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Let $M$ be an infinite first-order structure and $I$ a total order. After choosing a Skolem expansion of $M$, an elementary extension contains distinct elements $(a_i{)}_{i\in I}$ forming an order-indiscernible sequence in that expanded language. Their Skolem hull is an elementary substructure in the expanded language, and its reduct is a first-order model of $\mathrm{Th}(M)$. Every order automorphism of $I$ extends uniquely to a structure automorphism of the hull preserving the chosen Skolem expansion, by transporting terms in the generators.

Uniqueness need not hold among all structure automorphisms of the reduct. For an infinite structure in the pure logical equality language, take two distinct nullary Skolem function values $c_0,c_1$ in the hull. For an infinite order-indiscernible sequence, neither can equal a generator: the expanded formula $x=c_j$ would otherwise hold at every generator, contradicting their distinctness. Swapping $c_0,c_1$ fixes every generator and preserves the pure logical equality reduct, while failing to preserve the Skolem expansion. Thus the identity order automorphism already has two extensions in that reduct.

An elementary extension is an extension preserving all first-order truth with parameters, hence giving elementary equivalence. An extension $N\supseteq M$ is elementary when every formula with parameters from $M$ has the same truth value in both structures.

The saturated model condition in degree $\kappa$ means that a first-order model $M$ realizes every one-variable complete type over a parameter set $A\subseteq M$ of size less than $\kappa$. Equivalently every finitely satisfiable such complete type is realized, since it can be extended to a complete type. Without a specified degree, saturated usually means $|M|$-saturated. Countable saturation means ${\aleph }_1$-saturation, including complete types over countably many parameters.

Here is a general saturated elementary extension theorem, with proof: for every infinite regular cardinal $\kappa$, every first-order structure $M$ has a $\kappa$-saturated elementary extension. At a stage $M_{\alpha }$, list all complete types over its subsets of size less than $\kappa$. There is a set of them, since their formulas lie in a set-sized language. Add a witness constant for each complete type to the elementary diagram of $M_{\alpha }$, together with every formula in that complete type evaluated at its witness constant. A finite part mentions finitely many complete types and finitely many formulas from each; each finite conjunction is realizable in $M_{\alpha }$, and the witnesses for different complete types are unconstrained relative to each other. Thus the finite part is satisfiable. The compactness theorem produces an elementary extension $M_{\alpha +1}$ realizing all those complete types. At limits take elementary unions, justified by the elementary chain theorem.

Iterate for $\kappa$ stages and put $N={\bigcup }_{\alpha <\kappa }{M}_{\alpha }$. If $A\subseteq N$ has size below $\kappa$, regularity bounds the stages at which its elements enter, so $A\subseteq M_{\alpha }$ for some $\alpha <\kappa$. A complete type over $A$ consistent with $N$ is finitely satisfiable in $M_{\alpha }$: each finite conjunction is an existential formula with parameters from $M_{\alpha }$, and elementarity transfers its truth from $N$. The complete type is therefore realized in $M_{\alpha +1}$. This provesFor a singular desired degree, apply the theorem at a larger regular cardinal. If $M$ is infinite and an uncountable regular $\kappa$ satisfies ${\kappa }^{<\kappa }=\kappa$, with $|L|<\kappa$ and $|M|\le \kappa$, the construction can keep every stage of size $\kappa$. There are then at most $\kappa$ parameter sets and complete types, and downward Löwenheim-Skolem gives that size after each extension; adding $\kappa$ distinct elements initially ensures final size exactly $\kappa$. The resulting model is saturated in the unqualified sense. The cardinal-arithmetic hypothesis belongs to this size refinement, not to the general existence theorem.

An explicit ultraproduct version is useful too. In a countable language let $U$ be a nonprincipal ultrafilter on $\omega$ and $N={\prod }_{i<\omega }{M}_i/U$, where each factor is nonempty. Enumerate a countable finitely satisfiable complete type, including its chosen parameter representatives, as ${\phi }_1(x),{\phi }_2(x),\dots$. By the Łoś theorem, the set $A_n$ of coordinates where the first $n$ formulas have a simultaneous witness belongs to $U$. PutThese are decreasing members of $U$. At coordinate $i$, let $k(i)$ be the greatest $n\le i$ with $i\in B_n$, if there is one. Choose a witness to the first $k(i)$ formulas there, and an arbitrary element when there is none. For fixed $n$, every coordinate in $B_n$ has $k(i)\ge n$, so the chosen function satisfies ${\phi }_n$ on a $U$-large set. Its ultraproduct class realizes the whole complete type. A countable language over a countable parameter set has only countably many formulas, so this proves countable saturation of a nonprincipal ultraproduct over omega.

For NFU, we give the rank-indiscernible construction of an NFU model explicitly. NFU has a unary predicate $S$ for sets, atoms have no members, extensionality applies to sets, and comprehension applies to every stratified formula. A stratification assigns integer types to variables so that equality uses equal types and $x\in y$ requires $\mathrm{type}(y)=\mathrm{type}(x)+1$. A unary set predicate adds no type difference.

Start with the actual set structure $B=(V_{\omega +\omega },\in )$. This structure satisfies extensionality and every pure-membership separation instance: the subset defined in a set $a$ of rank below $\omega +\omega$ still has rank below $\omega +\omega$. We do not assert that $B$ satisfies replacement or all of ZFC. Expand it by Skolem functions. In it use the sequence of rank objects $V_{\omega +n+6}$, $n<\omega$. For two such objects $d<e$ in sequence order, we havewith quantifiers in $B$. The second statement holds because every subset of $V_{\alpha }$ belongs to $V_{\alpha +1}$, and the later rank is at least $\alpha +1$.

Introduce constants $d_i$, $i\in \mathbb{Z}$, and require them to be order indiscernibles in the Skolem language, satisfying these two assertions whenever $i<j$. Also require $V_{\omega +5}\subseteq d_i$ for every $i$; this fixed rank object is first-order definable in $B$. Every finite part of these requirements together with $\mathrm{Th}(B^{\ast })$ is satisfiable: color finite increasing tuples of the original rank sequence by the truth values of the finitely many formulas mentioned, and use Ramsey's theorem to get a homogeneous subsequence. The rank assertions hold for every increasing tuple there. The compactness theorem gives a model of the full theory. In its Skolem hull $N$ generated by the $d_i$, the integer shift extends to an automorphism $j$ with $j(d_i)=d_{i+1}$, by the term-transport proof of the Ehrenfeucht-Mostowski theorem. Thus $N$ still satisfies extensionality and all pure-membership separation instances, andThis is a set-sized first-order model; no satisfaction predicate for a proper-class universe is being assumed.

Externally let $D_i=\{x\in N:N\models x\in d_i\}$. The automorphism bijects $D_i$ onto $D_{i+1}$. Interpret a typed structure with sort $i$ equal to $D_i$. At sort $i+1$, declare $y$ a set exactly when $N\models y\subseteq d_i$; otherwise it is an atom. Define adjacent-sort membership byThe guard is necessary: a typed atom might have some ordinary $N$-members in $D_i$, and those must not become its typed members. Set-extensionality follows from extensionality in $N$, because the ordinary members of a typed set all lie in $D_i$. For any typed formula on sort $i$, its quantifiers can be restricted to the corresponding $d_l$, its set predicates replaced by subset assertions, and its memberships replaced by the guarded relation. This is an ordinary pure-membership formula of $N$ with finitely many parameters. Separation gives its extension $X\subseteq d_i$, and the displayed closure assertion gives $X\in d_{i+1}$. It is therefore a typed set witnessing comprehension. The automorphism shifts these sorted domains and preserves the set flags and guarded memberships.

Collapse the types onto $D_0$. DefineObjects not satisfying $S$ have no $E$-members. If two sets have the same $E$-members, their images under $j$ are subsets of $d_0$ with the same ordinary members, so are equal by extensionality of $N$; injectivity of $j$ gives equality of the original objects.

To verify every stratified comprehension instance, assign a variable $v$ its stratification type $t(v)$ and translate its value $a\in D_0$ to $j^{t(v)}(a)\in D_{t(v)}$. Equality is preserved. An atomic $E(x,y)$ translates to the guarded adjacent-sort membership because $t(y)=t(x)+1$; the unary predicate $S(y)$ translates to $j^{t(y)}(y)\subseteq d_{t(y)-1}$. Induction on formulas, using the domain bijections for quantifiers, preserves truth. If the free variable $x$ has type $t$, typed comprehension produces its extension $X\subseteq d_t$, with $X\in d_{t+1}$. Put $b=j^{-(t+1)}(X)\in D_0$. Then $S(b)$ holds, and for every $a\in D_0$,This is exactly the required NFU set. No formula containing the external automorphism $j$ has been used in a separation or comprehension scheme; $j$ only transports values after the pure formula is formed.

The construction also permits the usual infinity requirement. The ordinary $\omega$ and its successor graph $f$ are definable in $N$, hence fixed by $j$, and lie below $V_{\omega +5}$, so belong to all the required domains. Under $E$, the members of $\omega$ are still its ordinary $N$-members. The new Kuratowski pair satisfiesSince $j(f)=f$, membership of this pair in $f$ under $E$ is equivalent to $⟨x,y{⟩}_N\in f$. Thus $f$ is an internal injection of $\omega$ into itself omitting zero, giving a Dedekind-infinite set.

We have built a model of NFU, with infinity if it is included in the formulation, in ordinary ZFC metatheory:In particular $\mathrm{Con}(\mathrm{ZFC})\Rightarrow \mathrm{Con}(\mathrm{NFU})$. This proof does not assume $\mathrm{Con}(\mathrm{ZFC})$ as an additional internal axiom: its starting structure is the set $V_{\omega +\omega }$, not a model of ZFC.

A useful form of the Ehrenfeucht-Mostowski theorem says that, for an infinite first-order structure $M$ and any total order $I$, there is a model of $\mathrm{Th}(M)$ generated as a Skolem hull by distinct order indiscernibles $(a_i{)}_{i\in I}$. Every order automorphism of $I$ extends uniquely to an automorphism of the specified Skolem expansion of that hull. The model can be obtained inside an elementary extension of a Skolem expansion of $M$; the generated hull itself need not contain $M$.

Choose a Skolem expansion $M^{\ast }$ by iteratively adjoining witness functions: for each existential formula in the current language choose a function returning a witness whenever one exists, with an arbitrary value otherwise. Repeat for countably many rounds. Every formula in the final language uses finitely many symbols, so has a witness function from a later round. Thus the expansion is Skolemized for its own formulas, as required for elementarity of its hull. Add constants $a_i$ for $i\in I$ to the complete theory of $M^{\ast }$, require them to be distinct, and impose the schemefor every formula in the Skolem language. If an elementary extension of $M^{\ast }$ is desired, also add its elementary diagram with separate names for its elements.

Every finite part of this theory is satisfiable. Choose infinitely many distinct elements $b_0,b_1,\dots$ of $M$. For each of the finitely many arities in that fragment, color increasing index tuples by the finite vector of truth values of the formulas occurring. Repeated applications of the infinite Ramsey theorem give an infinite subset on which all these vectors are constant. Interpret the finitely many $a_i$ in increasing-index order from that subset. This satisfies the finite indiscernibility requirements and distinctness, while the elementary diagram is satisfied by the named original structure. The compactness theorem gives a structure $N^{\ast }$ satisfying the whole theory.

Let $H$ be the closure of the $a_i$ under the Skolem functions, including nullary ones. It is elementary in $N^{\ast }$: whenever an existential formula with parameters from $H$ is true in $N^{\ast }$, its Skolem witness lies in $H$. This is the Tarski-Vaught test, whose proof is induction on formulas, using precisely this witness property for the existential step. Hence the reduct of $H$ is a model of $\mathrm{Th}(M)$.

For an order automorphism $\sigma :I\to I$, defineThis is well-defined. If two terms denote the same element, write their equality as a formula on the combined increasing tuple of their indices; indiscernibility preserves that formula after transport by $\sigma$. The same argument preserves every relation and function. Applying ${\sigma }^{-1}$ gives the inverse, so $\hat{\sigma }$ is an automorphism. Since all elements are terms in the generators, it is the unique automorphism of this Skolem expansion extending $a_i\mapsto a_{\sigma (i)}$. This proves the theorem.

One may require the indiscernibles to lie in any specified infinite sequence of the starting structure, or to satisfy additional finite-tuple properties holding along that sequence: the same finite satisfiability proof chooses its homogeneous subsequence there. This strengthened form is what the rank-domain construction of NFU uses. Uniqueness is asserted in the Skolem expansion, not for all automorphisms of its reduct.

Let $U$ be all universal sentences true in $C$. The theory $T\cup U$ is consistent. Indeed, if a finite conjunction $u$ of sentences from $U$ were inconsistent with $T$, then $T\models \neg u$. A finite conjunction of universal sentences is universal, so its negation is equivalent to an existential sentence. By the hypothesis $C$ would satisfy $\neg u$, whereas it satisfies $u$.

Choose $A\models T\cup U$. Now expand the language by names for every element of $C$ and consider its elementary diagram $\mathrm{ElDiag}(C)$. Its universal consequences in the original language are exactly $U$: any such consequence is true in the named copy of $C$, while any original-language sentence true in $C$ already belongs to its elementary diagram. Part (b), applied to $A$ and this expanded theory, gives a structure embedding of $A$ into a model $D$ of $\mathrm{ElDiag}(C)$. The named copy of $C$ is an elementary substructure of $D$. Identifying that copy with $C$, we obtainThus the required target is an elementary extension of $C$, although $C$ itself need not be a model of $T$.

Let $S$ be the set of all existential sentences entailed by $T$. Certainly every model of $T$ satisfies $S$. Conversely, take $C\models S$. Part (c) gives $A\models T$ embedded in an elementary extension $D$ of $C$. After identifying $A$ with its image, $D$ is a superstructure of a model of $T$. The assumed closure under superstructures implies $D\models T$. Since $C\preccurlyeq D$, it follows that $C\models T$ as well. HenceThis supplies an existential axiomatization. Conversely, an existential sentence is preserved under embeddings because its finite tuple of witnesses and the quantifier-free facts about them persist, explaining why this is the appropriate syntactic class.

A complete type over $A\subseteq M$ is a maximal consistent set of first-order formulas in a fixed finite tuple of variables with parameters from $A$, consistent with the theory of that parameter expansion. A first-order model $M$ is a saturated model of degree $\kappa$ if every such type with $|A|<\kappa$ is realized in $M$. It suffices to require this for one-variable types and realize a finite tuple successively. Without a subscript, saturated commonly means $|M|$-saturated; some contexts specify only finite-parameter saturation. We state the degree explicitly.

The saturated elementary extension theorem says that every first-order structure has a $\kappa$-saturated elementary extension for any specified infinite $\kappa$. It suffices to prove this for regular $\kappa$, since a larger regular degree implies the requested degree. At a single stage $M$, list all complete one-variable types over all subsets of $M$ of size below $\kappa$. This collection is a set. Adjoin a witness constant for each type to the elementary diagram of $M$ and require that constant to satisfy its type. Every finite subset is satisfiable in $M$: each type's finite conjunction has a witness there, and the finitely many new constants have no cross-constraints. The compactness theorem gives an elementary extension realizing all these types simultaneously.

Repeat this step along an elementary chain $(M_{\alpha }{)}_{\alpha <\kappa }$, taking unions at limit stages, and let $N$ be its union. The elementary chain theorem follows here by induction on first-order formulas: an existential statement true in the union has a witness in a later stage; elementarity brings an existential witness back to the earlier stage containing its parameters. Thus each stage is elementary in $N$.

If $A\subseteq N$ has size below regular $\kappa$, all its elements lie in a single $M_{\alpha }$, by taking the supremum of their fewer-than-$\kappa$ stage indices. Any complete type over $A$ consistent with $N$ is already consistent over $M_{\alpha }$, since these structures have the same parameter theory. It was therefore realized in $M_{\alpha +1}$. This proves $\kappa$-saturation of $N$.

A corresponding theorem about full saturation is available with a stated cardinal-size hypothesis: if $\lambda$ is uncountable regular, $|\mathcal{L}|<\lambda$, and ${\lambda }^{<\lambda }=\lambda$, then every infinite first-order model of size at most $\lambda$ has an elementary extension of size $\lambda$ which is $\lambda$-saturated. First adjoin $\lambda$ pairwise distinct constants to the elementary diagram; every finite fragment is satisfiable because the original first-order model is infinite. A Skolem hull of their realizations and the original domain has size $\lambda$. There are at most $\lambda$ small parameter sets and at most $2^{|\mathcal{L}|+|A|+{\aleph }_0}\le \lambda$ types over each of them. Compactness and the downward elementary-submodel construction keep each realization stage of size $\lambda$. Taking a Skolem hull of the preceding stage and the chosen witnesses closes under at most $\lambda$ finite terms and proves this size bound and elementarity. The same length-$\lambda$ chain is consequently a fully saturated model of size $\lambda$. The cardinal hypothesis is not silently asserted to hold for every cardinal.

We now prove consistency of NFU by the permitted alternative method of indiscernibles and a domain-shifting structure automorphism. This can start from an explicit set structure available in ZFC, so we need not assume that there is a first-order model of all ZFC. Letand give $F$ the value $\varnothing$ off the natural numbers. Regard $(H,\in ,F,\omega )$ as a first-order structure, with a named object $\omega$. The graph of $F$ is an external set in the metatheory; it need not be a member of $H$. This structure is extensional and satisfies every separation instance in the pure membership language: any subset of $x\in H$ has rank below $\omega +\omega$ and belongs to $H$. It also satisfies, for natural $m<n$,since a subset of $V_{\omega +m}$ has rank at most $\omega +m$ and hence belongs to $V_{\omega +n}$.

Apply the indiscernible construction proved in question 7(i) to a Skolem expansion of this set structure, requiring $(c_i{)}_{i\in \mathbb{Z}}$ to be increasing members of its named natural-number object. For every finite fragment use an increasing sequence of ordinary natural numbers in the initial structure in the Ramsey argument. The resulting generated elementary hull $M$ has an external structure automorphism $j$ with $j(c_i)=c_{i+1}$, preserving $F$ and the named $\omega$. Put $D_i=F^M(c_i)$ and $D=D_0$. Then $j(D_i)=D_{i+1}$, and $M$ satisfies the displayed subset-containment assertion between successive domains. No claim that $M$ satisfies replacement is required.

All membership and subset statements about these objects are interpreted inside $M$. The hull is necessarily externally nonstandard: the natural-number objects $\dots ,c_{-2},c_{-1},c_0$ give an external descending chain. The external structure automorphism is not an object that must be definable within that first-order model.

On the external domain $D$ defineThis is the rank-shifting NFU model with explicit set predicate. The guard $S(y)$ is essential: an object not contained in $D$ could still have some ordinary members in $D$, and must not thereby become a nonempty atom. The objects failing $S$ have no $E$-members. If two objects satisfying $S$ have the same $E$-members, their images under $j$ are subsets of $D$ with the same members, so extensionality in $M$ and injectivity of $j$ make the original objects equal. Thus set-extensionality and the atom condition hold.

It remains to prove every comprehension instance. Use the domains $D_i$ just constructed, equivalently $D_i=j^i(D)$, for integer sorts. The map $j$ is a bijection from the objects of $D_i$ to the objects of $D_{i+1}$, and every subset of $D_i$ that exists in $M$ is an element of $D_{i+1}$. In sort $i$ define the set predicate by $y\subseteq D_{i-1}$; from sort $i$ to $i+1$ define membership by ordinary membership guarded by $y\subseteq D_i$.

For a stratified formula, assign integer types to its variables with equal types for equality and adjacent types for membership. Shift all types uniformly if necessary so the finitely many used types are nonnegative. Replace a variable of type $i$ by $j^i$ of its untyped value, quantify it over $D_i$, and use the typed relations just described. Applying $j^i$ to the untyped definition shows that each atomic first-order formula has the same truth value under this translation; induction on connectives and quantifiers proves the same for the whole first-order formula. Importantly the translated first-order formula is an ordinary first-order formula of $M$ with finitely many domain and transformed-parameter constants, not a first-order formula containing the external structure automorphism as a predicate.

If the comprehension variable has type $i$, separation in $M$ produces its extension $X\subseteq D_i$. The subset-containment assertion gives $X\in D_{i+1}$. Put $y=j^{-(i+1)}(X)\in D$. Then $j(y)=j^{-i}(X)\subseteq D$, hence $S(y)$, and for every $x\in D$,This verifies stratified comprehension with arbitrary parameters. The empty extension gives a distinguished empty set; the always-true first-order formula gives a universal set. The unstratified Russell predicate is not an allowed comprehension instance. ThusStarting from the actual set structure $H$ and applying compactness constructs a set first-order model, so this is a consistency proof in ZFC, not merely an assumption of Con(ZFC). It proves the requested consistency with an explicit interpretation; it does not depend on treating the much stronger full extensionality of NF as already consistent. The external $j$ is a tool of the construction, not an extra symbol permitted in NFU's comprehension scheme.

One useful form of the Ehrenfeucht-Mostowski theorem is this: for an infinite first-order structure $A$ and any total order $I$, there is a first-order model of $\mathrm{Th}(A)$ generated in a chosen Skolem expansion by an order-indiscernible sequence $(c_i{)}_{i\in I}$ of distinct elements. Every order automorphism of $I$ induces an structure automorphism of this expanded hull. One can additionally obtain an elementary extension containing $A$ by including its elementary diagram. Uniqueness of the induced structure automorphism is asserted in the Skolem expansion, not among arbitrary structure automorphisms of the reduct.

Choose a Skolem expansion $A^{\ast }$: for every existential first-order formula choose a witness function, satisfying $\exists y\phi (y,\bar{x})\Rightarrow \phi (f_{\phi }(\bar{x}),\bar{x})$. Iterate the expansion if necessary so witnesses are available for first-order formulas in the resulting language. Add constants $c_i$ and require $c_i\ne c_j$ for distinct indices. For every first-order formula $\phi (x_1,\dots ,x_m)$ of the expanded language and every two increasing index tuples, addTogether with $\mathrm{Th}(A^{\ast })$, this is the indiscernibility theory.

Every finite fragment is satisfiable. It mentions only finitely many first-order formulas and index constants. Choose a countably infinite sequence of distinct elements of $A^{\ast }$. For each relevant arity, colour increasing tuples from that sequence by the finite truth vector of all first-order formulas of that arity in the fragment. Successive applications of the infinite Ramsey theorem for r-sets leave an infinite subset homogeneous for each of these finitely many finite colourings. Interpret the finitely many index constants by distinct elements of that subset in increasing enumeration order. All the required truth-value equivalences then hold, and the original structure satisfies the finite part of its own theory.

If an actual elementary extension of $A$ is desired, include constants for its elements and the elementary diagram of $A^{\ast }$. A finite fragment still contains only finitely many parameter instances; include those truth vectors in the same colourings. Thus finite satisfiability is unchanged. The compactness theorem now supplies a first-order model with the entire indiscernible sequence, and in the diagram version an elementary copy of $A$.

Take the Skolem hull of the generators, including the diagram constants when present. The witness functions show directly that the hull is elementary: whenever an existential first-order formula with hull parameters is true in the ambient first-order model, its Skolem witness belongs to the hull. Induction on first-order formulas, or the Tarski-Vaught test, gives elementarity. Its reduct is therefore a first-order model of $\mathrm{Th}(A)$. In the version without diagram constants its cardinality is at most $\max (|I|,|\mathcal{L}|,{\aleph }_0)$.

For an order automorphism $\rho :I\to I$, defineThis is well defined. If two terms represent the same element, express their equality as a first-order formula on the increasing union of their generator indices. Indiscernibility and preservation of the index order carry that equality to the transported terms. The same argument for every relation shows preservation of the expanded structure; applying it to ${\rho }^{-1}$ supplies the inverse. Every element is a term in the generators, so no other structure automorphism of the specified expansion can have the same action on them. HenceFor an application requiring the generators to lie in a definable infinite ordered class, impose that class predicate and its increasing-order first-order formulas too. The finite-fragment proof uses an increasing sequence in that class, so all these additional conditions are preserved. In particular the integer index order and its shift give the structure automorphism used in the NFU construction. The constants are order indiscernibles, not necessarily indiscernibles under arbitrary permutations of the index set.

Let $M$ be an externally countable model of ZFC, and let $\kappa =({\omega }_1{)}^M$. In $M$ form the forcing orderThis is the finite-function collapse to countable size. For every $n\in {\omega }^M$ and $\alpha \in {\kappa }^M$, the internal setsare dense: extend a finite function by assigning a missing input, and use a fresh input to put any required value in its range.

Externally enumerate all the dense subsets of $\mathbb{P}$ which $M$ recognizes as dense. There are only countably many, because $M$ is countable. Recursively choose $p_{i+1}\le p_i$ in the $i$th such dense set and let $G$ be the upward closure of this descending chain. This produces a generic filter over $M$. By the forcing theorem, its generic extension $M[G]$ satisfies ZFC. The canonical forcing name $\dot{g}=\bigcup \dot{G}$ is forced to be a function with domain $\omega$ and range $\kappa$: the dense sets $D_n$ and $E_{\alpha }$ ensure exactly these assertions. ThereforeIts new first uncountable ordinal is accordingly different from $\kappa$.

For a transitive $M$, ordinary evaluation of names gives the familiar literal extension. The question only assumes a countable model, so one must also cover externally ill-founded models. In that case use the quotient of the internally defined forcing names by forced equality modulo $G$, with membership defined by forced membership. The internal forcing identities and the forcing theorem hold in $M$ and give a well-defined first-order structure satisfying every standard ZFC axiom. Check names embed the original membership structure, so identify their images with $M$. Its domain is a quotient of a subset of the countable domain of $M$, hence is externally countable. Thus the required extension exists even without assuming transitivity. It is a membership extension, not an elementary extension: the assertion that the parameter $\kappa$ is uncountable changes truth value.

For nonempty first-order structures $(M_i{)}_{i\in I}$ in a common language and an ultrafilter $D$ on $I$, identify product functions byThe ultraproduct ${\prod }_i{M}_i/D$ has these equivalence classes as elements. Functions are interpreted coordinatewise, and a relation holds of classes exactly when its coordinatewise truth set belongs to $D$. Closure under finite intersections makes these interpretations independent of representatives.

The Łoś theorem assertsProve it by induction on first-order formulas. Atomic first-order formulas hold by the definitions and induction on terms. Conjunction uses intersections, and negation uses the fact that an ultrafilter contains exactly one of a set and its complement. For an existential first-order formula, a product witness gives a coordinate witness on a $D$-large set. Conversely, on the $D$-large set where a coordinate witness exists, choose one in each factor, and choose arbitrary values elsewhere. Its class is a product witness by the induction hypothesis. This is the only witness-selection step; the surrounding argument is made in a choice metatheory. The special case with all factors equal is an ultrapower, and constant functions give an elementary embedding.

Here is a direct compactness theorem proof. Suppose every finite subset of a theory $T$ has a model. Index factors by the finite subsets $i\subseteq T$, choosing $M_i\models i$. For finite $F\subseteq T$, the cone $C_F=\{i:F\subseteq i\}$ is nonempty, and $C_F\cap C_H=C_{F\cup H}$. Extend the generated proper filter to an ultrafilter $D$. For every sentence $\sigma \in T$, its truth set contains $C_{\{\sigma \}}$, so the Łoś theorem gives ${\prod }_i{M}_i/D\models T$. Finite satisfiability therefore implies satisfiability, directly from the ultraproduct construction rather than from a syntactic completeness argument.

For saturated models, a structure is $\kappa$-saturated if every complete type over fewer than $\kappa$ parameters is realized. A useful concrete case is countable saturation of a nonprincipal ultraproduct over omega. In a countable language, enumerate any finitely satisfiable type over countably many parameters as ${\phi }_1(x),{\phi }_2(x),\dots$, and represent its parameters by product functions. By the Łoś theorem, the set $A_n$ of coordinates where the first $n$ first-order formulas have a simultaneous witness belongs to the nonprincipal ultrafilter $D$. PutThese are decreasing $D$-large sets. At coordinate $i$, let $h(i)$ be the largest $n\le i$ with $i\in B_n$, or zero if none exists, and choose a witness for the first $h(i)$ first-order formulas. For fixed $n$, every coordinate in $B_n$ has $h(i)\ge n$, so the resulting product function satisfies ${\phi }_n$ on a $D$-large set. Its class realizes the whole type. Thus such an ultraproduct is ${\aleph }_1$-saturated.

There is also an existence construction at every prescribed degree $\kappa$. Given an infinite model $M$, choose a regular cardinal $\lambda \ge \kappa$ and construct an elementary chain $(M_{\alpha }{)}_{\alpha \le \lambda }$. At a successor stage, add a new constant for a realization of every type over every subset of $M_{\alpha }$ of size below $\kappa$, together with the elementary diagram of $M_{\alpha }$. Every finite part is satisfiable: only finitely many types and finitely many first-order formulas from each are involved, and each finite type fragment has a witness in $M_{\alpha }$. Compactness therefore supplies a simultaneous elementary extension. At limits take unions. To justify elementarity of a union, any existential first-order formula with parameters in an earlier stage has a witness in that stage whenever it has one at a later stage, by elementarity; this is the Tarski-Vaught test.

Any fewer-than-$\kappa$ parameters of $M_{\lambda }$ occur together in some $M_{\alpha }$, by regularity of $\lambda$. A type over them which is finitely satisfiable in $M_{\lambda }$ is finitely satisfiable in $M_{\alpha }$, again by elementarity, and was realized in $M_{\alpha +1}$. Hence every infinite structure has a $\kappa$-saturated elementary extension for any prescribed cardinal $\kappa$. No bound on the size of this extension is asserted; full saturation at the model's own cardinality has additional cardinal-arithmetic issues.

For the Ehrenfeucht-Mostowski theorem, start with an infinite structure $M$ and a total order $I$. Choose a Skolem expansion $M^{\ast }$, and adjoin constants $c_i$ for $i\in I$. Require the elementary diagram of $M^{\ast }$, distinctness of these new constants, and the order-indiscernible sequence schema: any two increasing tuples of the $c_i$ of the same length satisfy the same first-order formulas in the expanded language. Every finite fragment mentions finitely many new constants and finitely many first-order formulas. Enumerate a countably infinite subset of $M$. For each of the finitely many first-order formulas, color its increasing tuples by its truth value in $M^{\ast }$, including whatever fixed parameters occur in the fragment. Successive applications of the infinite Ramsey theorem leave an infinite subset homogeneous for all these colorings. Assign the finitely many new constants to distinct elements of this subset in their index order. Their finite indiscernibility requirements and the elementary diagram are then simultaneously satisfied.

The compactness theorem supplies an elementary extension $N^{\ast }$ containing the required distinct indiscernibles. Their Skolem hull is an elementary substructure: an existential first-order formula true of its elements has its chosen Skolem-function witness in the hull, so the Tarski-Vaught test applies. Any order automorphism $\sigma$ of $I$ acts on this hull byIndiscernibility makes this well defined, since every equality of two term representations is preserved. The same argument preserves every first-order formula; the inverse is given by ${\sigma }^{-1}$. Thus it is an automorphism, uniquely determined in the language of the chosen Skolem expansion by its values on the generators. This proves both the indiscernible-model and automorphism forms of the Ehrenfeucht-Mostowski theorem. Uniqueness is not claimed for automorphisms of the reduct which need not preserve the chosen Skolem functions.

For simple typed set theory with atoms, write $S_i$ for the set predicate at sort $i$. Members have sort one lower; urelements have no typed members, extensionality applies to sets, and every well-typed first-order formula defines a set at the next sort. Typical ambiguity adds $\varphi \leftrightarrow {\varphi }^{+}$ for each closed sentence, where $+$ raises every sort by one. We prove finite satisfiability using Ramsey theorem, rather than assuming that adjacent sorts have the same cardinality.

Let $X_n=V_{\omega +2n}$. If $n<m$, then $\mathcal{P}(X_n)\subseteq X_m$. For increasing integers $h_0<h_1<h_2<\cdots$, interpret sort $i$ by $D_i=X_{h_{i+1}}$, interpret $S_i$ by the objects of $\mathcal{P}(X_{h_i})$, and interpret adjacent-sort membership by actual membership restricted to these designated sets. All other objects of a sort are typed urelements. Every subset of $D_i$ is a designated set in $D_{i+1}$, so comprehension holds for every first-order formula, including first-order formulas with quantifiers and parameters at other sorts. Two designated sets with the same members are equal, while the other objects have no typed members. Thus this is a full model of the typed axioms for every increasing sequence of levels. The extra preceding level $h_0$ gives a consistent interpretation of the bottom set predicate as well.

Take finitely many ambiguity sentences and let $r$ bound their highest sort before raising. Color increasing $(r+2)$-tuples $(h_0,\dots ,h_{r+1})$ by the vector of truth values of these sentences in the corresponding finite sorted structure. There are finitely many colors. The infinite Ramsey theorem supplies an homogeneous set of indices. Choose the level sequence from it. The sentence $\varphi$ is evaluated in the window $(h_0,\dots ,h_{r+1})$, while ${\varphi }^{+}$ is evaluated in the next window $(h_1,\dots ,h_{r+2})$; homogeneity makes their truth values equal. Sentences using fewer sorts simply ignore the unused final levels. Thus every finite family of ambiguity axioms has a model satisfying all the typed axioms. Apply many-sorted first-order compactness theorem to obtainThis is a relative consistency construction in the ordinary set-theoretic metatheory. Allowing urelements is essential here: skipping ranks leaves objects outside the represented power sets, and those objects must be permitted as urelements.

The Ehrenfeucht-Mostowski theorem asserts that an infinite first-order structure has, for any prescribed total order $I$, an elementary extension containing distinct order-indiscernibles $(c_i{)}_{i\in I}$. After a Skolem expansion, the Skolem hull of those generators is a model of the original complete theory. Every order automorphism of $I$ extends to a unique automorphism of this specified Skolem hull in the expanded language.

Choose a Skolem expansion $M^{\ast }$ of the original infinite structure: for each existential formula, select a witness function, including nullary functions when needed. Add constants $c_i$ for $i\in I$ and the elementary diagram of a structure of $M^{\ast }$, using separate names for its elements. Require $c_i\ne c_j$ for distinct indices, and for each formula $\phi$ in the Skolem language requirewhenever both index tuples are strictly increasing. These are the order-indiscernible sequence requirements.

Every finite fragment is satisfiable in $M^{\ast }$. Choose a countably infinite sequence of distinct elements of $M^{\ast }$. The fragment mentions only finitely many formulas and arities. Color increasing $k$-tuples of that sequence by the finite vector of truth values of its relevant $k$-ary formulas. The Ramsey theorem supplies an infinite homogeneous subsequence; repeat for the finitely many arities. Assign the finitely many named $c_i$ to distinct elements of the final homogeneous subsequence in increasing index order. The fixed finite part of the elementary diagram of a structure already holds in $M^{\ast }$. By the compactness theorem proved in Question 5, the full set of requirements has a model $N^{\ast }$, giving the required elementary extension and order-indiscernible sequence.

Let $H$ comprise all values of Skolem-language terms in finitely many $c_i$. It is closed under the witness functions. If $N^{\ast }\models \exists x\psi (x,\bar{h})$ for parameters in $H$, the specified witness function supplies a witness in $H$. Induction on formulas therefore gives $H⪯N^{\ast }$; its reduct is a model of $\mathrm{Th}(M)$. For infinite $I$, its cardinality is at most $\max (|I|,|L^{\ast }|,{\aleph }_0)$ and at least $|I|$.

For an order automorphism $\alpha :I\to I$, defineRewrite any finite list of indices in increasing order, allowing a term to repeat or omit variables. The order-indiscernible sequence property transfers every equality between terms and every relation among their values, making this definition well defined and elementary. Applying ${\alpha }^{-1}$ gives its inverse. All of $H$ is generated by the $c_i$ and the named Skolem functions, so an automorphism extending $\alpha$ must have precisely this action. Thus $\boxed{\mathrm{Aut}(I)\hookrightarrow \mathrm{Aut}(H^{\ast })}$, with uniqueness in the Skolem expansion. This uniqueness need not hold in the unexpanded reduct.

A useful precise form of the Ehrenfeucht-Mostowski theorem is this: for an infinite structure $M$ and any linear order $I$, a model of $\mathrm{Th}(M)$ can be generated as a Skolem hull by distinct order indiscernibles $(a_i{)}_{i\in I}$. Every order automorphism of $I$ extends to an automorphism of that hull in the chosen Skolem language. One may first name all elements of $M$, giving an elementary extension containing the original structure. The generation and uniqueness assertions concern this specified expansion, not arbitrary automorphisms of its reduct.

Choose a Skolem expansion $M^{\ast }$: for each existential formula add a function selecting a witness whenever one exists. Introduce constants $c_i$ for $i\in I$. Take the theory of $M^{\ast }$, distinctness of these constants, and all equivalencesThe formulas here are in the fixed expanded language, without the newly added $c_i$ as extra parameters. If naming $M$, use its complete elementary diagram and allow those original names in the formulas.

Every finite fragment has a model by Ramsey's theorem. Only finitely many formulas and constants occur. Take a sequence of distinct elements of $M$. For each arity appearing, color the increasing tuples from this sequence by the finite vector of truth values of all formulas of that arity. Successive Ramsey thinning gives an infinite subset homogeneous for each of these finitely many colorings. Assign the finitely many constants in increasing order to distinct elements of this subset. This satisfies every displayed equivalence in the fragment; the original theory is satisfied because the underlying structure remains $M^{\ast }$. Compactness now gives a model of the full theory, with the required order-indiscernible sequence.

The substructure generated by the constants under the Skolem functions is elementary: any existential formula with parameters which are terms in the generators has its Skolem witness in that same hull. If $g$ is an order automorphism of $I$, defineIndiscernibility makes this well-defined, by applying it to the equation between any two term representations using the increasing tuple of all their indices. The same argument preserves every relation and function; $g^{-1}$ supplies the inverse. It is unique in the generated Skolem structure. This completes the Ramsey proof, rather than stopping at the existence of indiscernibles.

For simple typed set theory with atoms, use sorts $D_0,D_1,\dots$ and membership only from $D_i$ to $D_{i+1}$. A set predicate distinguishes sets from atoms, extensionality compares sets of the same positive type, and comprehension produces in type $i+1$ every subset of $D_i$ defined by a well-typed formula. Atoms have no members. First-order models use Henkin semantics; they are not required to have their next domain equal to the full external power set. Typical ambiguity is the schema $\varphi \leftrightarrow {\varphi }^{+}$ for each closed sentence, where every sort index in ${\varphi }^{+}$ is raised by one.

We show finite satisfiability of this entire typed theory with ambiguity. Choose a list of ranks ${\rho }_0<{\rho }_1<\cdots$ above $\omega$, spaced by at least two. For a finite increasing selection of ranks $r_{-1}<r_0<\cdots <r_m$, letwhere $S_i$ is the set of objects declared to be sets at type $i$. Membership from $D_i$ to $D_{i+1}$ is ordinary membership restricted to objects of $S_{i+1}=\mathcal{P}(D_i)$; all other type-$i+1$ objects are atoms. The spacing puts $\mathcal{P}(D_i)$ inside $D_{i+1}$. Thus comprehension and set extensionality hold exactly, and atoms have no typed members. There are infinitely many objects at every visible type, so any usual typed infinity requirement also holds.

The auxiliary rank $r_{-1}$ provides a uniform interpretation if a set predicate is allowed at type zero; no type-minus-one domain is used in the theory. This avoids artificially declaring all type-zero objects to be atoms, which would not be invariant under type-raising. If the language has no type-zero set predicate, simply ignore this auxiliary interpretation.

A finite set of ambiguity instances uses finitely many closed sentences of bounded highest type, say at most $m$. Color increasing $(m+2)$-tuples of ranks by their finite truth vector in the structure with visible domains $D_0,\dots ,D_m$ and the auxiliary preceding rank. Ramsey's theorem gives an infinite homogeneous rank set. Choose $m+3$ consecutive ranks from it and build the hierarchy on all these ranks. The first window evaluates each original sentence; shifting the window one place evaluates its uniform type-raise. Homogeneity makes the truth vectors equal. All finite typed axioms and the selected ambiguity instances therefore hold together. Choosing $m$ large enough covers every sort needed by the finite fragment. Many-sorted compactness, equivalently ordinary first-order compactness after encoding sorts, yieldsThis is a model existence argument in the ambient set theory; no equality of adjacent type cardinalities is claimed. The gaps between ranks supply atoms, and the ambiguity is about truth of sentences, not a bijection between domains.

The compactness theorem states that a set of first-order sentences has a model if every finite subset has a model. The Upward Lowenheim-Skolem theorem says that an infinite structure $M$ in a language $L$ has an elementary extension of every prescribed infinite cardinality $\kappa \ge \max (|M|,|L|,{\aleph }_0)$.

Add constants $c_a$ for all $a\in M$ and the elementary diagram of a structure of $M$: every sentence with these constants true in $M$. Add $\kappa$ further constants $d_i$ and sentences $d_i\ne d_j$ for $i\ne j$. Any finite subset is satisfiable in $M$, by choosing the finitely many new constants distinctly, since $M$ is infinite. Compactness gives a model containing an elementary copy of $M$ and at least $\kappa$ elements.

For the exact cardinality, start with that copy and all the $d_i$, a set of cardinality $\kappa$. For every existential formula with parameters from this set that holds in the large model, choose one witness. Also close under the language's functions. Repeat countably many times. At each stage there are at most $\kappa$ formulas and finite parameter tuples, so the union still has cardinality $\kappa$. Every existential formula true in the large model with parameters in the union has a witness in the union; an induction on formulas proves that this substructure is elementary. This establishes the exact-size statement without assuming an unproved downward theorem.

A dense total order with $a<b$ is not a well-order: the nonempty set $\{x:a<x\}$ has no least element, since density supplies a point strictly between $a$ and any proposed least element.

In the strict-order language, the required axioms areThese axiomatize dense total orders, allowing endpoints. If more than one point is required, add $\exists x\exists y(x<y)$. For a non-strict poset language replace $x<y$ by $x\le y\wedge x\ne y$.