[Paper Review] Separating Models by Formulas and the Number of Countable Models

Mohammad Assem|arXiv (Cornell University)|Nov 23, 2012

Advanced Topology and Set Theory3 references3 citations

TL;DR

This paper investigates the separation of countable models in model theory using formulas from countable fragments of $L_{ ho_1eta}$, establishing that if a first-order theory has an uncountable family of pairwise separable countable models, then it must have $2^{\aleph_0}$ non-isomorphic countable models. The key contribution is a topological and logical characterization of model distinguishability via Borel equivalence relations and cardinality comparisons in a Polish space of structures.

ABSTRACT

We indicate a way of distinguishing between structures, for which, two structures are said to be separable.Being separable implies being non-isomorphic. We show that for any first order theory $T$ in a countable language, if it has an uncountable set of countable models that are pairwise separable, then actually it has such a set of size $2^{\aleph_0}$. Our result follows trivially assuming the Continuum Hypothesis ($CH$). We work here in $ZFC$ (only without $CH$).

Motivation & Objective

• To formalize the notion of model separability using formulas from countable fragments of $L_{\omega_1\omega}$.
• To analyze the topological structure of the space $X_L$ of countably infinite $L$-structures under a Polish topology induced by such fragments.
• To characterize when two structures are non-isomorphic via differences in the cardinalities of their definable sets.
• To establish that the equivalence relation $E_F$ of indiscernibility in a fragment $F$ is Borel measurable in the product topology.
• To prove that the existence of an uncountable family of pairwise separable models implies the existence of $2^{\aleph_0}$ non-isomorphic countable models.

Proposed method

• Define $X_L = \prod_{i \in I} 2^{(\mathbb{N}^{n_i})}$ as the space of all countably infinite $L$-structures over a countable relational language $L$.
• Introduce a fragment $F$ of $L_{\omega_1\omega}$ closed under subformulas, negation, quantifiers, and finite conjunctions/disjunctions.
• Define $Mod(\varphi, \bar{s})$ as the set of structures in $X_L$ satisfying a formula $\varphi$ with a given tuple $\bar{s}$ of natural numbers.
• Equip $X_L$ with the topology $t_F$ generated by basic open sets $Mod(\varphi, \bar{s})$, which is shown to be Polish.
• Construct bijections $\mu_\varphi: \mathbb{N} \to {}^{|\Delta\varphi|}\omega$ to encode finite and infinite sets of tuples for cardinality comparisons.
• Use these bijections to express equality of cardinalities of definable sets via injective functions and preimage conditions, leading to a Borel characterization of $E_F$.

Experimental results

Research questions

• RQ1 Under what conditions can two countable models be distinguished using formulas from a countable fragment of $L_{\omega_1\omega}$?
• RQ2 Is the equivalence relation $E_F$ of indiscernibility in a fragment $F$ Borel measurable in the Polish topology $t_F$?
• RQ3 What is the relationship between the existence of uncountably many pairwise separable models and the total number of non-isomorphic countable models?
• RQ4 Can the cardinality of definable sets in models be used to characterize isomorphism types via logical formulas?
• RQ5 How does the topological structure of $X_L$ with $t_F$ reflect model-theoretic properties like isomorphism and separability?

Key findings

• The equivalence relation $E_F$ of indiscernibility in a countable fragment $F$ of $L_{\omega_1\omega}$ is Borel measurable in the product topology $(X_L, t_F) \times (X_L, t_F)$.
• Two structures are separable in $F$ if there exists a formula $\varphi \in F$ such that the cardinalities of their $\varphi$-definable sets differ.
• For any formula $\varphi$ that is not a sentence, the cardinality of its definable sets can be characterized via injective functions on $\mathbb{N}$ using the chosen bijection $\mu_\varphi$.
• The condition $|X| = |Y| \in \omega$ is logically equivalent to the existence of injective functions $f, g$ such that $f^*(g^{-1}(Y)) = X$ and $g^*(f^{-1}(X)) = Y$, which is expressible in a Borel way.
• If a first-order theory $T$ has an uncountable family of pairwise separable countable models in any countable fragment of $L_{\omega_1\omega}$, then it has $2^{\aleph_0}$ non-isomorphic countable models.
• The result holds even when the set of models is $G_{\delta}$ in $X_L$, indicating robustness of the cardinality conclusion under topological constraints.