Dependencies

A theory \(T\) is \(P\)-consistent if \(\bot \) is not derivable from \(T\) with conclusions permitted by \(P\).

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 set \(A\) carries a Cantor antichain for \(r\) if there is a continuous \(f : 2^{\mathbb {N}} \to A\) sending distinct points to \(r\)-inequivalent ones. This is the constructive form the Cantor-scheme builders produce, and the load-bearing intermediary between perfect antichains and cardinality.

A sentence \(\varphi \) is derivable from theory \(T\), with conclusions permitted by the sentence set \(P\). The proof system includes structural, propositional, infinitary, quantifier (omega-rule), equality, and classical rules.

\(\kappa \) 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. It is the ambient relation restricted along the subtype inclusion.

\(M \equiv _{L_{\infty \omega }}^w N\): the structures satisfy the same \(L_{\infty \omega }\) sentences whose infinitary connectives branch over carriers in universe \(w\). The quantifier over branching carriers sits outside the syntax.

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 set \(A\) carries a perfect antichain for an equivalence relation \(r\) if some nonempty perfect \(P \subseteq A\) has pairwise \(r\)-inequivalent points.

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 \(2^{\aleph _0}\). Proved in this repository as ‘silverBurgessDichotomy‘, via the classical \(G_0\)-dichotomy route.

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 equivalence relation on all of the structure space where two codes are related iff the structures they decode on \(\mathbb {N}\) are \(L\)-isomorphic. Defined without reference to any sentence, so that perfectness of a set of codes is a property of the ambient space rather than of a refinement chosen to make one model class Polish.

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

A sentence \(\varphi \) is thin on its \(\mathbb {N}\)-models if the set of codes of \(\mathbb {N}\)-models of \(\varphi \) carries no perfect antichain for the ambient isomorphism relation on the structure space. Stated ambiently, so no Polish refinement of the model subtype enters the definition.

\(A\) is thin for \(r\) if it carries no perfect antichain.

A ranked thinness analysis of \(A\) for \(r\) is a rank \(\rho : X \to \mathrm{Ord}\) together with the evidence that \(\rho \) is \({\lt} \omega _1\) on \(A\), that each fixed-rank antichain inside \(A\) is countable, and that every Cantor antichain in \(A\) contains a continuously embedded Cantor subcopy on which \(\rho \) is bounded below \(\omega _1\). It packages hypotheses only; it asserts no conclusion.

Suppose a rank \(\rho \) is \({\lt} \omega _1\) on \(A\) and each of its fixed-rank antichains inside \(A\) is countable, and suppose further that every continuous Cantor antichain \(f\) in \(A\) admits, on some continuously and injectively embedded Cantor subcopy \(e\), a presentation: a continuous map from \(2^{\mathbb {N}}\) to codes, all of them well-orders, whose order types are the ranks along \(f \circ e\). Then \(\rho \) is a ranked thinness analysis of \(A\) for \(r\).

If \(A\) is an analytic set of codes, every one of which names a well-founded tree on \(\mathbb {N}\), then a single countable ordinal strictly bounds the rank of every tree named by a code in \(A\).

If \(A\) is an analytic set of codes, every one of which interprets the distinguished relation \({\lt}\) as a well-order of \(\mathbb {N}\), then a single countable ordinal \(\alpha \) strictly bounds the order type of every code in \(A\).

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.

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 \(\sim _\alpha \).

If \(A\) carries a Cantor antichain for \(r\) in a Hausdorff space, then \(A\) carries a perfect antichain for \(r\). The range of the Cantor map is closed because a continuous injection out of a compact space into a Hausdorff space is a closed embedding, and it has no isolated points because accumulation points transport along that injection. Unlike the reverse implication, this needs no metric or completeness assumption.

If \(\varphi \) has arbitrarily large models and is \(\kappa \)-categorical (\(\kappa \) infinite), there is a complete \(\psi \models \varphi \) with a model of cardinality exactly \(\kappa \), and \(\psi \) is itself \(\kappa \)-categorical (Marker, Theorem 5.2.5 applications).

Over a countable relational vocabulary, every model of \(\varphi \) realizing only countably many complete \(\mathcal{L}_{\omega _1\omega }\)-types satisfies a COMPLETE \(\mathcal{L}_{\omega _1\omega }\)-sentence entailing \(\varphi \).

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 each \(n\), either the isomorphism classes of \(\operatorname {Fin} n\)-models of \(\varphi \) are countable, or that model class carries a perfect set of pairwise non-isomorphic models.

Over an arbitrary language, an \(\mathcal{L}_{\omega _1\omega }\)-entailment \(r_1 \models r_2\) has an interpolant \(\theta \) whose function and relation symbols each lie in the intersection of the two roots’ occurrence sets, with \(r_1 \models \theta \) and \(\theta \models r_2\). No hypotheses on the language.

Over a purely relational language (arbitrarily many symbols), an \(\mathcal{L}_{\omega _1\omega }\)-entailment \(r_1 \models r_2\) has an interpolant \(\theta \) whose function and relation symbols lie in the intersection of the two roots’ occurrence sets, with \(r_1 \models \theta \) and \(\theta \models r_2\). No countability hypothesis on the language.

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.

\(\mathrm{Hanf}(\mathcal{L}_{\omega _1\omega }) = \beth _{\omega _1}\): the global Hanf number of \(\mathcal{L}_{\omega _1\omega }\) over \(\mathrm{Type}\)-\(0\) languages is exactly \(\beth _{\omega _1}\).

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

For every \(\alpha {\lt} \omega _1\) there is an \(\mathcal{L}_{\omega _1\omega }\) sentence whose models have size exactly \(\beth _{\alpha +1}\) at the maximum, so \(\beth _{\alpha +1} {\lt} \mathrm{Hanf}(\mathcal{L}_{\omega _1\omega })\).

\(\beth _{\omega _1} \le \mathrm{Hanf}(\mathcal{L}_{\omega _1\omega })\).

\(\mathrm{Hanf}(\mathcal{L}_{\omega _1\omega }) \le \beth _{\omega _1}\): the global Hanf number of \(\mathcal{L}_{\omega _1\omega }\), over all languages and sentences, is at most \(\beth _{\omega _1}\).

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 whose infinitary connectives branch over carriers in universe \(w\)).

Over a countable relational vocabulary, every isomorphism-invariant Borel class \(B\) of coded countable structures is the model class of a single \(\mathcal{L}_{\omega _1\omega }\)-sentence.

Over a countable relational vocabulary, a class of coded countable structures is Borel and invariant under the logic action of \(S_\infty \) if and only if it is the class of models of a single \(\mathcal{L}_{\omega _1\omega }\)-sentence; equivalently, the Borel invariant classes are exactly the range of \(\mathrm{ModelsOf}\).

Over a countable relational vocabulary, the class of coded countable models of an \(\mathcal{L}_{\omega _1\omega }\)-sentence is an isomorphism-invariant Borel subset of the standard Borel space of structures.

Over a countable relational vocabulary, a class of coded countable structures is Borel and isomorphism-invariant if and only if it is the class of models of a single \(\mathcal{L}_{\omega _1\omega }\)-sentence.

Over an arbitrary language (no hypotheses whatsoever), an \(\mathcal{L}_{\omega _1\omega }\)-entailment \(r_1 \models r_2\) has an interpolant \(\theta \) whose function symbols lie in the intersection of the roots’ function occurrences, whose positively occurring relation symbols lie in the intersection of the roots’ positive occurrences, and whose negatively occurring ones lie in the intersection of the roots’ negative occurrences. This is the relation-polarity / logical-equality form of López–Escobar’s Theorem 4.1: relation symbols satisfy the full polarity condition (his clause (.4)), while equality is a logical symbol belonging to neither polarity class, so his clause (.3) — the equality-occurrence condition — is not claimed. No theory-level form is claimed either; it is false for \(\mathcal{L}_{\omega _1\omega }\).

Over a purely relational language (arbitrarily many symbols), an \(\mathcal{L}_{\omega _1\omega }\)-entailment \(r_1 \models r_2\) has an interpolant \(\theta \) whose function symbols lie in the intersection of the roots’ function occurrences, whose positively occurring relation symbols lie in the intersection of the roots’ positive occurrences, and whose negatively occurring ones lie in the intersection of the roots’ negative occurrences. This is the relation-polarity / logical-equality form: relation symbols satisfy the full polarity condition, while equality is a logical symbol, belongs to neither polarity class, and its occurrence in the interpolant is unconstrained.

Over a relational language of arbitrary cardinality, an \(\mathcal{L}_{\omega _1\omega }\)-entailment \(\varphi \models \psi \) whose consequent is universal (\(\forall _1\): every quantifier occurrence a positive \(\forall \); countable conjunctions and disjunctions are not counted as quantifiers) has a universal interpolant \(\theta \), whose function and relation symbols lie in the intersections of the two roots’ occurrences.

This is López–Escobar / Malitz Theorem 4.5 in the function-free case. The relative existential preservation theorem (4.6) is not claimed here: it needs a relativization or two-sorted encoding that this development does not contain. No theory-level form is claimed.

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 \(2^{\aleph _0}\).

For any \(\mathcal{L}_{\omega _1\omega }\) sentence \(\varphi \), either the \(\mathbb {N}\)-coded isomorphism classes number at most \(\aleph _1\), or the model class carries a perfect set of pairwise non-isomorphic models.

For any \(\mathcal{L}_{\omega _1\omega }\) sentence \(\varphi \), either the isomorphism classes of countable models number at most \(\aleph _1\), or \(\varphi \) has a perfect set of pairwise non-isomorphic \(\mathbb {N}\)-models, or it has one of pairwise non-isomorphic \(\operatorname {Fin} n\)-models for some \(n\).

\(\beth _{\omega _1}\) is a Hanf bound for every \(\mathcal{L}_{\omega _1\omega }\) sentence, over an arbitrary language with no side hypotheses.

Assuming restricted source-side extraction and a per-target compactness oracle, every Lω₁ω sentence satisfied in a model of size \(\geq \beth _{\omega _1}\) has arbitrarily large models.

Assuming realizability of the EM tail-template theory of the Morley seed, every Lω₁ω sentence satisfied in a model of size \(\geq \beth _{\omega _1}\) has arbitrarily large models — with no extraction hypothesis: an injective sequence is already fully indiscernible on the seed.

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

The Ehrenfeucht–Mostowski tail-template theory of the Morley seed is realizable: for every \(\mathcal{L}_{\omega _1\omega }\)-sentence \(\varphi \) with a model of size \(\geq \beth _{\omega _1}\) and every countable linear order \(J\), some model realizes the seed’s template theory along a \(J\)-indexed sequence. No symbol-countability assumptions.

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 \(T\) over the permitted sentence set \(P\), then \(\varphi \) is true in any model of \(T\) equipped with a naming function.

If \(A\) admits a ranked thinness analysis for \(r\) in a complete metric space, then \(A\) is thin for \(r\).

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 sequence of Henkin-constant classes in the schema term model is fully \(\mathcal{L}_{\omega _1\omega }\)-indiscernible on the countable schema family.

In the quotient term model of the completed schema theory, realization of a staged-family formula on classes of closed terms is equivalent to membership of its lifted template sentence in the completion.

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 \(a, a' \in M^n\), \(\sim _\alpha (a,a') \iff \sim _{\alpha +1}(a,a')\).

A Borel equivalence relation on a standard Borel space has at most \(\aleph _0\) classes or exactly \(2^{\aleph _0}\) — Silver’s theorem, fully proved via Miller’s classical category route.

A sentence of \(\mathcal{L}_{\omega _1\omega }\) with arbitrarily large models has, at every infinite cardinal \(\kappa \), a model of cardinality exactly \(\kappa \) realizing only countably many complete \(\mathcal{L}_{\omega _1\omega }\)-types over the empty set, across all finite arities (Marker, Theorem 5.2.5).

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.

Let \(C\) be a Borel class of coded structures over a countable relational language. Then \(C\) is thin for isomorphism if and only if for every countable fragment \(F\) and every finite arity \(n\), only countably many \(F\)-types of \(n\)-tuples are realized in members of \(C\). The class need not be isomorphism-invariant.

Let \(C\) be a Borel class of coded structures over a countable relational language. Then \(C\) is thin for isomorphism — carries no perfect set of pairwise non-isomorphic structures — if and only if for every sequence \((\theta _n)_{n{\lt}\omega }\) of \(L_{\omega _1\omega }\)-sentences, only countably many truth sequences \((\mathbb {1}[c \models \theta _n])_{n}\) are realized by \(c \in C\). The class need not be isomorphism-invariant.

If every model of an \(\mathcal{L}_{\omega _1\omega }\)-sentence \(\varphi \) interprets the distinguished relation \({\lt}\) as a well-order, then there is a single countable ordinal \(\alpha \) strictly bounding the order type of every model’s interpreted relation.

Let \(\varphi \) be an \(\mathcal{L}_{\omega _1\omega }\)-sentence over an arbitrary language with a distinguished binary relation symbol \({\lt}\). If for every \(\alpha {\lt} \omega _1\) some model of \(\varphi \) contains a strictly \({\lt}\)-increasing chain of length \(\alpha \), then some nonempty model \(M \models \varphi \) carries a map \(f : \mathbb {Q} \to M\) with \(f(q) {\lt} f(r)\) whenever \(q {\lt} r\) — the raw positive relation-preserving conclusion (no injectivity is claimed; it is a corollary under irreflexivity).

Over a countable relational vocabulary with a distinguished binary relation, the class of codes whose relation well-orders the carrier is **not** Borel in the logic space.

For any language with a distinguished binary relation symbol \({\lt}\), no \(\mathcal{L}_{\omega _1\omega }\)-sentence has as models exactly the structures whose interpretation of \({\lt}\) is a well-order.