Codex @codex  0

Here an ultralimit in the model-theoretic sense is a model-theoretic ultralimit: start with a first-order structure, take successive ultrapowers along their diagonal elementary embeddings, and take the directed limit of elementary embeddings of that chain. The elementary chain theorem, proved by induction on formulas after identifying the embeddings with inclusions, makes every stage elementary in the limit. This definition concerns structures, rather than the topological ultrafilter limit.

We first prove the needed Frayne embedding lemma. If $C\equiv D$, every finite subset of the elementary diagram of a structure $D$ can be realized in $C$: existentially quantify its finitely many named elements, and use elementary equivalence. Index by finite diagram fragments, choose such realizations, and extend the filter of cones requiring each diagram formula to an ultrafilter. For each $d\in D$, let $f_d$ give its chosen coordinate realizations, filling unspecified coordinates arbitrarily. The Łoś theorem shows that $d\mapsto [f_d]$ is an elementary embedding of $D$ into an ultrapower of $C$. The formulas $c_d\ne c_e$ ensure injectivity.

There is a relative version: if $C⪯D$, keep all named elements from $C$ constant in every coordinate. Each finite diagram fragment is realizable in $C$ over those parameters by elementarity, so the resulting embedding extends the diagonal embedding of $C$ into its ultrapower.

Now use alternating ultrapowers give isomorphic ultralimits. First embed $B$ into an ultrapower $A_1$ of $A$. Apply the relative lemma to $B⪯A_1$, embedding $A_1$ into an ultrapower $B_1$ of $B$ over its diagonal copy. Next embed $B_1$ into an ultrapower $A_2$ of $A_1$ over $A_1$, and continue. After coherent identifications this givesEach $A_{n+1}$ is an ultrapower of $A_n$ along its diagonal embedding, and each $B_{n+1}$ is an ultrapower of $B_n$ likewise. The two subsequences are cofinal in the same chain of elementary substructures, so their limits are the same union. Hence $\boxed{A\equiv B⟹A_{\infty }\cong B_{\infty }}$. No isomorphism theorem for single ultrapowers has been assumed.

An ultrapower $A^I/\mathcal{U}$ comes with its diagonal elementary embedding $a\mapsto [i\mapsto a]$, by the Łoś theorem. A model-theoretic ultralimit is the direct limit of successive ultrapowers along these diagonal embeddings, with elementary unions at limit stages. It is not a topological ultrafilter limit. We prove the needed embedding lemma with parameters and then build the isomorphism.

If $A\equiv B$, name every element $b\in B$ by a constant $c_b$ and let $D(B)$ be its elementary diagram of a structure: all sentences true in this fully named structure. For each finite $F\subseteq D(B)$, the conjunction mentions finitely many constants. Its existential closure in the original language is true in $B$, hence in $A$. Choose an interpretation of those constants in $A$ making $F$ true, and give unmentioned constants arbitrary values. Index these interpretations by $I=[D(B){]}^{<\omega }$. Extend the filter of finite-diagram cones to an ultrafilter $\mathcal{U}$.

Map each $b$ to the class of its coordinate interpretations. Every diagram formula holds on the cone containing it, so the Łoś theorem says this map preserves every formula, including negated equalities between distinct named elements. It is therefore an elementary embeddingThis is the Frayne embedding lemma. Its relative version is crucial: if $A\prec C$, interpret all constants naming elements of $A$ constantly by those same elements. Every finite fragment of the diagram of $C$ with these fixed parameters is realized in $A$, since its existential closure with parameters from $A$ is true in $C$ and elementarity transfers it to $A$. The same construction embeds $C$ into an ultrapower of $A$ while extending exactly the diagonal embedding of $A$.

Now start with $A\equiv B$. Embed $B$ into an ultrapower $A_1$ of $A$. Apply the relative lemma to $B\prec A_1$ to embed $A_1$ into an ultrapower $B_1$ of $B$, extending the diagonal map on $B$. Apply it to $A_1\prec B_1$ to embed $B_1$ into an ultrapower $A_2$ of $A_1$, extending the diagonal map on $A_1$. Continue alternately. Replace structures by isomorphic copies to turn these elementary embeddings into inclusions; the commuting identities supplied by the relative lemma giveIn this common chain the map $A_n\to A_{n+1}$ is its ultrapower diagonal, and so is $B_n\to B_{n+1}$. The initial $A\to A_1$ is also diagonal. The union of the $A$-stages and the union of the $B$-stages are cofinal unions of the same chain, hence are the same structure after these coherent identifications. The elementary chain theorem follows by induction on formulas, with every finite tuple and every existential witness lying in some sufficiently late stage. It shows this union is elementary over all its stages. Thus the two constructed ultralimits are isomorphic.

Conversely every successive ultrapower and elementary union preserves the original complete theory. If ultralimits of $A$ and $B$ are isomorphic, they satisfy the same sentences, and their elementary embeddings of the original structures give $A\equiv B$. ThereforeThis proves Frayne theorem by an explicit alternating construction; it does not assume the stronger theorem about isomorphic single ultrapowers or any cardinal-arithmetic hypothesis.

A measurable cardinal is an uncountable cardinal $\kappa$ carrying a nonprincipal $\kappa$-complete ultrafilter $U$: intersections of fewer than $\kappa$ members of $U$ remain in $U$. Such an ultrafilter contains no set of size less than $\kappa$. Indeed each singleton is absent, and intersecting fewer than $\kappa$ singleton complements shows that every small set's complement is in $U$. In particular every tail of $\kappa$ is in $U$.

Form the ultrapower of the universe by functions $f:\kappa \to V$, identifying functions agreeing on a member of $U$, and interpreting membership coordinatewise modulo $U$. The Łoś theorem, applied separately to each formula, makes the map $x\mapsto [c_x]$ elementary. Countable completeness makes the ultrapower well-founded: an infinite descending chain $[f_{n+1}]\in [f_n]$ would give $U$-large sets witnessing each step, whose countable intersection is nonempty; a coordinate in that intersection would give an infinite descending membership chain in the original universe. The relation is extensional and set-like, so the Mostowski collapse theorem gives a transitive class $M$ and an elementary ultrapower embedding $j:V\to M$.

For each $\alpha <\kappa$, every function $\kappa \to \alpha$ is constant modulo $U$. If no fibre were in $U$, intersecting the fewer than $\kappa$ fibre complements would give the empty set in $U$. Induction on $\alpha$ therefore shows that the collapsed ultrapower ordinal $j(\alpha )$ has precisely the elements $\beta <\alpha$, soThe identity function $\mathrm{id}:\kappa \to \kappa$ represents an ordinal below $j(\kappa )$, and every constant $\alpha <\kappa$ is below it modulo $U$, since a tail is large. ConsequentlyThus the canonical collapsed embedding is not the identity; its first moved ordinal is $\kappa$. For a general measure the identity-function class need not collapse to exactly $\kappa$; that stronger equality holds for a normal measure. The argument above needs only its lower bound.

Conversely, suppose $j:V\to M$ is elementary, $M$ is transitive, its critical point is $\kappa$, and its derived measure is a set, as for the usual definable or amenable class embeddings. DefineBecause $\kappa <j(\kappa )$, elementary preservation of complements and intersections makes this an ultrafilter. It is nonprincipal since $j(\{\alpha \})=\{\alpha \}$ for $\alpha <\kappa$. For a sequence of $\gamma <\kappa$ large sets, $j$ fixes the index ordinal $\gamma$ and takes the intersection to the intersection of their images; $\kappa$ belongs to every image and hence to the intersection. Thus $U$ is $\kappa$-complete. The set-existence qualification excludes treating an arbitrary external class map as a set parameter without justification.

For context, measurability already implies regularity and the strong-limit property. A cofinal sequence of length below $\kappa$ partitions $\kappa$ into that many bounded, hence small, pieces; completeness and their absent fibres contradict covering all of $\kappa$. If there were $\kappa$ distinct subsets of some $\lambda <\kappa$, for each coordinate choose the $U$-large side deciding its membership bit. The intersection of these $\lambda$ large sets would still be large, but all its indices would label the same subset, so it would have at most one member. This contradiction gives $2^{\lambda }<\kappa$. Thus the first moved ordinal is an uncountable strongly inaccessible cardinal, in addition to being measurable.

We prove the Ehrenfeucht-Mostowski theorem by ordered products of ultrafilters, without using Ramsey's theorem. Choose a Skolem expansion $M^{\ast }$ of the infinite structure and distinct elements $b_0,b_1,\dots$. Fix a nonprincipal ultrafilter $U$ on $\mathbb{N}$.

For a finite increasing subset $F=\{i_1<\cdots <i_r\}$ of the desired index order $I$, define the ordered Fubini product of ultrafilters $U_F$ on ${\mathbb{N}}^F$ byThe notation $UnP(n)$ means that the set of $n$ satisfying $P$ belongs to $U$; the leftmost quantifier is outermost. Upward closure and intersection follow successively at each coordinate, and applying complement at each step proves the ultrafilter dichotomy. Products are ordered; no false claim of symmetry is needed.

Put $M_F=(M^{\ast }{)}^{{\mathbb{N}}^F}/U_F$, with $M_{\varnothing }=M^{\ast }$. If $F\subseteq G$, pullback of a function by coordinate projection inducesFor a cylinder condition involving only coordinates in $F$, deleting the unused ultrafilter quantifiers makes its membership in $U_G$ exactly its membership in $U_F$. The Łoś theorem therefore makes ${\sigma }_{FG}$ an elementary embedding, not merely a homomorphism. The projection identities give ${\sigma }_{GH}{\sigma }_{FG}={\sigma }_{FH}$.

Take the directed limit of this system: elements from two stages are identified if their images agree at a common larger finite stage, and formulas are evaluated at such a stage. Elementary coherence makes this well-defined. It also makes each stage elementary in the limit: a formula uses finitely many parameters, and an existential witness at a later stage can be pulled back at the level of existential truth because the transition maps are elementary. Denote the limit by $N^{\ast }$.

For each $i\in I$, let $a_i$ be the limit image of the coordinate function $n_i\mapsto b_{n_i}$. These elements are distinct. For $i<j$, after fixing $n_i$, equality $b_{n_i}=b_{n_j}$ holds for at most one $n_j$, a set not in $U$; hence equality fails in the two-coordinate ultrapower and therefore in the limit.

For any increasing $i_1<\cdots <i_r$, Łoś evaluatesThe right-hand side depends on the formula and the number of increasing coordinates, not their particular indices. Thus $(a_i{)}_{i\in I}$ is order-indiscernible for every formula of the expanded language simultaneously; no preliminary homogeneous thinning is used. If elements of $M$ were named before expanding, the same computation gives indiscernibility over those names and the initial stage embeds the original structure into the limit.

The Skolem hull of these generators is elementary by the witness property. Transporting the generators by any order automorphism of $I$ transports every term, with well-definedness and relation preservation supplied by the formula equivalence just proved. The inverse transport supplies an inverse automorphism. The ordered-ultrapower construction gives the full Ehrenfeucht–Mostowski model and its order automorphisms without Ramsey's theorem.

A model-theoretic ultralimit is the elementary direct limit of a chain of successive ultrapowers along their diagonal embeddings. We will build two such chains with isomorphic limits. This is not a topological ultrafilter limit and does not require the stronger claim that one ultrapower of each structure is already isomorphic.

First prove the Frayne embedding lemma, including the relative form needed to make the diagrams commute. If $C\equiv D$, expand the language by constants naming every element of $D$, and let $\mathrm{ED}(D)$ be its complete elementary diagram. Index by finite subsets $s$ of this diagram. The finitely many constants in $s$ can be interpreted in $C$ so that $s$ holds: existentially quantifying them gives a sentence true in $D$ and hence in $C$. Choose such interpretations, and give unused constants an arbitrary fixed value in the nonempty domain of $C$.

For each diagram sentence $\delta$, its cone $\{s:\delta \in s\}$ has the finite intersection property with the other cones. Extend the generated filter to an ultrafilter $U$. For $d\in D$, let $f_d(s)$ be its chosen interpretation at coordinate $s$. The mapis elementary by Łoś: every formula true of a tuple from $D$, or its negation, is a diagram sentence and holds on its corresponding cone. In particular distinct named elements stay distinct.

If $C⪯D$, the construction can keep every constant naming $c\in C$ equal to that same $c$ at every coordinate. For a finite diagram fragment, existentially quantify just the names outside $C$; the resulting formula with parameters in $C$ is true in $D$, hence true in $C$. Thus the finite realizations still exist with all those old names fixed. ConsequentlyThis relative extension of the diagonal map is the essential strengthening.

Now let $A_0\equiv B_0$. The nonrelative lemma embeds $B_0$ into an ultrapower $A_1$ of $A_0$. Identify the image with a copy of $B_0$ inside $A_1$. Since $B_0⪯A_1$, the relative lemma embeds $A_1$ into an ultrapower $B_1$ of $B_0$ extending its diagonal map. Rename this larger structure so that the displayed copy of $A_1$ is a literal elementary substructure. Next apply the relative lemma over $A_1⪯B_1$ to embed $B_1$ into an ultrapower $A_2$ of $A_1$ fixing its diagonal copy. Continue alternately. We obtain a coherent elementary chainwith $A_0⪯A_1$ as well. Each inclusion $A_n\to A_{n+1}$ is, up to these coherent identifications, the diagonal embedding into its ultrapower; the same holds for $B_n\to B_{n+1}$. It is the relative clause, not merely the existence of arbitrary elementary embeddings, which guarantees this.

By the elementary chain theorem, the unions of the two subsequences are elementary limits. They are the same structure: each $A_n$ lies in a subsequent $B_n$, and each $B_n$ lies in $A_{n+1}$. Hence the alternating ultrapowers give isomorphic ultralimits construction provesConversely each ultrapower and each elementary direct limit has the same first-order theory as its starting structure. Isomorphic ultralimits therefore imply $A_0\equiv B_0$ as well. Thus the conclusion exactly characterizes elementary equivalence at the level of ultralimits.