Proof. Suppose the vocabulary of T is a single binary relation symbol R for simplicity, and suppose the second-order sentence $\phi $ axiomatizes T. Let N be a model of $\phi $ of least cardinality and assume, toward a contradiction, that $\phi $ also has a model M with . Let $\theta $ be the sentence

$$\begin{align*}\exists P \exists R' (\theta'(P) \land \phi'(P, R')), \end{align*}$$

where

• • P is a unary predicate, not occurring in $\phi $ , and $R'$ is a binary relation symbol not occurring in $\phi $ .

• • $\phi '(P,R')$ is a modification of the sentence $\phi $ , where the first-order quantifiers $\exists x \dots $ are relativised to P as $\exists x (P(x) \land {\dots })$ , and each occurrence of R is replaced by $R'$ .

• • $\theta '(P)$ says that the cardinality of P is smaller than the ambient domain of the model (for example, that there is no injective function with range contained in P).

As $\phi $ is complete and $M \models \theta $ (by taking $(P, R')$ isomorphic to N), also $N \models \theta $ , so there is a model of $\phi $ of cardinality smaller than that of N, which is a contradiction. Thus all models of $\phi $ have the same cardinality $ \left \lvert {N} \right \rvert = \kappa $ .

Lastly, the observation that $\kappa $ is second-order characterizable follows from existentially quantifying all the symbols in the vocabulary of $\phi $ to get a sentence in the empty vocabulary (for example, $\exists R \phi $ serves to characterize $\kappa $ in this case).

Proof. Suppose $\phi $ is a complete second-order sentence in a vocabulary with a single binary relation symbol R (for simplicity), so $\phi $ has models in only one cardinality by Lemma 1.

Now let $M_0$ be the $<_{L[U]}$ -least model of $\phi $ . Suppose first that $ \left \lvert {M_0} \right \rvert> \kappa $ : in this case we can mimic the categoricity argument for L as follows. Let $\theta $ be the sentence

$$\begin{align*}\exists E \exists u \exists m \exists P \exists R'(\theta'(E,u) \land \phi'(P,R') \land \theta_{least}(E,u,m) \land \theta_{isom}(E,m,P,R')), \end{align*}$$

where

• • $E, R'$ are binary predicate symbols, P a unary predicate symbol and $u, m$ are first-order variables, none occurring in $\phi $ (the intuition is that E is $\in $ , u is a normal ultrafilter, m is a structure in the vocabulary of $\phi $ , P is the domain of m and $R'$ is the single binary relation of m).

• • $\theta '(E,u)$ states E is well-founded and extensional (so that E has a transitive collapse, and the domain of the model equipped with E can be thought of as a transitive set), and its collapse is a level of $L[u]$ having a normal measure u as an element.

• • $\phi '(P,R')$ is (as before) a modification of the sentence $\phi $ where each first-order quantifier $\exists x \dots $ is relativised to P as $\exists x(P(x) \land \dots )$ , and each occurrence of R is replaced by $R'$ .

• • $\theta _{least}(E,u,m)$ says $m <_{L[u]} m'$ for any other $m' = (Q,S)$ also satisfying $\phi '(Q,S)$ (using the formula defining the canonical well-order of $L[u]$ with u as a parameter).

• • $\theta _{isom}(E,m,P,R')$ states that $m = (P,R')$ , and that $(P,R')$ is isomorphic to the ambient model (so there is an injection F with range P such that $R(x,y) \leftrightarrow R'(F(x),F(y))$ for all $x,y$ ).

If $M \models \theta $ with witnesses E, u and $m = (P,R')$ , and $\pi \colon (M,E) \to (N,\in )$ is the transitive collapse, then $\pi (u) = U$ is the unique normal measure U on $\kappa $ , $N = L_\alpha [U]$ for some $\alpha $ and $\pi (m)$ is the $<_{L[U]}$ -least model $M_0$ of $\phi $ , so M is isomorphic to $M_0$ .

Conversely, let $\alpha $ be least such that $M_0 \in L_\alpha [U]$ . Then $\kappa < \alpha < \left \lvert {M_0} \right \rvert ^+$ and $U \in L_\alpha [U]$ , so we may pick a bijection $\pi \colon M_0 \to L_\alpha [U]$ and let E, u and $m = (P,R')$ be the preimages of $\in $ , U and $M_0$ under $\pi $ to witness $M_0 \models \theta $ .

Thus the above sentence $\theta $ is such that $M \models \theta $ if and only if M is isomorphic to the $<_{L[U]}$ -least model of $\phi $ . Now if $M \models \phi $ , also $M \models \theta $ by completeness of $\phi $ , so M is isomorphic to $M_0$ and $\phi $ is categorical.

Suppose now that $ \left \lvert {M_0} \right \rvert = \lambda < \kappa $ . In this case we cannot find a binary relation E on $M_0$ and $u \in M_0$ such that u is a normal measure in the transitive collapse of $(M_0,E)$ , so we modify the previously produced sentence $\theta $ . This argument relies on a straightforward modification of the $\Delta ^1_3$ well-order of reals in $L[U]$ . We make the further assumption that the domain of $M_0$ is a cardinal (and that $M_0$ is the $<_{L[U]}$ -least among such models), and let $\theta $ be the sentence

$$\begin{align*}\exists E \exists W \exists m \exists P \exists R'(\theta'(E,W) \land \phi'(E,P,R') \land \theta_{least}(E,W,m) \land \theta_{isom}(E,m,P,R')), \end{align*}$$

where

• • $E, R'$ are binary and $W,P$ unary predicate symbols, and m a first-order variable, none occurring in $\phi $ .

• • $\theta '(E,W)$ states E is well-founded and extensional, whose transitive collapse is an iterable $L[U]$ -premouse $(L_\alpha [W],\in ,W)$ for some $\alpha $ , where W is an $L[W]$ -ultrafilter on some $\gamma $ , where $\gamma $ is an ordinal greater than the cardinality of the ambient model.

• • $\phi '(E,P,R')$ is the sentence $\phi '(P,R')$ from before, with the additional stipulation that the extent of the predicate P is a cardinal.

• • $\theta _{least}(E,W,m)$ says $m <_{L[W]} m'$ for any other $m' = (Q,S)$ also satisfying $\phi '(E,Q,S)$ (using the formula defining the canonical well-order of $L[W]$ with W as a predicate).

• • $\theta _{isom}(E,m,P,R')$ remains unchanged from earlier.

We claim that $\theta $ is a sentence such that $M \models \theta $ if and only if M is isomorphic to the $<_{L[U]}$ -least model of $\phi $ (among models whose domain is a cardinal). So suppose $M \models \theta $ with witnesses $E, W$ and $m = (P,R')$ , and let $\pi \colon (M,E,W) \to (N,\in ,W')$ be the transitive collapse. Then $W' = \pi " \, W$ is a N-ultrafilter on some $\gamma> \lambda $ and $N = L_\alpha [W']$ for some $\alpha> \gamma $ , and $\pi (m)$ is the $<_{L[W']}$ -least model of $\phi $ in $L_\alpha [W']$ , to which M is isomorphic.

To see why $\pi (m)$ is $M_0$ , let $j \colon L[U] \to L[F]$ and $k \colon L_\alpha [W'] \to L_\delta [F]$ be long enough iterations of $L[U]$ and $L_\alpha [W]$ respectively such that they become comparable. Then $\operatorname {\mathrm {crit}}(j) = \kappa> \lambda $ and $\operatorname {\mathrm {crit}}(k) = \gamma> \lambda $ , so $j(M_0) = M_0$ and $k(\pi (m)) = \pi (m)$ . By elementarity, both $M_0$ and $\pi (m)$ are now the $<_{L[F]}$ -least model of $\phi $ among models whose domain is a cardinal, so $\pi (m) = M_0$ and M is isomorphic to $M_0$ .

Conversely, to see $M_0 \models \theta $ amounts to finding an appropriate premouse $(L_\alpha [W], \in , W)$ . Let $\delta $ be a large enough cardinal such that $M_0, U \in L_\delta [U]$ , and that $(L_\delta [U],\in ,U)$ is an iterable premouse. Then let N be the Skolem hull of $\lambda \cup \left \lbrace {M_0} \right \rbrace $ in $L_\delta [U]$ of cardinality $\lambda $ , and let $\pi \colon (N,\in , U \cap N) \to (L_\alpha [W], \in , W)$ be the transitive collapse. Now $(L_\alpha [W],\in ,W)$ is also an iterable premouse with $ \left \lvert {L_\alpha [W]} \right \rvert = \lambda $ , W is a $L_\alpha [W]$ -ultrafilter on some $\gamma = \pi (\kappa )> \lambda $ , and $\pi (M_0) = M_0$ , so by elementarity $M_0$ is the $<_{L[W]}$ -least model of $\phi $ as required. So $\theta $ is a sentence such that $M \models \theta $ if and only if M is isomorphic to $M_0$ , implying as before that $\phi $ is categorical.

Finally, observe that the case $ \left \lvert {M_0} \right \rvert = \kappa $ is impossible, since the measurable cardinal $\kappa $ is $\Pi ^2_1$ -indescribable. Thus $\phi $ is categorical.

Proof. For $\alpha < \kappa $ let $M_\alpha = (\lambda + \alpha , <)$ , so in a structure of cardinality $\lambda $ , $M_\alpha $ is straightforwardly definable from $\alpha $ (as $\lambda $ is second-order characterizable). These models have the property that $M_\alpha \cong M_\beta $ implies $M_\alpha = M_\beta $ .

For a second-order sentence $\phi $ in vocabulary $(<)$ , let

$$\begin{align*}S_\phi = \left\lbrace {\alpha < \kappa : M_\alpha \models \phi} \right\rbrace , \end{align*}$$

and let $T_0$ be the set of sentences $\phi $ such that $S_\phi \in U$ . As U is an ultrafilter, $T_0$ is a complete theory (so for any $\phi $ , exactly one of $\phi \in T_0$ or $\lnot \phi \in T_0$ hold), and by the $\sigma $ -completeness of U the intersection $X = \bigcap \left \lbrace {S_\phi : \phi \in T_0} \right \rbrace \in U$ is nonempty. The set X is such that for any $\alpha , \beta \in X$ , the structures $M_\alpha $ , $M_\beta $ have the same second-order theory $T_0$ , so it remains to see that the theory $T_0$ is recursively axiomatizable.

For a second-order sentence $\phi $ in vocabulary $(<)$ , let E be a binary relation symbol and u a first-order variable, neither occurring in $\phi $ , and let $\phi ^+$ be the second-order sentence

$$\begin{align*}\exists E \exists u (\theta'(E,u) \land (\exists x \in u)(\forall \alpha \in x) \, M_\alpha \models \phi), \end{align*}$$

where $\theta '(E,u)$ says E is well-founded and extensional, and its transitive collapse is a level of $L[u]$ containing $\lambda $ and having a normal measure u as an element. Note that $\phi ^+$ is a sentence in the empty vocabulary. Intuitively, $\phi ^+$ states that $M_\alpha \models \phi $ for a U-big set of ordinals $\alpha < \kappa $ , so for any structure N with we have the equivalences

$$ \begin{align*} N \models \phi^+ &\iff \left\lbrace {\alpha < \kappa : M_\alpha \models \phi} \right\rbrace = S_\phi \in U \\ &\iff M_\alpha \models \phi \text{ for some } \alpha \in X \\ &\iff \phi \in T_0. \end{align*} $$

The import of the vocabulary of $\phi ^+$ being empty is that for a structure N, the truth of $N \models \phi ^+$ depends only on , so we get that for all structures N with ,

$$\begin{align*}N \models \phi^+ \iff M_\alpha \models \phi^+ \text{ for some } \alpha \in X \iff \phi^+ \in T_0 \end{align*}$$

so also $\phi \leftrightarrow \phi ^+ \in T_0$ for all second-order sentences $\phi $ in vocabulary $(<)$ .

Now define the recursive set of sentences

$$\begin{align*}T = \left\lbrace {\phi \leftrightarrow \phi^+ : \phi \text{ is a second-order sentence in vocabulary } (<)} \right\rbrace. \end{align*}$$

Observe that any model N of the theory T has cardinality $\lambda $ , since taking $\theta _\lambda $ to be the second-order characterization of $\lambda $ , we have $M_\alpha \models \theta _\lambda $ for all $\alpha < \kappa $ , so $N \models \theta _\lambda ^+$ and thus $N \models \theta _\lambda $ since $\theta _\lambda ^+ \leftrightarrow \theta _\lambda \in T$ .

To see that T axiomatizes $T_0$ , suppose $N \models T$ so , and that $\phi $ is a second-order sentence in the vocabulary $(<)$ , so either $\phi \in T_0$ or $\lnot \phi \in T_0$ . In the former case we have $S_\phi \in U$ so $N \models \phi ^+$ , so $N \models \phi $ , and in the latter case we have $S_{\lnot \phi } \in U$ so $N \models \lnot \phi $ . Thus $\operatorname {\mathrm {Th}}_2(N) = T_0$ , so T recursively axiomatizes $T_0$ as desired.

Proof. Suppose $\phi $ is a complete second-order sentence with a countable model. Then all models of $\phi $ are countable by Lemma 1. Suppose $\phi $ is on the level $\Sigma ^1_n$ of second-order logic and its vocabulary is, for simplicity, just one binary predicate symbol P. Let R be the $\Sigma ^1_n$ (lightface) set of real numbers coding models of $\phi $ . By PD and its consequence, the Projective Uniformization theorem [Reference Moschovakis24, Theorem 6C5], there is a $\Sigma ^1_{n+1}$ (even $\Sigma ^1_n$ if n is even) element r in R. Suppose r codes the model M of $\phi $ . We show that every model of $\phi $ is isomorphic to M. Suppose N is a model of $\phi $ . Let $\theta $ be the following second-order sentence:

$$ \begin{align*}\begin{array}{l} \exists Q_+\exists Q_{\times}(\theta_1(Q_+,Q_{\times})\wedge\exists A (\theta_2(Q_+,Q_{\times},A)\wedge\\[4pt] \exists B(\theta_3(Q_+,Q_{\times},A,B)\wedge \exists F\theta_4(F,B)))), \end{array}\end{align*} $$

where

• • $\theta _1(Q_+,Q_{\times })$ is the standard second-order characterization of $({\mathbb {N}},+,\times )$ .

• • $\theta _2(Q_+,Q_{\times },A)$ says that the set A satisfies the $\Sigma ^1_{n+1}$ definition of r in terms of $Q_+$ and $Q_{\times }$ .

• • $\theta _3(Q_+,Q_{\times },A,B)$ says in a domain N that $(N,B)$ is the binary structure coded by A in terms of $Q_+$ and $Q_{\times }$ .

• • $\theta _4(F,B)$ is the second-order sentence which says that F is a bijection and

  $$ \begin{align*}\forall x\forall y(P(x,y)\leftrightarrow B(F(x),F(y))).\end{align*} $$

Thus, $\theta $ essentially says “I am isomorphic to the model coded by r.” Trivially, $M\models \theta $ . Recall that $M\models \phi $ . Since $\phi $ is complete, $\phi \models \theta $ . Therefore our assumption $N\models \phi $ implies $N\models \theta $ and therefore $N\cong M$ .

Proof. For any $x\subseteq \omega $ let

$$ \begin{align*}M_x=(V_\omega\cup\{y\subseteq\omega : y\equiv_T x\},\in),\end{align*} $$

where $y\equiv _Tx$ means that y and x are Turing-equivalent. We denote the second-order theory of $M_x$ by $\operatorname {\mathrm {Th}}_2(M_x)$ . By construction, $x\equiv _Ty$ implies $\operatorname {\mathrm {Th}}_2(M_x)=\operatorname {\mathrm {Th}}_2(M_y)$ . On the other hand, if $x\not \equiv _T y$ , then clearly $M_x\ncong M_y$ . If $\phi $ is a second-order sentence, then ‘ $M_x\models \phi $ ’ is a projective property of x, closed under $\equiv _T$ , and hence by Turing Determinacy for projective sets [Reference Martin22] has a constant truth value on a cone of reals x. By intersecting the cones we get a cone C of reals x on which $\operatorname {\mathrm {Th}}_2(M_x)$ is constant. For any second-order $\phi $ let $\phi ^+$ be the second-order sentence

$$ \begin{align*}"M_y\models\phi \text{ for a cone of} y"\end{align*} $$

and $\hat {\phi }$ the sentence $\phi \leftrightarrow \phi ^+$ . Let us consider the recursive second-order theory T consisting of $\hat {\phi }$ , when $\phi $ ranges over second-order sentences in the vocabulary of the structures $M_x$ . We may immediately conclude that T is complete, for if a second-order sentence $\phi $ is given, then by the choice of C either $M_x\models \phi $ for $x\in C$ or $M_x\models \neg \phi $ for $x\in C$ . In the first case $\hat {\phi }\models \phi $ and in the second case $\hat {\phi }\models \neg \phi $ . Therefore, $T\models \phi $ or $T\models \neg \phi $ . There are a continuum of non Turing equivalent reals in the cone C. Hence there are a continuum of non-isomorphic $M_x$ with $x\in C$ .

Proof. The pertinent consequence of $(*)$ is the quasihomogeneity of NS $_{\omega _1}$ , the nonstationary ideal on $\omega _1$ (see [Reference Woodin31, Section 5.8], particularly Definition 5.102). We take “NS $_{\omega _1}$ is quasihomogeneous” to be the following statement: if $X \subseteq {\mathcal {P}}(\omega _1)$ is ordinal definable from parameters in ${\mathbb {R}} \cup \{$ NS $_{\omega _1}\}$ , and X is closed under equality modulo NS $_{\omega _1}$ , and X contains one bistationary (i.e., stationary and co-stationary) subset of $\omega _1$ , then X contains every bistationary subset of $\omega _1$ .

We focus on the $\omega _1$ -like dense linear orders $\Phi (A)$ classified by Conway in [Reference Conway7], see also [Reference Nadel and Stavi25, Section 3]. Recall that for $A \subseteq \omega _1$ , the linear order $\Phi (A)$ is the concatenation $\Phi (A) = 1+\eta + \sum _{\alpha < \omega _1} \eta _\alpha $ , where $\eta $ denotes the order-type of the rationals, and

$$\begin{align*}\eta_\alpha = \begin{cases} \eta, & \alpha \notin A \\ 1+\eta, & \alpha \in A. \end{cases} \end{align*}$$

To wit, $\Phi (A)$ encodes the set A in that the initial segment $1 + \eta + \sum _{\alpha < \gamma } \eta _\alpha $ has a supremum in $\Phi (A)$ just in case $\gamma \in A$ (we add the initial $1+\eta $ block so that $\Phi (A)$ has a left endpoint whether $0 \in A$ or not).

These linear orders have the property that $\Phi (A) \cong \Phi (B)$ if and only if $A \mathrel {\triangle } B \in \text {NS}_{\omega _1}$ . To give a sketch of the proof, let $\Phi (A)_\gamma $ denote the initial segment $1+\eta +\sum _{\alpha < \gamma } \eta _\alpha $ consisting of the first $\gamma $ many blocks. Now if $f \colon \Phi (A) \to \Phi (B)$ is an isomorphism, the set

$$\begin{align*}C = \left\lbrace {\gamma < \omega_1 : f" \, \Phi(A)_ \gamma = \Phi(B)_\gamma} \right\rbrace \end{align*}$$

is a club satisfying $A \cap C = B \cap C$ . Conversely, if $A \cap C = B \cap C$ for a club $C = \langle \gamma _\alpha : \alpha < \omega _1 \rangle $ enumerated in increasing order, we may choose isomorphisms between $\Phi (A)_{\gamma _{\alpha +1}} \setminus \Phi (A)_{\gamma _\alpha }$ and $\Phi (B)_{\gamma _{\alpha +1}} \setminus \Phi (B)_{\gamma _\alpha }$ for each $\alpha < \omega _1$ , which give a piecewise definition for an isomorphism between $\Phi (A)$ and $\Phi (B)$ (the role of the club C is to guarantee that either both, or neither, of the above pieces of $\Phi (A)$ and $\Phi (B)$ have left endpoints).

Now for a second-order sentence $\phi $ in vocabulary $(<)$ , the set

$$\begin{align*}X_\phi = \left\lbrace {S \subseteq \omega_1 : S \text{ bistationary}, \Phi(S) \models \phi} \right\rbrace \end{align*}$$

is ordinal definable, and closed under equality modulo NS $_{\omega _1}$ , so the quasihomogeneity of NS $_{\omega _1}$ implies that $X_\phi $ contains either every bistationary subset of $\omega _1$ , or none of them.

This shows the models $\Phi (S)$ for bistationary $S \subseteq \omega _1$ all have the same complete second-order theory, which is thus non-categorical. This theory is axiomatized by the second-order sentence in vocabulary $(<)$ expressing “I am isomorphic to $\Phi (S)$ for some bistationary $S \subseteq \omega _1$ ”, so it is finitely axiomatizable, as required.

Proof. By Martin, AD implies $\omega _1 \to (\omega _1)^\omega $ , and moreover the homogeneous set given by $\omega _1 \to (\omega _1)^\omega $ can be taken to be a club (see [Reference Kleinberg14]). We may then intersect $\omega $ many homogeneous clubs for $\omega $ many colorings to obtain $\omega _1 \to (\omega _1)^\omega _{2^\omega }$ , and the homogeneous subset can still be taken to be a club.

We focus on models of the form $M_X = (\omega _1, <, X)$ for $X \in [\omega _1]^\omega $ . The second-order theory $\operatorname {\mathrm {Th}}_2(M_X)$ in the vocabulary $(<,X)$ can be encoded by a real $f(X) \in 2^\omega $ consisting of the Gödel numbers of the sentences true in $M_X$ . This gives a coloring $f \colon [\omega _1]^\omega \to 2^\omega $ , so we find a homogeneous club subset $H_0 \subseteq \omega _1$ such that $f(X)$ does not depend on $X \in [H_0]^\omega $ . Hence the models $M_X$ with $X \in [H_0]^\omega $ all have the same complete second-order theory $T_0$ , which is thus non-categorical.

The theory $T_0$ is axiomatized by

$$\begin{align*}T = \left\lbrace {\phi \leftrightarrow \phi^+ : \phi \text{ is a second-order sentence}} \right\rbrace , \end{align*}$$

where for a given second-order sentence $\phi $ in vocabulary $(<,X)$ , the sentence $\phi ^+$ expresses “there exists a club $C \subseteq \omega _1$ such that $M_X \models \phi $ for all $X \in [C]^\omega $ ”.

For a given second-order sentence $\phi $ , if $M_X \models \phi $ for each $X \in [H_0]^\omega $ , then $H_0$ serves to witness that $\phi ^+$ holds, so $T \models \phi $ . Conversely, if $\phi ^+$ holds, there is a club C such that $M_X \models \phi $ for every $X \in [C]^\omega $ , and taking $X \in [C \cap H_0]^\omega $ we see also that $M_X \models \phi $ for all $X \in [H_0]^\omega $ . Thus $T \models \phi $ for exactly those $\phi $ such that $M_X \models \phi $ for all $X \in [H_0]^\omega $ , so we see that T is a recursive axiomatization of the theory $T_0$ as desired.

Proof. Recall $(*)$ states that $L({\mathcal {P}}(\omega _1)) = L({\mathbb {R}})^{{{\mathbb {P}}_{\text {max}}}}$ and AD holds in $L({\mathbb {R}})$ . As ${{\mathbb {P}}_{\text {max}}}$ is homogeneous and does not add reals under AD (see [Reference Woodin31, Lemmas 4.40 and 4.43]), $\omega _1 = \omega _1^{L({\mathbb {R}})}$ and $[\omega _1]^\omega = ([\omega _1]^\omega )^{L({\mathbb {R}})}$ .

We again look at models $M_X = (\omega _1, <, X)$ for $X \in [\omega _1]^\omega $ , and working in $L({\mathbb {R}})$ , define a coloring $f \colon [\omega _1]^\omega \to 2^\omega $ by

$$\begin{align*}f(X) = r \quad \iff \quad L({\mathbb{R}}) \models {{\mathbb{P}}_{\text{max}}} \Vdash "\check r \text{ codes } \operatorname{\mathrm{Th}}_2(M_{\check X})". \end{align*}$$

That f is a well-defined total function follows from the homogeneity of ${{\mathbb {P}}_{\text {max}}}$ . By AD $^{L({\mathbb {R}})}$ we find a club $H_0 \in L({\mathbb {R}})$ , $H_0 \subseteq \omega _1$ homogeneous for f. Stepping out of $L({\mathbb {R}})$ , we see that the models $M_X$ , $X \in [H_0]^\omega $ all have the same complete second-order theory $T_0$ (in $L({\mathbb {R}})^{{\mathbb {P}}_{\text {max}}} = L({\mathcal {P}}(\omega _1))$ and in V both).

Working now in V, we again define

$$\begin{align*}T = \left\lbrace {\phi \leftrightarrow \phi^+ : \phi \text{ is a second-order sentence}} \right\rbrace , \end{align*}$$

where for a given second-order sentence $\phi $ , the sentence $\phi ^+$ expresses “there exists a club $C \subseteq \omega _1$ such that $M_X \models \phi $ for all $X \in [C]^{\omega }$ ”. The proof concludes analogously to the preceding theorem.

We note that $(*)$ implies $ \left \lvert {\omega _1^\omega } \right \rvert $ to be $\omega _2$ , so $T_0$ has $\omega _2$ many non-isomorphic models as claimed.

Proof. We modify a construction from the proof of [Reference Woodin33, Lemma 4.33] to our context. Let us call a pair $(A,B)$ of sets of ordinals an interlace, if $A\cap B=\emptyset $ , above every element of A there is an element of B, and vice versa. Suppose we have disjoint sets $X, Y, Z\subseteq \omega _1$ such that $(X\cup Y,Z)$ is an interlace. Let $z\sim z'$ in Z if $(z,z')\cap (X\cup Y)=\emptyset $ . Let $[z_n]$ , $n<\omega $ , be the $\omega $ first $\sim $ -equivalence classes in Z in increasing order. The triple $(X,Y,Z)$ is said to code the set $a\subseteq \omega $ if for all $n<\omega $ :

$$ \begin{align*}n\in a\iff \min\{\alpha\in X\cup Y:[z_n]<\alpha<[z_{n+1}]\}\in X.\end{align*} $$

It suffices to prove that for every $a\subseteq \omega $ in V there is a triple $(X,Y,Z)\in L[A]$ such that $(X\cup Y,Z)$ is an interlace, and $(X,Y,Z)$ codes a. To this end, suppose $a\in {\mathcal {P}}(\omega )^V$ . We define three alternate generics $G_0$ , $G_1$ , $G_2 \subseteq {\mathbb {P}}$ and an ordinal $\delta < \omega _1$ so that the initial segments $C_{G_0} \cap \delta $ , $C_{G_1} \cap \delta $ and $C_{G_2} \cap \delta $ code a. Moreover, there is a club $D \subseteq C_G$ such that $D \cap \alpha \in L[A]$ for all $\alpha < \omega _1$ , and we ensure the initial segments $D_{G_0} \cap \delta $ , $D_{G_1} \cap \delta $ and $D_{G_2} \cap \delta $ code a in the same manner as $C_{G_0}$ , $C_{G_1}$ and $C_{G_2}$ do, whence $a \in L[A]$ .

Suppose $\dot {A}$ is a ${\mathbb {Q}}$ -name for A in V, $\dot {D}$ is a ${\mathbb {P}} \times {\mathbb {Q}}$ -name for D, and $\dot {F}$ a ${\mathbb {P}} \times {\mathbb {Q}}$ -name for a function $\omega _1\to \omega _1$ which lists the elements of $\dot {D}$ in increasing order. In $V[G]$ , we may choose a sequence $\langle f_\alpha : \alpha <\omega _1 \rangle $ of countable partial functions from ${\mathbb {Q}}$ to $\omega _1$ such that $\text {dom}(f_\alpha )$ is a maximal antichain in ${\mathbb {Q}}$ and $q \Vdash \dot {F}(\alpha ) = \check {f}_\alpha (q)$ for each $\alpha < \omega _1$ and $q \in \text {dom}(f_\alpha )$ . We then pick ${\mathbb {P}}$ -names $\dot {f}_\alpha \in V$ for each $f_\alpha $ , $\alpha < \omega _1$ .

Suppose (without loss of generality) that the weakest condition in ${\mathbb {P}}\times {\mathbb {Q}}$ forces what is assumed about $\dot {A}$ , $\dot {D}$ , $\dot {F}$ and $\dot {f}_\alpha $ , $\alpha < \omega _1$ . Since $\Vdash \dot {D}\subseteq C_{\dot {G}}$ , we have $\Vdash \operatorname {\mathrm {ran}}(\dot {f}_\alpha )\subseteq C_{\dot {G}}$ .

For $\delta <\omega _1$ , we say $p \in {\mathbb {P}}$ bounds $\operatorname {\mathrm {ran}}(\dot {f}_\delta )$ if

(2) $$ \begin{align} p\Vdash \operatorname{\mathrm{ran}}(\dot{f}_\delta)\subseteq c_p\setminus\delta, \end{align} $$

and we let $W_\delta \subseteq {\mathbb {P}}$ be the set of such conditions p. It is easy to see that $W_\delta $ is dense by first picking a condition p that decides $\operatorname {\mathrm {ran}}(\dot {f}_\delta )$ , and then extending $c_p$ by an ordinal greater than $\sup (\operatorname {\mathrm {ran}}(\dot {f}_\delta ))$ . It follows that if p bounds $\operatorname {\mathrm {ran}}(\dot {f}_\delta )$ , then $p \Vdash \dot D \cap c_p \setminus \delta \neq \emptyset $ .

We construct descending $\omega $ -sequences $(p_n), (q_n)$ and $(r_n)$ in ${\mathbb {P}}$ as follows. We let $p_0=q_0=r_0$ be the weakest condition in ${\mathbb {P}}$ , and $\delta _0 = 0$ . Suppose $p_n, q_n$ , $r_n$ and $\delta _n$ have been defined already. Let $r_{n+1} \leq r_n$ be such that $\min (c_{r_{n+1}} \setminus c_{r_n})> \delta _n$ and $r_{n+1} \in W_{\delta _n}$ . Now there are two cases:

1. 1. Case $n\in a$ :

   1. (a) Let $p_{n+1}\le p_n$ be such that $\min (c_{p_{n+1}}\setminus c_{p_n})>\max (c_{r_{n+1}})$ and $p_{n+1}\in W_{\delta _n}$ .

   2. (b) Let $q_{n+1}\le q_n$ be such that $\min (c_{q_{n+1}}\setminus c_{q_n})>\max (c_{p_{n+1}})$ and $q_{n+1}\in W_{\delta _n}$ .

2. 2. Case $n\notin a$ :

   1. (a) Let $q_{n+1}\le q_n$ be such that $\min (c_{q_{n+1}}\setminus c_{q_n})>\max (c_{r_{n+1}})$ and $q_{n+1}\in W_{\delta _n}$ .

   2. (b) Let $p_{n+1}\le p_n$ be such that $\min (c_{p_{n+1}}\setminus c_{p_n})>\max (c_{q_{n+1}})$ and $p_{n+1}\in W_{\delta _n}$ .

Lastly set $\delta _{n+1} = \max (c_{p_{n+1}} \cup c_{q_{n+1}})$ , which defines $p_{n+1}$ , $q_{n+1}$ , $r_{n+1}$ and $\delta _{n+1}$ .

Then, as $p_{n+1}$ (and thus also $p_{n+2} \leq p_{n+1}$ ) bounds $\operatorname {\mathrm {ran}}(\dot {f}_{\delta _n})$ , we have that

$$\begin{align*}p_{n+2} \Vdash \emptyset \neq \dot{D} \cap [\delta_n, \delta_{n+1}) \subseteq C_{\dot G} \cap [\delta_n, \delta_{n+1}) = \check{c}_{p_{n+1}} \setminus \check{c}_{p_n} \end{align*}$$

for all $n < \omega $ . Respectively, $q_{n+2}$ and $r_{n+2}$ force the same with $c_{q_{n+1}} \setminus c_{q_n}$ and $c_{r_{n+1}} \setminus c_{r_n}$ in place of $c_{p_{n+1}} \setminus c_{p_n}$ .

Let $p_\omega =\inf _n p_n, q_\omega =\inf _n q_n, r_\omega =\inf _n r_n$ , and let $\delta =\sup _n \delta _n$ . Let each of $G_0, G_1, G_2 \subseteq {\mathbb {P}}$ be generic over $V[H]$ with $p_\omega \in G_0$ , $q_\omega \in G_1$ , and $r_\omega \in G_2$ . Lastly, let

$$\begin{align*}X=\dot{D}_{G_0\times H}\cap\delta, \quad Y=\dot{D}_{G_1\times H}\cap\delta, \quad Z=\dot{D}_{G_2\times H}\cap\delta. \end{align*}$$

As $\Vdash _{{\mathbb {P}}\times {\mathbb {Q}}}\dot {D}\cap \delta \in L[\dot {A}]$ and $\dot {A}_{G_0\times H}=\dot {A}_H$ , we have $V[G_0\times H]\models X\in L[A]$ . By absoluteness, $V[H]\models X\in L[A]$ . Similarly, $V[H]\models Y,Z\in L[A]$ . Now by construction, $(X\cup Y,Z)$ is an interlace and $(X,Y,Z)$ codes a, so $a\in L[A]$ .

Proof. Suppose first $A \subseteq \omega _1$ nearly constructs every club on $\omega _1$ . Let $\dot A \in V[G]$ be a ${\mathbb {Q}}$ -name for A, and let $\beta < \omega _2$ be large enough that $\dot A \in V[G_{<\beta }]$ , where $G_{<\beta }$ is the restriction of G to the $\beta $ first factors of the iteration ${\mathbb {P}}_{\omega _2}$ . Let also $G_\beta $ and $G_{>\beta }$ denote the $\beta $ th component and the rest of the generic respectively, so that $G = G_{<\beta } * G_\beta * G_{> \beta }$ .

Now letting $W = V[G_{<\beta }]$ , $A \in W[H]$ nearly constructs $C_{G_\beta }$ , the $\beta $ th fast club added by the iteration over V, so applying Lemma 12 over W we get that ${\mathcal {P}}(\omega )^V \subseteq {\mathcal {P}}(\omega )^W \subseteq L[A]$ .

The preceding shows that ${\mathcal {P}}(\omega )^V \subseteq \bigcap \left \lbrace {{\mathcal {P}}(\omega )^{V[G \times H]} \cap L[A] : A \in \mathcal A} \right \rbrace $ . To see the reverse inclusion holds, let $A_0 \in {\mathcal {P}}(\omega _1)^V$ code the set $([\omega _1]^\omega )^V$ and we verify $A_0$ nearly constructs every club on $\omega _1$ . To wit, if $C \subseteq \omega _1$ is a club in $V[G \times H]$ , then as ${\mathbb {Q}}$ is a forcing with c.c.c., there is a club $D \in V[G]$ with $D \subseteq C$ . Since ${\mathbb {P}}_{\omega _2}$ does not add $\omega $ -sequences, $D \cap \alpha \in V$ for all $\alpha < \omega _1$ . This shows $A_0 \in \mathcal A$ and that all reals constructed from $A_0$ are in V, as desired.

Proof. Assume CH, without loss of generality. As said above, we start with some preparatory countably closed forcing ${\mathbb {P}}_{\omega _2}$ obtaining a generic extension $V[G]$ . Then we add $\aleph _1$ Cohen reals obtaining a further generic extension $V[G][H]$ . In this model we consider for every $x\subseteq \omega $ the model $M_x$ as defined in (1). Clearly, the cardinality of $M_x$ is $\aleph _1$ . We shall now show that if x is Cohen-generic over $V[G]$ , e.g., one of the $\aleph _1$ many coded by H, then the complete second-order theory of $M_x$ is finitely axiomatizable (in second-order logic). To end the proof of the theorem, we show that if x and y are mutually Cohen-generic over $V[G]$ , then $M_x$ and $M_y$ are second-order equivalent but non-isomorphic.

We let the preparatory forcing ${\mathbb {P}}_{\omega _2}=\langle {\mathbb {P}}_\alpha :\alpha <\omega _2\rangle $ be the countable support iteration of $\langle \dot {Q}_\alpha :\alpha <\omega _2\rangle $ , where ${\mathbb {P}}_\alpha \Vdash "\dot {Q}_\alpha $ is the fast club forcing ${\mathbb {P}}_{{\scriptsize {\text {fast}}}}"$ . Let G be ${\mathbb {P}}_{\omega _2}$ -generic over V and $G_\alpha =G\cap {\mathbb {P}}_\alpha $ . In $V[G]$ we force with ${\mathbb {Q}} = \text {Fn}(\omega _1 \times \omega , 2, \omega )$ adding a generic H. Note that $\aleph _1^{V[G][H]}=\aleph _1^V$ and ${\mathcal {P}}(\omega )^{V[G]}={\mathcal {P}}(\omega )^V$ . Working in $V[G][H]$ , let the second-order sentence $\phi (R,E)$ , where R is unary and E is binary, say in a model M:

1. (1) $E^M$ is a well-founded relation satisfying $ZFC^- +$ “every set is countable”. This should be also true when relativized to $R^M$ .

2. (2) $|M|=\aleph _1$ .

3. (3) If $P'\in R^M$ denotes (in M) the set $\text {Fn}(\omega ,2,\omega )$ of conditions for adding one Cohen real, then there is $K\subseteq P'$ such that K is $P'$ -generic over $R^M$ and $M\models "V=R[K]"$ .

4. (4) If $a\subseteq \omega $ and the transitive collapse of M is N, then the following conditions are equivalent:

   1. (a) $a\in R^N$ .

   2. (b) If $A\subseteq \omega _1$ nearly constructs every club on $\omega _1$ , that is, for every club $C\subseteq \omega _1$ there is a club $D\subseteq C$ such that $D\cap \alpha \in L[A]$ for every $\alpha <\omega _1,$ then $a\in L[A]$ .

Note that we can express $"D\cap \alpha \in L[A]"$ , or equivalently $"\exists \beta (|\beta |=\aleph _1\wedge D\cap \alpha \in L_\beta [A]"$ , in second-order logic on M since second-order logic gives us access to all structures of cardinality $|M|$ ( ${} = \aleph _1$ ).

Claim: The following conditions are equivalent in $V[G][H]$ :

1. (i) $M\models \phi (R,E)$ .

2. (ii) $M\cong M_x$ for some real x which is Cohen generic over V.

We continue the proof of Theorem 14. The sentence $\phi (R,E)$ is non-categorical in $V[G][H]$ because if we take two mutually generic (over $V[G]$ ) Cohen reals $r_0$ and $r_1$ , then $M_{r_0}$ and $M_{r_1}$ are non-isomorphic models of $\phi (R,E)$ . To prove that $\phi (R,E)$ is complete, suppose $(M,R^M,E^M)$ and $(N,R^N,E^N)$ are two models of $\phi (R,E)$ . Without loss of generality, they are of the form $(M,R^M,\in )$ and $(N,R^N,\in )$ , where M and N are transitive sets. By construction, they are of the form $M_{r_0}$ and $M_{r_1}$ where both $r_0$ and $r_1$ are Cohen generic over $HC^V$ , hence over $HC^{V[G]}$ . They are subsumed by the generic H. By homogeneity of Cohen forcing $\text {Fn}(\omega ,2,\omega )$ the models are second-order equivalent.

Proof. Without loss of generality, we assume GCH up to $\kappa $ . We first force a second-order definable well-order of the bounded subsets of $\kappa $ with a reverse Easton type iteration of length $\kappa $ described in [Reference Menas23, Theorem 20].

Let $S=\langle \kappa _\xi : \xi <\lambda \rangle $ be a strictly increasing sequence of aleph-fixpoints cofinal in $\kappa $ such that $\kappa _0> \lambda $ . Let $\pi :\kappa \times \kappa \to \kappa $ be the Gödel pairing function, and W be a well-order of $V_\kappa $ . Suppose $A\subseteq \mu $ for a cardinal $\mu $ . We write $A\sim V_\mu $ if

$$\begin{align*}(V_\mu,\in)\cong(\mu,\{(\alpha,\beta):\pi(\alpha,\beta)\in A\}). \end{align*}$$

Let the poset $E(\mu ,A)$ be the iteration (product) of the posets ${\mathbb {R}}_\alpha $ , $\alpha <\mu $ , where

$$\begin{align*}{\mathbb{R}}_\alpha= \left\{\begin{array}{@{}ll} \text{Fn}(\aleph_{\mu+\alpha+3}\times \aleph_{\mu+\alpha+1},2,\aleph_{\mu+\alpha+1}), &\text{if } \alpha=\omega\cdot\beta \text{ and } \beta\in A\\ \text{Fn}(\aleph_{\mu+4}\times \aleph_{\mu+2},2,\aleph_{\mu+2}), &\text{if } \alpha=1 \text{ and } \mu = \kappa_\xi, \xi<\lambda\\ (\{0\},=)&\text{ otherwise}\end{array}\right. \end{align*}$$

with Easton support; i.e., $E(\mu ,A)$ consists of functions $p\in \prod _{\alpha <\mu } {\mathbb {R}}_\alpha $ such that, denoting the support $\{\alpha : p(\alpha )\ne \emptyset \}$ of p by $\text {supp}(p)$ , $|\text {supp}(p)\cap \gamma |<\gamma $ for all regular $\gamma $ . (The first option for ${\mathbb {R}}_\alpha $ corresponds to encoding the set A into the continuum function in the interval $(\aleph _\mu , \aleph _{\mu +\mu })$ , and the second option corresponds to encoding an element of the sequence $\kappa _\xi $ .)

We now define an iteration $\langle {\mathbb {P}}_\alpha :\alpha <\lambda \rangle $ with the property that ${\mathbb {P}}_\alpha $ does not change beth fixed points $\beta =\beth _\beta $ for any $\beta $ . We let ${\mathbb {P}}=\langle {\mathbb {P}}_\alpha :\alpha <\lambda \rangle $ be the following iteration: If $\alpha $ is a limit ordinal, we use direct limits for regular $\alpha $ and inverse limits for singular $\alpha $ . Suppose then $\alpha =\beta +1$ . Let $\dot {A}$ be the W-first ${\mathbb {P}}_\beta $ -name $\dot {A}$ in $V_\kappa $ such that ${\mathbb {P}}_\beta \Vdash \dot {A}\sim V_{\check \kappa _\beta }$ . Then ${\mathbb {P}}_\alpha ={\mathbb {P}}_\beta \star E(\check {\kappa _\beta },\dot {A})$ . Let G be ${\mathbb {P}}$ -generic over V and $G_\alpha =G\cap {\mathbb {P}}_\alpha $ .

In the forcing extension $V[G]$ , the sequence $\langle \kappa _\xi : \xi < \lambda \rangle $ satisfies the definition

$$\begin{align*}V[G] \models \left\lbrace {\kappa_\xi : \xi < \lambda} \right\rbrace = \left\lbrace {\mu < \kappa : \mu \text{ is an aleph-fixpoint and } 2^{\aleph_{\mu+2}} = \aleph_{\mu+4}} \right\rbrace. \end{align*}$$

Also, for every $\xi <\lambda $ there is a set $A\subseteq \kappa _\xi $ which codes a bijection $f_A \colon \kappa _\xi \to (V_{\kappa _\xi })^{V[G]}$ . The set A itself satisfies

$$\begin{align*}V[G]\models A=\{\alpha<\kappa_\xi: 2^{\aleph_{\kappa_\xi+\omega\cdot\alpha+1}}=\aleph_{\kappa_\xi+\omega\cdot\alpha+3}\}\text{:} \end{align*}$$

we remark that since $\kappa _\xi $ and $\kappa _{\xi +1}$ were chosen to be aleph-fixpoints, the encoding intervals $(\aleph _{\kappa _\xi }, \aleph _{\kappa _\xi +\kappa _\xi })$ and $(\aleph _{\kappa _{\xi +1}}, \aleph _{\kappa _{\xi +1}+\kappa _{\xi +1}})$ are disjoint. From A we can read off $f_A$ as the transitive collapse of $(\kappa _\xi , \left \lbrace {(\alpha ,\beta ) : \pi (\alpha ,\beta ) \in A} \right \rbrace )$ , and a well-order $<_\xi ^*$ of $(V_{\kappa _\xi })^{V[G]}$ :

$$\begin{align*}V[G]\models x_0 <_\xi^* x_1 \iff \alpha_0 < \alpha_1 < \kappa_\xi, \end{align*}$$

where $\alpha _i = \min \left \lbrace {\alpha : f_A(\alpha ) = x_i} \right \rbrace $ for $i = 0,1$ .

Now working in $V[G]$ , fix a collection $\mathcal F \subseteq {\mathcal {P}}(\kappa )$ , and we set out to define a well-order not on the whole of $\mathcal F$ but a certain subset of it. Define a relation $\lhd $ on $\mathcal F$ by

$$\begin{align*}X \lhd Y \iff X \cap \kappa_\xi <^*_\xi Y \cap \kappa_\xi \text{ for all but boundedly many } \xi < \lambda. \end{align*}$$

As $\lambda $ is uncountable, $\lhd $ is well-founded, so the set

$$\begin{align*}\mathcal W = \left\lbrace {X \in \mathcal F : X \text{ is minimal in } \lhd} \right\rbrace \end{align*}$$

is nonempty, and if $X, Y \in \mathcal W$ with $X \neq Y$ , then both $X \cap \kappa _\xi <^*_\xi Y \cap \kappa _\xi $ and $Y \cap \kappa _\xi <^*_\xi X \cap \kappa _\xi $ occur for unboundedly many $\xi < \lambda $ .

We claim that $ \left \lvert {\mathcal W} \right \rvert < \kappa $ . To see this, suppose to the contrary that $ \left \lvert {\mathcal W} \right \rvert \geq \kappa $ and define a coloring $c \colon [\mathcal W]^2 \to \lambda $ by $c( \left \lbrace {X,Y} \right \rbrace ) = \pi (\xi _1, \xi _2)$ where $\xi _1$ is the least $\xi < \lambda $ such that $X \cap \kappa _\xi <^*_\xi Y \cap \kappa _\xi $ , and $\xi _2$ is the least $\xi < \lambda $ such that $Y \cap \kappa _\xi <^*_\xi X \cap \kappa _\xi $ , and $\xi _1 < \xi _2$ . Since $ \left \lvert {\mathcal W} \right \rvert \geq \kappa> (2^\lambda )^+$ , by the Erdős-Rado theorem there is a set $H \subseteq \mathcal W$ homogeneous for c of color $\pi (\xi _1, \xi _2)$ and cardinality $\lambda ^+$ . But this is a contradiction, since ordering H in $<^*_{\xi _1}$ -increasing order yields an infinite decreasing sequence in the well-order $<^*_{\xi _2}$ , so $ \left \lvert {\mathcal W} \right \rvert < \kappa $ .

Now for each $X \in \mathcal W$ , define $g_X \colon \lambda \to \kappa $ such that $g_X(\xi )$ is the index of $X \cap \kappa _\xi $ in the well-order $<^*_\xi $ . Then the set $\bigcup \left \lbrace {\operatorname {\mathrm {ran}}(g_X) : X \in \mathcal W} \right \rbrace $ has some order-type $\gamma < \kappa $ (as its cardinal is also smaller than $\kappa $ ), and we can let $h \colon \bigcup \left \lbrace {\operatorname {\mathrm {ran}}(g_X) : X \in \mathcal W} \right \rbrace \to \gamma $ be the transitive collapse map.

Then for $X \in \mathcal W$ , the function $h \circ g_X \colon \lambda \to \gamma $ is an element of a large enough $V_{\kappa _\xi }^{V[G]}$ , and obviously $h \circ g_X \neq h \circ g_Y$ if $X \neq Y$ , so we can well-order $\mathcal W$ by

$$\begin{align*}X_0 \prec X_1 \iff \zeta_0 < \zeta_1, \text{ or } \zeta_0 = \zeta_1 \text{ and } \alpha_0 < \alpha_1, \end{align*}$$

where $\zeta _i = \min \left \lbrace {\zeta < \lambda : h \circ g_{X_i} \in V_{\kappa _\zeta }^{V[G]}} \right \rbrace $ , $\alpha _i = \min \left \lbrace {\alpha : f_{A_i}(\alpha ) = h \circ g_{X_i}} \right \rbrace $ and $A_i \subseteq \kappa _{\zeta _i}$ is the set encoding a well-order, for $i = 0,1$ . All this is second-order definable in $V[G]$ in any structure of size $\kappa $ , if the collection $\mathcal F$ is. This allows us to pick a distinguished element of $\mathcal F$ as the $\prec $ -least $\lhd $ -minimal element.

Suppose now that $\phi $ is a complete second-order sentence with a model M of cardinality $\kappa $ , and let $\mathcal F$ consist of the set of $X \subseteq \kappa $ encoding a model of $\phi $ . Note that over a model of cardinality $\kappa $ we can write a formula $\phi _\lhd (X,Y)$ expressing $X \lhd Y$ for $X, Y \in \mathcal F$ , a formula $\phi _{\mathcal W}(X)$ expressing $X \in \mathcal W$ , and a formula $\phi _\prec (X,Y)$ expressing $X \prec Y$ if X and Y are $\lhd $ -minimal.

Let $M \models \Phi $ now say that $X \subseteq M$ encodes a model isomorphic to M (and thus satisfies $\phi $ ), and for any $Y \subseteq M$ that also encodes a model of $\phi $ , $\lnot \phi _\lhd (Y,X)$ , and moreover if for all $Z \subseteq M$ that encode a model of $\phi $ also $\lnot \phi _\lhd (Z,Y)$ , then $X = Y$ or $\phi _\prec (X,Y)$ . That is, $X \in \mathcal W$ and if also $Y \in \mathcal W$ then $X = Y$ or $X \prec Y$ , which uniquely specifies X. As the model of $\phi $ with the least code in this sense satisfies $\Phi $ and $\phi $ is complete, $\phi $ implies $\Phi $ and thus that all models of $\phi $ are isomorphic, so $\phi $ is categorical.

Proof. Let T be a recursively axiomatized second-order theory with only countably many non-isomorphic countable models. Let A be the $\Pi ^1_\omega $ (i.e., an intersection of a recursively coded family of sets each of which is $\Pi ^1_n$ for some n) set of reals that code a model of T. Since A is a countable union of equivalence classes of the $\Sigma ^1_1$ -equivalence relation of isomorphism, we may conclude that A is $\mathbf {\Sigma }^1_1$ .

We wish to show that A is $\Pi ^1_2(r_0)$ in a parameter $r_0$ which is a $\Pi ^1_\omega $ singleton. For this, we mimic a proof of Louveau ([Reference Louveau and Stern19, Theorem 1]) to show:

Proof. Let A be a $\mathbf {\Sigma }^1_1$ set that is also $\Pi ^1_\omega $ , say $A = \bigcap _n A_n$ with each $A_n$ being $\Pi ^1_n$ . Let also $U \subseteq (\omega ^\omega )^2$ be a universal $\Sigma ^1_1$ set.

We define for each n a game $G_n$ on $\omega $ where players I and II take turns to play the digits of reals $\alpha $ and $\gamma $ respectively (there is no need to let II pass turns here). Then II wins a play of $G_n$ if $\alpha \in A \implies \gamma \in U$ and $\alpha \notin A_n \implies \gamma \notin U$ .

$$\begin{align*}\begin{array}{c|ccccc} \text{I} & n_0 & & n_1 & & \cdots \\ \hline \text{II} & & m_0 & & m_1 & \cdots \end{array} \quad \begin{matrix} \alpha \\ \gamma \end{matrix} \end{align*}$$

As in Louveau’s proof, II has a winning strategy as follows: since A is $\mathbf {\Sigma }^1_1$ , we have $A(x) \iff U(y,x)$ for some y, so II wins by playing the digits of $\langle y, \alpha \rangle $ (as I is playing the digits of $\alpha $ ). The complexity of the winning set for II in $G_n$ is $\Sigma ^1_\omega $ , so by Moschovakis’s strategic basis theorem ([Reference Moschovakis24, Theorem 6E.2]), II has a winning strategy $\sigma _n$ that is a $\Delta ^1_{\omega +1}$ -singleton. Note that the pointclass $\Sigma ^1_\omega $ , i.e., the collection of countable unions of recursively coded families of projective sets, is both adequate and scaled (see [Reference Rudominer27, Remark 2.2], essentially [Reference Steel30, Theorem 2.1]).

Then the set $B_n = \left \lbrace {y : (y * \sigma _n)_{\text {II}} \in U} \right \rbrace $ is a $\Sigma ^1_1(\sigma _n)$ set with $A \subseteq B_n \subseteq A_n$ (where $(y * \sigma _n)_{\text {II}}$ denotes the real $\gamma $ the strategy $\sigma _n$ produces as I plays $\alpha = y$ ), so altogether $A = \bigcap _n B_n$ is a $\Pi ^1_2(s_0)$ set where $s_0 = \langle \sigma _n : n < \omega \rangle $ is a $\Delta ^1_{\omega +1}$ -singleton.

We may reduce the complexity of the parameter down to being a $\Pi ^1_\omega $ singleton by the following theorem of Rudominer.

Therefore the set A is a $\Pi ^1_2(r_0)$ set where $r_0$ is a $\Pi ^1_\omega $ singleton. Let $\eta (r,s)$ be a second-order $\Pi ^1_2$ formula which defines the predicate $s\in A$ on $({\mathbb {N}},+,\times ,r_0)$ . Let $\theta _1(Q_+,Q_{\times })$ be the standard second-order characterization of $({\mathbb {N}},+,\times )$ , as above in the proof of Theorem 4. Let $\psi _n(Q_+,Q_{\times },s)$ , $n<\omega $ , be second-order formulas such that if $X_n$ is the set of reals s satisfying $\psi _n(Q_+,Q_{\times },s)$ in $({\mathbb {N}},+,\times )$ , then $\{r_0\}=\bigcap _nX_n$ . Let P be a new unary predicate symbol and

$$ \begin{align*}S=\{\theta_1(Q_+,Q_{\times})\}\cup\{\psi_n(Q_+,Q_{\times},P) : n<\omega\}.\end{align*} $$

Suppose M is a model of S. Without loss of generality, the arithmetic part of M consists of the standard $+$ and $\times $ on ${\mathbb {N}}$ . Let s be the interpretation of P in M. Then $s=r_0$ . Thus S is categorical. The theory S is recursive because the proofs of Theorems 21 and 22 are sufficiently uniform. In conclusion, M is categorically characterized by the recursive second-order theory S.

Now the countable models of T are interpretable in S in the following sense: a real s codes a model of T if and only if $M\models \eta (r_0,s)$ . We also get a translation of sentences: if $\phi $ is a second-order sentence in the vocabulary of T, letting $\hat \phi $ be the sentence $\exists X (\eta (r_0,X) \land X \models \phi )$ , we have that $\phi \in T$ if and only if $\hat \phi \in S$ .

Proof. Let $\alpha = \omega _2^{\text {HOD}}$ . Let $T(A_\alpha )$ be the theory constructed above. Finally, let M be a countable atomic model of $T(A_\alpha )$ . Since $\mathop {\text {HOD}}$ satisfies $ \left \lvert {T(A_\alpha )} \right \rvert = \omega _2$ , the theory $T(A_\alpha )$ has no atomic model in $\mathop {\text {HOD}}$ , but as being an atomic model is absolute, this shows that there is no model in $\mathop {\text {HOD}}$ isomorphic to M.

The isomorphism class of M is ordinal definable as the class of countable atomic models of $T(A_\alpha )$ , which is definable from $\alpha $ . Additionally, if $\alpha $ is second-order definable in the countably infinite structure of the empty vocabulary, we can define the theories T and $T(A_\alpha )$ in second-order logic expressing “I am isomorphic to a countable atomic model of $T(A_\alpha )$ ” with a single second-order sentence. This finitely axiomatizes the second-order theory of M.

Proof. We can use [Reference Koellner, Woodin, Foreman and Kanamori15, Theorem 3.10, Chapter 23]) to define $\mathop {\text {HOD}}^{L({\mathbb {R}})} \restriction \Theta ^{L({\mathbb {R}})}$ and $L_{\Theta ^{L({\mathbb {R}})}}({\mathbb {R}})$ from $\Theta ^{L({\mathbb {R}})}$ in second-order logic, which then allows us to define $\omega _2^{\scriptsize \mathop {\text {HOD}}}$ and M as in Theorem 25.

Open Problem 1. Assuming $V=L$ , is every recursively axiomatized complete second-order theory categorical?

Open Problem 2. Suppose $V = L[U]$ , $\kappa $ is the sole measurable cardinal of $L[U]$ , and T is a complete recursively axiomatized second-order theory that has a model of cardinality $\lambda < \kappa $ such that $\lambda $ is second-order characterizable. Is T categorical?

Open Problem 3. Assume $(*)$ . Is there a complete recursively axiomatized theory with exactly $\omega $ many models of cardinality $\aleph _1$ ?

Open Problem 4. Can we always force the categoricity of all finitely axiomatizable complete second-order theories with a model of cardinality $\kappa $ , where $\kappa $ is either a regular (non-measurable) limit cardinal, or singular of cofinality $\omega $ ?

Open Problem 5. Is every finitely axiomatizable complete second-order theory with a model of cardinality an $I_0$ -cardinal categorical (or, at least categorical among all models of that cardinality)?