Dependencies

A theory \(T\) is \(A\)-consistent if \(\bot \) is not derivable from \(T\) in fragment \(A\).

An admissible fragment of \(\mathcal{L}_{\omega _1\omega }\): a collection of sentences closed under subformulas, negation, quantification, and countable conjunction/disjunction, with a compactness axiom for finite subsets.

A sentence \(\varphi \) has arbitrarily large models if for every cardinal \(\kappa \) there is a model of \(\varphi \) of cardinality \(\geq \kappa \).

Back-and-forth equivalence \(\sim _\alpha (a,b)\) for tuples \(a \in M^n\), \(b \in N^n\) at ordinal \(\alpha \), defined by transfinite recursion: atomic agreement at \(0\), forth-and-back extension at successor, conjunction at limit.

A sentence \(\varphi \) is derivable from theory \(T\) in admissible fragment \(A\). The proof system includes structural, propositional, infinitary, quantifier (omega-rule), equality, and classical rules.

A full Barwise fragment: an admissible fragment containing all \(\mathcal{L}_{\omega _1\omega }\) sentences, equipped with the chain closure property for consistency.

\(\kappa \) is a Hanf bound for \(\varphi \) if having a model of cardinality \(\geq \kappa \) implies having arbitrarily large models.

The Hanf number of a sentence \(\varphi \): the least cardinal that is a Hanf bound.

The equivalence relation on coded \(\mathbb {N}\)-models of \(\varphi \) where two codes are related iff their decoded structures are \(L\)-isomorphic.

\(L_{\infty \omega }\)-equivalence: \(M\) and \(N\) satisfy the same \(L_{\infty \omega }^w\) sentences, where \(w\) is the universe of the index types.

The naming function for the extended language \(L[[M]]\): each element \(m \in M\) is named by the constant symbol \(\operatorname {con}(m)\).

A structure \(M\) omits a type \(p\) (a set of formulas in one free variable) if no element of \(M\) realizes all formulas in \(p\) simultaneously.

A potential isomorphism between \(L\)-structures \(M\) and \(N\): a nonempty family of finite partial maps (pairs of same-length tuples) preserving atomic type and closed under extension in both directions.

The Scott sentence of \(M\): the Scott formula for the empty tuple at the stabilization ordinal, \(\sigma _{\alpha _0}(\langle \rangle )\).

The Scott formula \(\sigma _\alpha (a)\) for a tuple \(a \in M^n\) and ordinal \(\alpha {\lt} \omega _1\): an \(\mathcal{L}_{\omega _1\omega }\)-formula mirroring the recursive definition of \(\sim _\alpha \).

The Scott height of a countable structure \(M\): the least ordinal \(\alpha \) such that \(\sim _\alpha (a,b) \Rightarrow \sim _{\alpha +1}(a,b)\) for all tuples \(a,b\) in any countable structures.

The Scott rank of a countable structure \(M\): \(\mathrm{sr}(M) = \sup _{m \in M}(\mathrm{er}(m) + 1)\), where \(\mathrm{er}(m)\) is the element rank.

On any standard Borel space, a Borel equivalence relation has either at most countably many equivalence classes or exactly \(2^{\aleph _0}\).

The stabilization ordinal of a countable structure \(M\): the least ordinal \(\alpha \) such that \(\sim _\alpha \) on \(0\)-tuples characterizes isomorphism among countable structures.

The coding space for countable \(L\)-structures on \(\mathbb {N}\): for each relation query, whether the relation holds on that tuple.

If every \(A\)-finite subset of a theory \(T \subseteq A\) has a model, then \(T\) has a model.

A countable theory in an admissible fragment that belongs to a consistency property has a countable model.

A countable \(A\)-consistent theory in a full Barwise fragment of a countable language has a countable model.

The set of code pairs where \(\sim _\alpha \) holds is measurable for \(\alpha {\lt} \omega _1\).

If all countable models of \(\varphi \) have Scott height \(\leq \alpha {\lt} \omega _1\), then isomorphism is equivalent to \(\sim _\alpha \).

The family of \(A\)-consistent subsets of a full Barwise fragment forms a ConsistencyPropertyEq.

If the Silver–Burgess dichotomy holds, then for any \(\mathcal{L}_{\omega _1\omega }\) sentence whose countable models all have Scott height \(\leq \alpha {\lt} \omega _1\), the total number of isomorphism classes of countable models is either \(\leq \aleph _0\) or exactly \(2^{\aleph _0}\).

If the Silver–Burgess dichotomy holds, then for any \(\mathcal{L}_{\omega _1\omega }\) sentence whose \(\mathbb {N}\)-models all have Scott height \(\leq \alpha {\lt} \omega _1\), the number of isomorphism classes among coded \(\mathbb {N}\)-models is either \(\leq \aleph _0\) or exactly \(2^{\aleph _0}\).

For any countable relational language \(L\), countable \(L\)-structure \(M\), tuple size \(n\), and tuple \(a \in M^n\), the set of refinement ordinals \(\{ \varepsilon {\lt} \omega _1\mid \exists (N, b)\, \sim _\varepsilon (a,b) \wedge \neg \sim _{\varepsilon +1}(a,b)\} \) is countable.

Given a countable model \(M\) of an \(\mathcal{L}_{\omega _1\omega }\) sentence \(\varphi \) in a countable language, one can construct a countable model in the same universe as \(M\) via language expansion \(L[[M]]\) and the model existence theorem.

Given a countable model \(M\) of a countable \(\mathcal{L}_{\omega _1\omega }\) theory \(T\) in a countable language, one can reconstruct a countable model in the same universe as \(M\) via language expansion \(L[[M]]\) and the model existence theorem.

For structures on \(\operatorname {Fin} n\), the isomorphism relation is the orbit of \(\operatorname {Sym}(\operatorname {Fin} n)\), a finite union of graphs of continuous maps, hence Borel.

Every \(\mathcal{L}_{\omega _1\omega }\) sentence has a Hanf bound (and therefore a Hanf number).

If all countable models of \(\varphi \) have Scott height \(\leq \alpha {\lt} \omega _1\), then the set of isomorphic pairs within \(\mathrm{Mod}(\varphi ) \times \mathrm{Mod}(\varphi )\) is measurable.

A sentence in a countable language that belongs to some consistency property with equality axioms has a countable model.

\(M\) and \(N\) admit a potential isomorphism if and only if they are \(L_{\infty \omega }^w\)-equivalent (satisfy the same \(L_{\infty \omega }\) sentences with index types in universe \(w\)).

If a countable set \(S\) of \(\mathcal{L}_{\omega _1\omega }\) sentences in a countable language belongs to a consistency property with equality axioms, then \(S\) has a countable model.

For any \(\mathcal{L}_{\omega _1\omega }\) sentence \(\varphi \), the set \(\mathrm{Mod}(\varphi )\) is clopenable in the structure space.

For any \(\mathcal{L}_{\omega _1\omega }\) sentence \(\varphi \), the subtype \(\mathrm{Mod}(\varphi )\) is a standard Borel space.

For any \(\mathcal{L}_{\omega _1\omega }\) sentence \(\varphi \), the number of isomorphism classes of countable models is either \(\leq \aleph _1\) or exactly \(2^{\aleph _0}\).

Conditional on the Morley-Hanf transfer hypothesis, \(\beth _{\omega _1}\) is a Hanf bound for every \(\mathcal{L}_{\omega _1\omega }\) sentence.

Given a countable consistent theory \(T\) in a countable language and a countable collection of non-isolated types, there exists a countable model of \(T\) that omits all the given types.

If \(\varphi \) is derivable from \(T\) in fragment \(A\), then \(\varphi \) is true in any model of \(T\) equipped with a naming function.

For a countable relational language, the set of codes in the structure space satisfying a given \(\mathcal{L}_{\omega _1\omega }\) sentence is measurable (Borel).

The Scott sentence of a countable structure \(M\) characterizes \(M\) up to isomorphism among countable structures (unconditional corollary of CRH).

Assuming the countable refinement hypothesis: the Scott sentence of a countable structure \(M\) characterizes \(M\) up to isomorphism among countable structures, i.e., \(N \models \sigma _M \iff M \cong N\).

For countable \(M\) and \(\alpha {\lt} \omega _1\): \(\sigma _\alpha (a)\) is realized by \(b\) in \(N\) if and only if \(\sim _\alpha (a,b)\).

For any countable \(L\)-structure \(M\), \(\mathrm{sh}(M) {\lt} \omega _1\).

For any countable \(L\)-structure \(M\), \(\mathrm{sr}(M) {\lt} \omega _1\).

For any countable \(L\)-structure \(M\), there exists \(\alpha {\lt} \omega _1\) such that \(M\) self-stabilizes completely at \(\alpha \): for every \(n\) and tuples \(a, a' \in M^n\), \(\sim _\alpha (a,a') \iff \sim _{\alpha +1}(a,a')\).

If \(M\) stabilizes completely at \(\alpha {\lt} \omega _1\) and \(\sim _\alpha (\bar{a},\bar{b})\) holds, then \(M \cong N\).

For a countable relational language, the structure space is a Polish space.