\bar{c})$, the definable ordering $<_{\varphi}$ linearly orders the universe. This is not exactly the assumption of Shelah in [8]: with his definition $<_{\varphi}$ linearly orders $\{x\mid\exists y\,\varphi(x,y,\bar{c})\}$, the domain of $\varphi$. The results presented here, in particular completeness and compactness, also apply to this modification of the semantics, it suffices to add, for each such $\varphi$, new relation symbols $R_{\varphi}$ and $H_{\varphi}$, and declare that for every $\bar{c}$, $R_{\varphi}(x,y,\bar{c})$ defines a linear ordering $<^{\prime}_{\varphi}$ on the universe and $H_{\varphi}(x,y,\bar{c})$ connects $<_{\varphi}$ and $<^{\prime}_{\varphi}$. This gives compactness. For the formulation of completeness (Theorem 4.3) one must adapt the axioms $\mathrm{SA}$ to the new situation.