The Second-order Version of Morley’s Theorem on the Number of Countable Models does not Require Large Cardinals

We induct on the complexity of the formulas. Observe that the homogeneity of Cohen forcing implies that if for some $p$, $p\Vdash_{Add(\omega,\kappa)}H(\omega_{1})\models_{\mathcal{L}_{\omega_{1},\omega}}\varphi(\bar{a})$, then $\Vdash_{Add(\omega,\kappa)}H(\omega_{1})\models_{\mathcal{L}_{\omega_{1},\omega}}\varphi(\bar{a})$. In particular, this means if we have proved the result for $\mathbf{\Sigma}^{1}_{\xi}$ formulas, then we get the result for $\mathbf{\Pi}^{1}_{\xi}$ formulas immediately.

Suppose we have proved the theorem for all formulas that are $\mathbf{\Sigma}^{1}_{\nu}$ for $\nu<\xi<\omega_{1}$ and $\xi$ is a limit, then $\varphi(\bar{x})$ is of the form $\bigvee_{n}\{\psi_{n}(\bar{x}):\psi_{n}\in\mathbf{\Sigma}^{1}_{\xi_{n}}\}$ for a sequence $\langle\xi_{n}<\xi:n\in\omega\rangle$. Let $G\times H\subseteq Add(\omega,\omega_{1})\times Add(\omega,\lambda)$. In $V[G\times H]$, suppose $H(\omega_{1})\models_{\mathcal{L}_{\omega_{1},\omega}}\varphi(\bar{a})$, equivalently, there is an $n\in\omega$, $H(\omega_{1})\models_{\mathcal{L}_{\omega_{1},\omega}}\psi_{n}(\bar{a})$. By the induction hypothesis, we know that $\Vdash_{Add(\omega,\kappa)}H(\omega_{1})\models_{\mathcal{L}_{\omega_{1},\omega}}\psi_{n}(\bar{a})$. As a result, $\Vdash_{Add(\omega,\kappa)}H(\omega_{1})\models_{\mathcal{L}_{\omega_{1},\omega}}\varphi(\bar{a})$. A similar argument shows that if $H(\omega_{1})\models_{\mathcal{L}_{\omega_{1},\omega}}\neg\varphi(\bar{a})$, then $\Vdash_{Add(\omega,\kappa)}H(\omega_{1})\models_{\mathcal{L}_{\omega_{1},\omega}}\neg\varphi(\bar{a})$.

Suppose we have proved the theorem for $\mathbf{\Sigma}_{\xi}^{1}$-formulas for $\xi<\omega_{1}$. Let $\exists x\psi(x,\bar{y})$ be a $\mathbf{\Sigma}_{\xi+1}^{1}$-formula. Suppose

where $\bar{a}\in V[G\times H]$. Given $(p,q)\in P=_{def}Add(\omega,\omega_{1})\times Add(\omega,\lambda)$ and $\dot{x}$ a $P\times Add(\omega,\kappa)$-name and $\dot{\bar{a}},\dot{\varphi}$ $P$-names such that $(p,q)$ forces the above holds for the respective names, find $\alpha<\omega_{1},A\subseteq\lambda,B\subseteq\kappa$ countable such that these conditions and names are in $Add(\omega,\alpha)\times Add(\omega,A)\times Add(\omega,B)$ or are $Add(\omega,\alpha)\times Add(\omega,A)\times Add(\omega,B)$-names.

Define an automorphism $\pi$ on $P\times Add(\omega,\kappa)$ as follows: $\pi(r,s,t)=(r^{*},s^{*},t^{*})$ if and only if

1. (1)

   $r\restriction\omega_{1}-[\alpha,\alpha+otp(B))=r^{*}\restriction\omega_{1}-[\alpha,\alpha+otp(B))$, $s=s^{*}$,

2. (2)

   $r^{*}(\alpha+i)=t(i)$, $t^{*}(i)=r(\alpha+i)$ for all $i\in B$,

3. (3)

   $t^{*}\restriction\kappa-B=t\restriction\kappa-B$.

In particular, $\pi$ fixes $(p,q)$, $\dot{\bar{a}}$ and $\dot{\varphi}$. Therefore, $(p,q,\emptyset)\Vdash_{P\times Add(\omega,\kappa)}\psi(\pi(\dot{x}),\dot{\bar{a}})$.

Let $\dot{\sigma}$ be an $Add(\omega,1)$-name such that $\Vdash_{Add(\omega,1)}``\neg\dot{\sigma}Er$ for any $r\in V$”.

Let $V^{*}=V[G]$ where $G\subseteq Add(\omega,\lambda)$ is generic over $V$. Work in $V^{*}$.

The proof is similar to that of Lemma 3.1 in [5], adapted to the “absolute” scenario. Therefore, we will only sketch the proof. The second-order theory $T$ considered here is second-order Peano Arithmetic with an additional unary predicate $R$. Let $\mathcal{L}$ be the natural language for this. Intuitively speaking, this predicate is coding a real that belongs to $A$. Namely, our model will be of the form of $\mathcal{A}=(\omega,+,\cdot,<,0,1,R^{\mathcal{A}})$ with $R^{\mathcal{A}}\in A$. There is a natural translation of a projective formula $\psi$ to a second-order $\mathcal{L}$-formula $\psi^{\mathcal{L}}$ that is truth-preserving (see [11, 8B.15] for more information). In particular, if $\varphi(x)$ is a projective definition of $A$, then the requirement that $R^{\mathcal{A}}\in A$ can be expressed as $\mathcal{A}\models``\exists Y((\forall n(n\in Y\leftrightarrow n\in R))\wedge\varphi^{\mathcal{L}}(Y))$”. As a result, the above procedure describes a second-order theory $T$ in the language $\mathcal{L}$ such that any $\mathcal{A}\models T$, $R^{\mathcal{A}}\in A$. Note that for any $\mathcal{A}_{0},\mathcal{A}_{1}\in Mod_{T}$ (namely, the collection of models for the theory $T$), $\mathcal{A}_{0}\cong\mathcal{A}_{1}$ if and only if $R^{\mathcal{A}_{0}}=R^{\mathcal{A}_{1}}$. Also the natural product topology on $Mod_{T}$ is generated by $U_{\sigma}=\{\mathcal{A}\in Mod_{T}:\sigma\sqsubset R^{\mathcal{A}}\}$ for $\sigma\in 2^{<\omega}$.

Assume that $|A|>\aleph_{1}$. Apply Second-order Absolute Morley. By the assumption that $|A|>\aleph_{1}$, $Mod_{T}$ must contain more than $\aleph_{1}$ many non-isomorphic models. In particular, there is a continuous injection $\pi^{\prime}:2^{\omega}\to Mod_{T}$ such that the images of $\pi^{\prime}$ are pairwise non-isomorphic. Let $\pi:2^{\omega}\to A$ be defined on $2^{\omega}$ such that $\pi(\sigma)=R^{\pi^{\prime}(\sigma)}$. We check that $\pi$ is a continuous injection. The fact that $\pi$ is an injection follows from the fact that for $\sigma_{0}\neq\sigma_{1}$, $\pi^{\prime}(\sigma_{0})\not\simeq\pi^{\prime}(\sigma_{1})$, which implies $R^{\pi^{\prime}(\sigma_{0})}\neq R^{\pi^{\prime}(\sigma_{1})}$. For any basic open set $[\sigma]_{A}=_{def}\{g\in A:\sigma\sqsubset g\}$ where $\sigma\in 2^{<\omega}$ and $f\in\pi^{-1}([\sigma]_{A})$, since $\pi^{\prime}$ is continuous and $f\in(\pi^{\prime})^{-1}(U_{\sigma})$, there is some $\tau\in 2^{<\omega}$ such that $f\in[\tau]=_{def}\{g\in 2^{\omega}:\tau\sqsubset g\}\subset(\tau^{\prime})^{-1}(U_{\sigma})$. For any $h\in[\tau]$, $\pi(h)=R^{\pi^{\prime}(f)}\sqsupset\sigma$. As a result, $f\in[\tau]\subset\pi^{-1}([\sigma]_{A})$. Hence $\pi$ is continuous.

It is well-known that it is consistent that there is a light-face projective well order of the reals and $2^{\aleph_{0}}=\aleph_{2}$ (see [4]). We use a projective well order of the reals to produce a projective set witnessing the failure of Lemma 1.11. More precisely, we will construct a projective Bernstein set $B$, i.e. such that neither $B$ nor $\mathbb{R}\setminus B$ includes a perfect set. In detail, let $\preceq$ be the projective well order. Recursively coding pairs of reals as reals, $\preceq$ induces a projective well order on pairs of reals that we will also call $\preceq$. For each real, we can ask if it codes a perfect subtree of $2^{<\omega}$ by some recursive bijection between $\omega$ and $2^{<\omega}$. Define $f:\mathbb{R}\to\mathbb{R}^{2}$ such that $f(x)=\langle y_{x},z_{x}\rangle$ if:

1. 1)

   for any $x^{\prime}\prec x$, $y_{x}$ and $z_{x}$ are not equal to either $y_{x^{\prime}}$ or $z_{x^{\prime}}$,

2. 2)

   if $x$ codes a perfect subtree $T_{x}$ of $2^{<\omega}$, then $y_{x}$ and $z_{x}$ are different branches through $T_{x}$, the tree determined by $x$,

3. 3)

   $\langle y_{x},z_{x}\rangle$ is the $\preceq$-least pair satisfying 1) and 2).

Since $\preceq$ is projective, so is $f$. Let

Then $y=y_{x}\in B$ implies $z_{x}\notin B$.