Continuous logic and the strict order property

TL;DR

This paper extends Shelah's theory to continuous logic, establishing that a continuous theory has the order property if and only if it has the independence property or the strict order property.

Contribution

It generalizes Shelah's classical result to the setting of continuous logic, linking OP, IP, and SOP.

Findings

A continuous theory has OP iff it has IP or SOP.

The result bridges classical and continuous model theory.

Provides a foundational understanding of order properties in continuous logic.

Abstract

We generalize a theory of Shelah for continuous logic, namely a continuous theory has OP if and only if it has IP or SOP.

Equations

$$\varphi (\mathcal{U},a_i)⩽\varphi (\mathcal{U},a_{i+1})\text{mbox}and\varphi (b_j,a_i)+ϵ<\varphi (b_i,a_j).$$

$$\psi_{n}({\mathcal{U}},a_{i})\mathbin{\ooalign{\hss\raise 3.87498pt\hbox{.}\hss\cr$-$}}\psi_{n}({\mathcal{U}},a_{i+1})\leqslant\frac{\epsilon}{n}\ \ \ \mbox{ and }\ \ \psi_{n}(b_{j},a_{i})+\epsilon<\psi_{n}(b_{i},a_{j}).$$

$$\underset{i\ne i_0,i_0+1}{\bigwedge }\tilde{\varphi }(a_i,x{)}^{{\eta }_0(i)}\wedge \tilde{\varphi }(a_{i_0},x)\ge s-\frac{ϵ}{2}\wedge \tilde{\varphi }(a_{i_0+1},x)\le r+\frac{ϵ}{2}$$

$$\underset{i\ne i_0,i_0+1}{\bigwedge }\tilde{\varphi }(a_i,x{)}^{{\eta }_0(i)}\wedge \tilde{\varphi }(a_{i_0},x)\le r+\frac{ϵ}{2}\wedge \tilde{\varphi }(a_{i_0+1},x)\ge s-\frac{ϵ}{2}$$

Full text

Continuous logic and the strict order property

**Karim Khanaki **

Department of science,

Arak University of Technology,

P.O. Box 38135-1177, Arak, Iran;

e-mail: khanaki@arakut.ac.ir

Abstract. We generalize a theory of Shelah for continuous logic, namely a continuous theory has OP if and only if it has IP or SOP.

Keywords: strict order property, continuous logic.

AMS subject classification: 03C45, 46E15, 46A50.

1 SOP in Continuous Logic

We assume that the reader is familiar with continuous logic from [2] and [3]. We introduce a notion of ‘strict order property’ for continuous logic as a complimentary to NIP: a theory has OP iff it has IP or SOP. We note that the usual translation of SOP in classical logic to continuous logic is not the ‘suitable’ notion, because it seems that Shelah’s theorem does not hold with this translation. So, we need to provide a different definition.

Definition 1.1**.**

(i) We say a formula $\phi(x,y)$ in continuous logic has the strict order property (SOP) if there exists a sequence $(a_{i}b_{i}:i<\omega)$ in the monster model $\mathcal{U}$ and $\epsilon>0$ such that for all $i<j$,

[TABLE]

We say that the theory $T$ has SOP if a formula $\phi(x,y)$ has SOP.

(ii) We say a theory $T$ has the weak strict order property (wSOP) if there are a formula $\phi(x,y)$ and $\epsilon>0$ such that for each natural number $n$ there are a formula $\psi_{n}(x,y)$ (of combination of instances $\phi(x,a)$) and an indiscernible sequence $(a_{i})_{i<\omega}$ and arbitrary sequence $(b_{i})_{i<\omega}$ such that for all $b\in{\mathcal{U}}$, the sequence $\psi_{n}(b,a_{i})$ has an eventual value and for all $i<j$,

[TABLE]

In this case, we say the formula $\phi(x,y)$ makes the weak strict order property (or makes wSOP).

(iii) We say a formula $\phi(x,y)$ has not the weak sequential completeness property (NSCP) if there exists an indiscernible sequence $(a_{i})_{i<\omega}$, an arbitrary sequence $(b_{i})_{i<\omega}$ and $\epsilon>0$ such that for all $b\in{\mathcal{U}}$, the sequence $\phi(b,a_{i})$ has an eventual value and for all $i<j$, $\phi(b_{j},a_{i})+\epsilon<\phi(b_{i},a_{j})$. We say that a theory $T$ has NSCP if a formula $\phi(x,y)$ has NSCP.

The acronym SOP (wSOP) stands for the (weak) strict order property and NSOP (NwSOP) is its negation. The acronym SCP stands for the negation of NSCP.

Question 1.2**.**

Is wSOP (or NSCP) an ‘expressible’ property? In the above definition, the notion ‘combination of instances $\phi(x,a)$’ is not expressible.

Remark 1.3**.**

(i) Clearly SOP implies wSOP. (Indeed, let $\psi_{n}=\phi$ for all $n$.) Also, wSOP implies NSCP. (Indeed, let $\phi=\psi_{1}$.) We will shortly show that SCP and NwSOP are the same. Of course, in classical ($\{0,1\}$-valued) logic, NSCP, wSOP and SOP are the same.

(ii) We will see shortly that OP implies IP or wSOP, but we could not prove that OP implies IP or SOP. The reason for this is that the usual argument of the proof of Shelah’s theorem does no hold for non-discrete-valued logics. So we believe that the correct notion of strict order property for continuous logic is wSOP.

(iii) We note that every formula of the form \psi(y_{1},y_{2})=\sup_{x}(\phi(x,y_{1})\mathbin{\ooalign{\hss\raise 3.87498pt\hbox{.}\hss\cr-}}\phi(x,y_{2})) defines a continuous pre-ordering (see Question 4.14 of [1] for the definition), in analogy with formulae of the form $\psi(y_{1},y_{2})=\forall x(\phi(x,y_{1})\to\phi(x,y_{2}))$ in classical logic. It is easy to see that for a theory $T$ (in continuous logic), some formula has SOP if and only if there is a formula in $T$ defining a pre-order (in the sense of [1]) with infinite chains.

(iv) In the definitions of NSCP and NwSOP we supposed that the sequence are eventually constant. The reason for this is that we want the sequence $\phi(x,a_{i}):S_{\phi}(\mathcal{U})\to\{0,1\}$ converges. In the definition of SOP, since the sequence $\phi(\mathcal{U},a_{i})$ is increasing, this requirement is guaranteed.

(v) Note that contrary to SOP, the property NSCP is not an ‘expressible’ property of formulas. In fact this property is from functional analysis: a Banach space $X$ is called weakly sequentially complete if every weak Cauchy sequence has a weak limit. Because of the importance of this concept, we reiterate it.

Definition 1.4**.**

(i) Let $X$ be a topological space and $F\subseteq C(X)$. We say that $F$ has the weak sequential completeness property (or short SCP) if the limit of each pointwise convergent sequence $\{f_{n}\}\subseteq F$ is continuous.

(ii) We say that a (bounded) family $F$ of real-valued function on a set $X$ has the relative sequential compactness in ${\mathbb{R}}^{X}$ (short RSC) if every sequence in $F$ has a pointwise convergent subsequence in ${\mathbb{R}}^{X}$.

The next result is another application of the Eberlein-Grothendieck criterion:

Fact 1.5**.**

Let $X$ be a compact space and $A\subseteq C(X)$ be bounded. Then $A$ is relatively weakly compact in $C(X)$ iff it has RSC and SCP.

Proof.

See Theorem 4.3 in [5]. $\Box$

Proposition 1.6**.**

If the set $\{\phi(x,a):a\in\mathcal{U}\}$ has the SCP, then $\phi(x,y)$ is NSOP.

Proof.

Suppose, for a contradiction, that $\{\phi(x,a):a\in\mathcal{U}\}$ has the SCP and $\phi$ is SOP. By SOP, there are $(a_{i}b_{i}:i<\omega)$ in the monster model $\mathcal{U}$ and $\epsilon>0$ such that $\phi({\mathcal{U}},a_{i})\leqslant\phi({\mathcal{U}},a_{i+1})$ and $\phi(b_{j},a_{i})+\epsilon<\phi(b_{i},a_{j})$ for all $i<j$. Let $b$ be a cluster point of $\{b_{i}\}_{i<\omega}$. By SCP, $\phi(S_{\phi}({\mathcal{U}}),a_{i})\nearrow\psi$ and $\psi$ is continuous. But $\lim_{i}\lim_{j}\phi(b_{j},a_{i})+\epsilon\leqslant\lim_{i}\lim_{j}\phi(b_{i},a_{j})$ and by continuity $\psi(b)+\epsilon\leqslant\psi(b)$, a contradiction. $\Box$

Corollary 1.7**.**

Suppose that $T$ is NIP and SCP. Then $T$ is stable.

Proof.

Use the Eberlein–Šmulian theorem. (See also 1.5 above.) $\Box$

Fact 1.8**.**

Suppose that $T$ is a theory. Then the following are equivalent:

• (i)

$T$ is NSOP.

• (ii)

For each indiscernible sequence $(a_{n})_{n<\omega}$ and formula $\phi(x,y)$, if the sequence $(\phi(x,a_{n}))_{n<\omega}$ is increasing on $S_{\phi}(\mathcal{U})$, then its limit is continuous.

Proof.

Immadiate by definition. $\Box$

1.1 Shelah’s theorem for continuous logic

Now we want to give a proof of Shelah’s theorem for continuous logic. First we show that SCP and NwSOP are the same. For this, we need some definitions. Let $M$ be a saturated enough structure and $\phi:M\times M\to{\mathbb{R}}$ a formula. For subsets $B,D\subseteq M$, we say that $\phi(x,y)$ has the *order property on * $B\times D$ (short OP on $B\times D$) if there are $\epsilon>0$ and sequences $(a_{i})\subseteq B$, $(b_{i})\subseteq D$ such that $|\phi(a_{i},b_{j})-\phi(a_{j},b_{i})|\geqslant\epsilon$ for all $i<j<\omega$. We will say that $\phi(x,y)$ has the NIP on $B\times D$, if for the set $A=\{\phi(a,y):S_{y}(D)\to{\mathbb{R}}~{}|a\in B\}$, any of the cases in Lemma 3.12 in [5] holds.

Proposition 1.9**.**

Suppose that $T$ is a theory. Then the following are equivalent:

• (i)

$T$* is NwSOP.*

• (ii)

$T$* is SCP.*

Proof.

(ii) $\Rightarrow$ (i) is by definition. For (i) $\Rightarrow$ (ii) we repeat the argument of Shelah’s theorem (see Proposition 4.6 of [5]).

Indeed, suppose that $T$ is NOT SCP; this means that there are an indiscernible sequence $(a_{n})_{n<\omega}$ and a formula $\phi(x,y)$ such that the sequence $(\phi(x,a_{n}))_{n<\omega}$ pointwise converges but its limit is not continuous. Since the limit is not continuous, $\tilde{\phi}(y,x)=\phi(x,y)$ has OP on $\{a_{n}\}_{n<\omega}\times S_{\phi}({\mathcal{U}})$. Since every sequence in $\{\phi(x,a_{n})\}_{n<\omega}$ has a pointwise convergent subsequence, $\tilde{\phi}(y,x)$ is NIP on $\{a_{n}\}_{n<\omega}\times S_{\phi}({\mathcal{U}})$. The following argument is classic (see [6] and [7]). Since $\tilde{\phi}(y,x)$ has OP, there are $r<s$ and a sequence $\{b_{N}\}\subseteq S_{\phi}({\mathcal{U}})$ such that $\tilde{\phi}(a_{i},b_{N})\leq r$ holds if $i<N$, and $\tilde{\phi}(a_{i},b_{N})\geq s$ in the otherwise. By NIP, for each $r<s$ and $\epsilon\in(0,s-r)$, there is some integer $n$ and $\eta:n\rightarrow\{0,1\}$ such that $\bigwedge_{i<n}\tilde{\phi}(a_{i},x)^{\eta(i)}$ is inconsistent, where for a formula $\varphi$, we use the notation $\varphi^{1}$ to mean $\varphi\leq r+\frac{\epsilon}{2}$ and $\varphi^{0}$ to mean $\varphi\geq s-\frac{\epsilon}{2}$. (Recall that unlike classical model theory, in continuous logic Trus is 0 and False is 1.) Starting with that formula, we change one by one instances of $\tilde{\phi}(a_{i},x)\geq s-\frac{\epsilon}{2}\wedge\tilde{\phi}(a_{i+1},x)\leq r+\frac{\epsilon}{2}$ to $\tilde{\phi}(a_{i},x)\leq r+\frac{\epsilon}{2}\wedge\neg\tilde{\phi}(a_{i+1},x)\geq s-\frac{\epsilon}{2}$. Finally, we arrive at a formula of the form $\bigwedge_{i<N}\tilde{\phi}(a_{i},x)\leq r+\frac{\epsilon}{2}\wedge\bigwedge_{N\leq i<n}\tilde{\phi}(a_{i},x)\geq s-\frac{\epsilon}{2}$. The tuple $b_{N}$ satisfies that formula. Therefore, for such $r<s$ and $\epsilon$, there is some $i_{0}<n$, $\eta_{0}:n\rightarrow\{0,1\}$ such that

[TABLE]

is inconsistent, but

[TABLE]

is consistent. Let us define $\varphi(\bar{a},x)=\bigwedge_{i\neq i_{0},i_{0}+1}\tilde{\phi}(a_{i},x)^{\eta_{0}(i)}$. Increase the sequence $(a_{i}:i<\omega)$ to an indiscernible sequence $(a_{i}:i\in\mathbb{Q})$. Then for $i_{0}\leq i<i^{\prime}\leq i_{0}+1$, the formula $\varphi(\bar{a},x)\wedge\tilde{\phi}(a_{i},x)\leq r+\frac{\epsilon}{2}\wedge\tilde{\phi}(a_{i^{\prime}},x)\geq s-\frac{\epsilon}{2}$ is consistent, but $\varphi(\bar{a},x)\wedge\tilde{\phi}(a_{i},x)\geq s-\frac{\epsilon}{2}\wedge\tilde{\phi}(a_{i^{\prime}},x)\leq r+\frac{\epsilon}{2}$ is inconsistent. Thus the formula $\psi(x,y)=\varphi(\bar{a},x)\wedge\tilde{\phi}(y,x)$ is the formula $\psi_{n}$ (for some $n$) in the definition of wSOP above. Note that for all $b$, the sequence $\psi(b,a_{i})$ has eventual true value; equivalently it converges. (Indeed, since the sequence $(\tilde{\phi}(a_{i},x):i<\omega)$ converges and we increased the sequence $(a_{i}:i<\omega)$ to the indiscernible sequence $(a_{i}:i\in\mathbb{Q})$, it is easy to verify that every sequence $(\tilde{\phi}(a_{j_{i}},x):i_{0}<j_{i}<j_{i+1}<i_{0}+1,~{}i<\omega)$ converse. Assume not, and for some $b$ the sequence $\tilde{\phi}(a_{j_{i}},b)$ diverges. Take the strictly increasing function $\tau:\omega\to\omega$ by $\tau(j_{i})=i$. By indiscernibility, the set of conditions $\{\tilde{\phi}(a_{i},x)=\tilde{\phi}(a_{j_{i}},b):i<\omega\}$ is consistence; but this means that for some $b$ the sequence $\tilde{\phi}(a_{i},b)$ diverges, a contradiction.) As $\epsilon$ is arbitrary, the proof is completed. $\Box$

The next result is a generalization of Shelah’s theorem ([9], Theorem 4.1) for continuous logic.

Corollary 1.10** (Shelah’s theorem for continuous logic).**

Suppose that $T$ is NIP and NwSOP. Then $T$ is stable.

Proof.

Let $\phi(x,y)$ be a formula, $(a_{n})_{n<\omega}$ an indiscernible sequence, and $(b_{n})_{n<\omega}$ an arbitrary sequence. Suppose that the double limits $\lim_{m}\lim_{n}\phi(b_{n},a_{m})$ and $\lim_{n}\lim_{m}\phi(b_{n},a_{m})$ exist. By NIP, there is a convergent subsequence $\phi(x,a_{m_{k}})$ such that $\phi(x,a_{m_{k}})\to\psi(x)$ on $S_{\phi}(\mathcal{U})$. Therefore, $\lim_{n}\lim_{k}\phi(b_{n},a_{m_{k}})=\lim_{n}\psi(b_{n})$ and $\lim_{k}\lim_{n}\phi(b_{n},a_{m_{k}})=\lim_{k}\phi(b,a_{m_{k}})=\psi(b)$ where $b$ is a cluster point of $\{b_{n}\}$. By NwSOP (or equivalently SCP), $\lim_{n}\psi(b_{n})=\psi(b)$. So the double limits are the same and thus $T$ is stable. (Compare Fact 1.5.) $\Box$

1.2 Universal models of Banach lattices

We show that a formuls in the language of Banach lattices has SOPn and so for many of cardinals there is not any universal model.

In [10], Shelah and Usvyatsov proved that the theory $T_{B}$ of all Banach spaces is quantifier-free-NSOP, i.e. there is not a quantifier-free formula such that defines a partial order with infinite chain. Also, they showed that a quantifier-free formula has SOP4 (even SOPn for $n\geq 4$). Using the Shelah’s result, this implies that for many cardinals there is not a universal model of Banach spaces. Note that since such the formula is quantifier-free, every subspace is an embedding, so universal model does not exist in the sense of Banach theorists. Of course, $T_{B}$ has SOP using a formula with a quantifier. Indeed, consider the formula $\phi(x,y)=\max(\|x+y\|,\|x-y\|)$. Let $s_{n}=e_{1}+\ldots+e_{n}$ where $(e_{n})$ is the standard basis of $c_{0}$. Now $\phi(e_{k},s_{n})+\frac{1}{2}<\phi(e_{k},e_{m})$ for all $n<k\leq m<\omega$ AND $\phi(x,e_{n})\leq\phi(x,e_{m})$ for all $x\in c_{0}$ and $n\leq m$. Let $\psi(x_{1},x_{2}):=\forall x(\phi(x,x_{1})\to\phi(x,x_{2}))$. Then $\psi(x,y)$ define a partial order with an infinite chain in the monster model of Banach spaces. (Recall that an incomplete theory has SOP if a complete extension of it has SOP. In this case, the Kojman–Shelah result holds still.)

On the other hand, in [4] it is showen that the class of C∗-algebras has SOP with a quantifier-free formula. (Note that its theory is incomplete.) So, using Kojma–Shelah, this implies non-existence of universal models in many cardinals. Here we want to show that the calls of Banach lattice has SOP4 with a quantifier-free formula. Indeed let $\phi(x,y)=\||x|+|y|\|$. Then $\phi(e_{k},s_{n})+\frac{1}{2}<\phi(e_{k},e_{m})$ for all $n<k\leq m<\omega$ AND $\phi(x,e_{n})\leq\phi(x,e_{m})$ for all $x\in c_{0}$ and $n\leq m$ (where $\phi(e_{k},s_{n})+\frac{1}{2}<\phi(e_{k},e_{m})$ for all $n<k\leq m<\omega$ AND $\phi(x,e_{n})\leq\phi(x,e_{m})$ for all $x\in c_{0}$ and $n\leq m$). Let $\psi(x_{1}x_{2},y_{1}y_{2}):=(\|x_{2}+y_{1}\|\leq 1\wedge\|x_{1}+y_{2}\|\geq 2)$. Now $\psi(e_{i}s_{i},e_{i+1}s_{i+1})$ holds for all $i<\omega$. This means that there is an infinite chain. It is easy to check that $\psi$ has SOP4, using the triangle property of norm. Note that $\psi$ is quantifier-free.

Since the above formula $\psi$ is quantifier-free we have:

Corollary 1.11**.**

Suppose there exists a universal Banach lattice (under isometry) in $\lambda=cf(\lambda)$. Then either $\lambda=\lambda^{<\lambda}$ or $\lambda=\mu^{+}$ and $2^{<\mu}\leq\lambda$.

Acknowledgements. I want to thank Alexander Usvyatsov for his comments and John T. Baldwin for his interest in reading of a preliminary version of this article and for his comments.