Proof. $(2) \implies (3)$ is clear, as the witnessing models for nearly supercompactness can be of minimal size. The implication $(3) \implies (2)$ follows from the fact that for $\theta = |H(\lambda )|$ , $\theta ^{<\kappa } = \theta $ . Then, using Hauser’s trick [Reference Hauser11], one can obtain an elementary embedding that respects a bijection between M and $\theta $ . Namely, let M be as in (2) and let $\bar {M}$ be a model with the same universe as M and an additional function symbol f which we interpret as a bijection between $\theta $ and M. Note that M satisfies the assertion that for every x of size $<\kappa $ , there is y, such that $y = f \operatorname{\textrm{"}} x$ . Apply the elementary embedding and using the nearly supercompactness hypothesis, we obtain a model $\bar {N}$ with a function symbol that we denote by $j(f)$ , and moreover $j \operatorname{\textrm{"}} M = j(f)\operatorname{\textrm{"}} (j\operatorname{\textrm{"}} \theta ) \in \bar {N}$ . Reducing the language by removing the function symbol of j, we obtain the result.

Let us show that (1) implies (2). Let $\kappa $ be $\lambda $ - $\Pi ^1_1$ -subcompact and let M witness that (2) is false. This means that M is transitive, $M^{<\kappa } \subseteq M$ , $|M| = |H(\lambda )|$ , and for every transitive model N and embedding $j \colon M \to N$ , either j is not elementary or $j\operatorname{\textrm{"}} M\notin N$ . Since we may assume that the model N has size $|H(\lambda )|$ (by taking an elementary substructure), this statement can be coded as a $\Pi ^1_1$ -statement on $H(\lambda )$ , using some predicate A in order to code the model M and its elementary diagram.

Applying (1), and Definition 2.6, we obtain cardinals $\bar {\kappa }$ and $\bar {\lambda }$ below $\kappa $ and a predicate $\bar {A}$ on $H(\bar {\lambda })$ that codes some transitive model $\bar {M}$ . We also obtain an elementary embedding $\tilde {j} \colon \langle H(\lambda ), \in \bar {A}\rangle \to \langle H(\lambda ), \in , A\rangle $ . By unwrapping the code A for M, we conclude that $\tilde {j}$ codes an elementary embedding $j \colon \bar {M} \to M$ . Recall that $\kappa $ is strongly inaccessible and $\bar \lambda < \kappa $ , so $|H(\bar \lambda )| = |\bar {M}| < \kappa $ . Since M is closed under sequences of size $<\kappa $ , $j\operatorname{\textrm{"}} \bar {M} \in M$ . Let us take an elementary substructure of M that contains $j\operatorname{\textrm{"}} \bar {M},\, \{j\operatorname{\textrm{"}} \bar {M}\}$ as elements of size $|H(\bar {\lambda })|$ and closed under $<\bar \kappa $ -sequences. The transitive collapse of this model is coded by some subset of $H(\bar {\lambda })$ witnessing that the above $\Pi ^1_1$ -statement fails in $H(\bar {\lambda })$ .

Let us prove that (2) implies (1). Let us assume that (2) holds, and let us show that $\kappa $ is $\lambda $ - $\Pi ^1_1$ -subcompact. Let $\Phi $ be a $\Pi ^1_1$ -statement that holds in the model $\langle H(\lambda ), \in , A\rangle $ . Applying the hypothesis, there is an elementary embedding with critical point $\kappa $ between some transitive model $M\supseteq H(\lambda ) \cup \{A, H(\lambda )\}$ and a transitive model N such that $j\operatorname{\textrm{"}} M \in N$ and $\lambda < j(\kappa )$ .

Since N is closed under basic operations on sets, $j \operatorname{\textrm{"}} H(\lambda ) = j(H(\lambda )) \cap j\operatorname{\textrm{"}} M \in N$ . By taking the transitive collapse of $j \operatorname{\textrm{"}} H(\lambda )$ inside N, we conclude that $H(\lambda ), A \in N$ .

Working in N, the following hold:

$$\begin{align*}N \models \text{"}\langle H(\lambda), \in, A\rangle \models \Phi\text{"}.\end{align*}$$

Let us note that the model N does not contain all subsets of $H(\lambda )$ , which means that in general the truth value of second-order formulas would not be absolute between N and V. So, the validity of the formula $\Phi ^N$ uses the fact that $\Phi $ is a $\Pi ^1_1$ -statement. In N, there is an elementary embedding $k = j\restriction H(\lambda )$ from the structure $\langle H(\lambda ), \in ,A\rangle $ to $\langle j(H(\lambda )), \in , j(A)\rangle $ with critical point $\kappa $ and $k(\kappa ) = j(\kappa )> \lambda $ . Thus, by elementarity of j, the same holds in M: there are $\bar {\kappa }, \bar {\lambda } < \kappa $ , $\bar {A} \subseteq H(\bar {\lambda })$ , and an elementary embedding $\bar {k} \colon \langle H(\bar \lambda ), \in , \bar {A}\rangle \to \langle H(\lambda ), \in , A\rangle $ . Moreover, we have $\langle H(\bar \lambda ), \in , \bar {A}\rangle \models \Phi $ .⊣

Proof. Let $j\colon M \to N$ be as in Lemma 2.8. Since $\lambda \in M$ , $j\operatorname{\textrm{"}} \lambda \in N$ .

Let us use the seed hull, as in [Reference Hamkins9],

$$\begin{align*}\mathbb{X}_{\{j\operatorname{\textrm{"}} \lambda\}} := \{j(f)(j\operatorname{\textrm{"}} \lambda) \mid f \in M,\, f \colon P_\kappa \lambda \to M\}.\end{align*}$$

Hamkins proves that $\mathbb {X}_{s} \prec N$ for every set of seeds s, and in particular $j \colon M \to \mathbb {X}_s$ is elementary. Hamkins’ proof is done in the context of an elementary embedding from V to some class, but it goes without change to our case, under the assumption that M has definable Skolem functions.

Let $\pi \colon \mathbb {X}_{\{j\operatorname{\textrm{"}} \lambda \}} \to N'$ be the transitive collapse, so $k = \pi \circ j \colon M \to N'$ is an elementary embedding. In order to show that $\operatorname {\textrm{crit}} k = \kappa $ , let us notice that all ordinals up to $\lambda $ belong to $\mathbb {X}_{\{j\operatorname{\textrm{"}} \lambda \}}$ , so $\operatorname {\textrm{crit}} \pi ^{-1} \geq \lambda $ . The main point is to verify that $N'$ is closed under $<\lambda $ -sequences. Let $\{ y_i \mid i < i_\star \} \subseteq N'$ , $i_\star < \lambda $ . Then, by the definition of $N'$ , for each i, there is a function $f_i \in M$ such that $y_i = \pi (j(f_i)(j\operatorname{\textrm{"}} \lambda ))$ . By the closure of M, $\vec {f} = \langle f_i \mid i < i_\star \rangle \in M$ .

Let us look at $k(\vec {f}) \in N'$ . Since k is elementary $N'$ satisfies enough set theory. Since $k \operatorname{\textrm{"}} \lambda = \pi (j\operatorname{\textrm{"}} \lambda ) \in N'$ , we conclude that

$$\begin{align*}k(\vec{f}) \operatorname{\textrm{"}} (k\operatorname{\textrm{"}} \lambda \cap k(i_\star)) = \{k(f_i) \mid i < i_\star\} \in N'.\end{align*}$$

Therefore, $A = \{k(f_i)(k\operatorname{\textrm{"}} \lambda ) \mid i < i_\star \} \in N'$ . Applying $\pi ^{-1}$ and using the fact that $\operatorname {\textrm{crit}} \pi ^{-1} \geq \lambda $ , we get

$$\begin{align*}\pi^{-1}(A) = \{j(f_i)(j\operatorname{\textrm{"}} \lambda) \mid i < i_\star\}.\end{align*}$$

Applying $\pi $ again the result follows.⊣

Proof. Clearly $(2)\implies (3)$ .

Let us show that $(1)\implies (2)$ . Let $T, p$ be as in the assumptions of $(2)$ . Let C be a club in $P_\kappa (T \cup p)$ such that for every $T' \cup p' \in C$ there is a model for $T'$ that omits $p'$ . Let us apply Lemma 2.8 for some $<\kappa $ -closed transitive model $M \prec H(\chi )$ for $\chi $ sufficiently large, $|M| = \lambda $ , and $T, p, C \in M$ . By applying Menas’ lemma in M, we obtain a function $F \in M$ , $F\colon T \cup p \to P_{\kappa }(T \cup p)$ such that $C_F \subseteq C$ . Using the hypothesis, we obtain an elementary embedding $j \colon M \to N$ where N is transitive and $j\operatorname{\textrm{"}} M \in N$ . Thus, $j(T) \cap j\operatorname{\textrm{"}} M = j\operatorname{\textrm{"}} T,\ j(p) \cap j\operatorname{\textrm{"}} M = j\operatorname{\textrm{"}} p,$ and $ j(C) \cap j\operatorname{\textrm{"}} M = j \operatorname{\textrm{"}} C$ are in N. Since $X = (j\operatorname{\textrm{"}} T) \cup (j\operatorname{\textrm{"}} p) = \bigcup j\operatorname{\textrm{"}} C \in N$ , $|C| = \lambda < j(\kappa )$ , and X is closed under $j(F)$ , we conclude that $(j\operatorname{\textrm{"}} T) \cup (j\operatorname{\textrm{"}} p) \in j(C_F) \subseteq j(C)$ .

So, in N there is a model $\mathcal {A}$ for the theory $j\operatorname{\textrm{"}} T$ that omits the type $j\operatorname{\textrm{"}} p$ . Although the language for the theory and the type is the value under j of the original language and might contain more symbols, the symbols that appear in $j\operatorname{\textrm{"}} T$ and $j\operatorname{\textrm{"}} p$ are only the j-images of the original symbols. Therefore, by applying $j^{-1}$ on those symbols we conclude that $\mathcal {A}$ is isomorphic to a model for T that omits p.

Let us now consider $(3)\implies (1)$ . Let M be a transitive model of size $\lambda $ which is closed under $<\kappa $ -sequences. We would like to find an elementary embedding with critical point $\kappa $ and a model N such that $j\operatorname{\textrm{"}} M \in N$ and $\lambda < j(\kappa )$ . Similarly to the proof of Lemma 2.3, we define a language that contains for every $x\in M$ a constant $c_x$ as well as two additional constants $d, s$ . We intend d to be the critical point $\kappa $ and s to be the set $j \operatorname{\textrm{"}} M$ .

The theory T contains the statement “d is an ordinal below $c_\kappa $ ,” and the statement “ $c_\alpha \in d$ ” for all $\alpha < \kappa $ . We also include in T the assertions “ $c_x \in s$ ” for all $x\in M$ and “ $|s| < c_\kappa $ ” (namely, that there is an injection from s to a bounded ordinal below $c_\kappa $ ). Finally, we include in T the full $\mathcal {L}_{\omega ,\omega }$ -elementary diagram of M.

We would also like to define a type that will be omitted. There are two offending objects that we would like to omit from our model: either witnesses for $s \neq j\operatorname{\textrm{"}} M$ or critical points below $\kappa $ . The type p is going to handle both cases. $p(x)$ is going to be the type of an element which is either in s but not $c_z$ for any $z\in M$ , or below d but not in s. Namely,

$$\begin{align*}p(x) = \{\text{"} x\in s \cup d \text{"}\} \cup \{\text{"} x \neq c_z\text{"} \mid z \in M\}.\end{align*}$$

We would like to show that indeed on a club in $P_{\kappa } (T \cup p)$ , there is a model for the sub-theory that omits the sub-type. Let $\theta $ be a sufficiently large regular cardinal, $M, T, p\in H(\theta )$ .

Let us fix a well order of $H(\theta )$ , $\leq _\theta $ . Pick some enumeration e of T and p and let $C'$ be the club of all elementary substructures $Y \prec \langle H(\theta ), \in , \leq _\theta \rangle $ that contains $M, T, p, e$ and satisfy $Y \cap \kappa \in \kappa $ . Let $\operatorname {\textrm{Hull}}$ denote the Skolem hull function in the structure $\langle H(\theta ), \in , \leq _\theta , M, T, p, e\rangle $ , defined using the well order $\leq _\theta $ . So $Y \in C'$ iff $Y = \operatorname {\textrm{Hull}}(A)$ for some A, $|A| < \kappa $ , and $Y \cap \kappa \in \kappa $ .

Let us assume moreover that the map $x \mapsto c_x$ is definable in $H(\theta )$ . Each such model Y would be closed under sub-formulas, so if a formula $\varphi $ in T or in p contains the constant $c_x$ as a sub-formula and $e(\varphi ) \in Y$ then $x \in Y$ . Thus, for any element $x \in M$ the constant $c_x$ appears as a sub-formula of a formula in $Y \cap (T \cup p)$ , if and only if x belongs to $Y \cap M$ .

Let $T' \cup p' \in C$ iff $T' \cup p' \in P_\kappa (T' \cup p')$ , $\operatorname {\textrm{Hull}}(T' \cup p') = Y \in C'$ , and $Y \cap (T \cup p) = T' \cup p'$ . One can easily describe a function $f \colon (T \cup p)^{<\omega } \to P_{\kappa }(T \cup p)$ such that C consists of all elements which are closed under f, so in particular C is a club.

Let us consider $T' \cup p'$ in C and let $X \prec M$ be a corresponding elementary submodel, $X = \operatorname {\textrm{Hull}}(T' \cup p') \cap M$ . We claim that the model M itself with the evaluations $c^M_a = a$ for every $a\in X$ , $d^M = X \cap \kappa $ , and $s^M = X$ realizes $T'$ while omitting $p'$ . First, $d^M$ is an ordinal below $c^M_\kappa $ . Moreover, by the closure assumption, if a constant $c_a$ appears in $T'$ then $a\in X$ . In particular, this model satisfies that whenever $c_a\in s$ appears in $T'$ then $a\in X$ . Since M is $<\kappa $ closed, there is some bijection between X and an ordinal below $\kappa $ in M. The other assertions in $T'$ follow similarly.

Let us consider $p'$ . If M does not omit $p'$ then there is some element $x \in M$ such that $x\in X \cup d^M$ , $x \neq c_z^M$ for every z such that the formula “ $x \neq c_z$ ” belongs to $p'$ . By the definition of $d^M$ , $d^M \subseteq X$ . By the closure of Y “ $x \neq c_z$ ” appears in $p'$ if and only if $z\in X$ , which is what we need.

Now, we may apply the hypothesis of the lemma and obtain a model N for T that omits p. As in the proof of Lemma 2.3, the embedding $j\colon M \to N$ which is defined by $j(z) = c_z^N$ is an elementary embedding with critical point $\kappa $ . By the type omission, $s^N = j \operatorname{\textrm{"}} M$ .

We are not done yet, since N might be ill founded.Footnote ²

Proof. Let M be a model, satisfying the hypothesis of the claim. Let $M' \prec H(\chi )$ ( $\chi $ large enough) be a larger model (so $M \subseteq M'$ ), closed under $<\kappa $ -sequences, such that $M \in M'$ , $\lambda + 1 \subseteq M'$ , and $|M'| = \lambda $ . Let $\bar {M}'$ be the transitive collapse of $M'$ . Since M is transitive, $M \in \bar {M}'$ .

Let $j \colon \bar {M}'\to N'$ be as in the hypothesis of the claim. Note that $j \restriction M \colon M \to j(M)$ is a member of $N'$ , as the intersection of $j \operatorname{\textrm{"}} \bar {M}'$ and $j(M)$ .

First, note that $j \operatorname{\textrm{"}} \lambda = j(\lambda ) \cap j\operatorname{\textrm{"}} \bar {M}'\in N'$ . Since $N'$ can compute the transitive collapse of $j\operatorname{\textrm{"}} \lambda $ , we conclude that $\textrm{Ord}^{N'} \supseteq \lambda + 1$ . Similarly, since ${}^{<\kappa }\lambda \subseteq \bar {M}'$ , we have ${}^{<\kappa }\lambda \in N'$ . Since $j \operatorname{\textrm{"}} M \in N'$ , we conclude that $j\operatorname{\textrm{"}} \lambda ^{<\kappa } \in N'$ as well.

Let $F \colon \lambda ^{<\omega } \to M$ be a function such that $F \restriction \lambda $ is a bijection between $\lambda $ and M, and F codes the Skolem functions of M, so for every $a \subseteq \lambda $ non-empty, $F \operatorname{\textrm{"}} [a^{<\omega }] \prec M$ .

Since M is closed under $<\kappa $ sequences and $|M|^{N'} = \lambda < j(\kappa )$ , we conclude that $j\operatorname{\textrm{"}} M \in j(M)$ and in particular there is some $\delta < j(\lambda )$ such $j(F)(\delta ) = j\operatorname{\textrm{"}} M$ . Let us consider the following subset of $N'$ :

$$\begin{align*}\tilde{N} = j(F) \operatorname{\textrm{"}} (j \operatorname{\textrm{"}} (\lambda^{<\kappa}) \cup \{\delta\}).\end{align*}$$

By the properties of F, $j\operatorname{\textrm{"}} M \subseteq \tilde {N}$ and $N' \models \tilde {N} \prec j(M)$ . Since $N' \in V$ , $V \models \tilde {N} \prec j(M)$ (see [Reference Hamkins and Yang10]).

Let us claim that $\tilde {N}$ is well founded. If not, then there is an $\omega $ -sequence of elements $a_n \in \tilde {N}$ such that $a_{n+1} \in ^{N'} a_n$ . Each $a_n$ is of the form $F(j(b_n), \delta )$ where $b \in \lambda ^{<\kappa }$ . Using the regularity of $\kappa $ , there is $c \in j\operatorname{\textrm{"}} \lambda ^{<\kappa } \subseteq N'$ such that c codes the $\omega $ -sequence $\langle j(b_n) \mid n < \omega \rangle $ . So, this sequence is a member of $N'$ . But this is absurd, as this would imply that the sequence $\langle a_n\mid n <\omega \rangle $ is a member of $N'$ , violating the fact that $N'\models $ Axiom of Foundation.

So, we conclude that $\tilde {N}$ is well founded and for every $x \in M$ , $j(x) \in \tilde {N}$ . Let $\pi \colon \tilde {N} \to N$ be the transitive collapse and let $k \colon M \to N$ be $k(x) = \pi (j(x))$ for $x \in M$ . It is straight-forward to verify that k is an elementary embedding, $\operatorname {\textrm{crit}} k = \kappa $ , and $k(\kappa )> \lambda $ . Moreover, $k \operatorname{\textrm{"}} M = \pi (j\operatorname{\textrm{"}} M) \in N$ .⊣

This concludes the proof of Theorem 3.3.⊣

Proof. $(2)\implies (3)$ is trivial.

Let us show that $(3) \implies (1)$ . Recall that $(1)$ is the statement: for every first-order theory T over a language of size $\mu $ and a type $p(x)$ , if for club many $T' \cup p' \in P_{\kappa }(T \cup p)$ there is a model M that satisfies $T'$ and omits $p'$ then there is a model of T that omits p.

Indeed, let us assume that T is a first-order theory over a relational language $\mathcal {L}$ with $\mu $ many symbols and p is a type. We will assume that T is Henkenized (for every sentence of the form $\psi := \exists x \varphi (x, r)$ , where r is a sequence of constants, there is a constant $c_{\varphi , r}$ such that c is a witness to the formula $\psi $ if and only if $\psi $ holds). So, finding a model for T that omits the type p is the same as finding a consistent complete extension, $\tilde {T}$ , in which for every constant c there is $\phi (x) \in p(x)$ such that $\neg \phi (c) \in \tilde {T}$ . Let us assume that there are club many $T'\cup p'$ such that there is a model M that realizes $T'$ and omits $p'$ . Let us construct a tree $\mathcal {T}$ as follows. Pick an enumeration e of length $\mu $ of all terms and formulas in the language $\mathcal {L}$ .

For every $a \in P_\kappa \mu $ , $\eta \in \mathcal {T}_a$ if and only if there is a model $M_\eta $ such that:

1. (1) for every $\alpha \in a$ , if $e(\alpha )$ is a sentence then $\eta (\alpha ) = 1 \iff M \models e(\alpha )$ and

2. (2) for every $e(\alpha ) \in T$ , $\eta (\alpha ) = 1$ .

Since we assumed that the language $\mathcal {L}$ is Henkenized, for every $a \subseteq b \in P_\kappa \mu $ , and $\eta \in \mathcal {T}_b$ , the function $\eta \restriction a$ defines a sub-model of the model $M_\eta $ , assuming that $e\operatorname{\textrm{"}} a$ is closed under sub-formulas and sub-terms. So, in this case, we say that $\eta \restriction a$ omits $p \cap e\operatorname{\textrm{"}} a$ if there is no constant $c \in e \operatorname{\textrm{"}} a$ such that for every $\varphi (x) \in p$ , $\eta (e^{-1}(\varphi (c))) = 1$ .

We are now ready to construct the ladder system L. Let $a \in P_\kappa \mu $ , such that the collection of formulas in $e \operatorname{\textrm{"}} a$ is closed under sub-formulas and apply substitution of a variable with a term in a formula. Let us define $\eta \in L \cap \mathcal {T}_a$ if there is a model $M_\eta $ that omits $e \operatorname{\textrm{"}} a \cap p$ . Note that if $a \cap \kappa $ is of uncountable cofinality, then there are club many $b \in P_{|a \cap \kappa |} a$ such that $\eta \restriction b$ is an assignment of a Henkenized theory, and $e\operatorname{\textrm{"}} b$ is closed under sub-formulas and omits $e \operatorname{\textrm{"}} b \cap p$ . Indeed, in order to omit the sub-types of p, one needs to verify that for every constant symbol c in $e \operatorname{\textrm{"}} b$ , there is a formula $\varphi \in p \cap e \operatorname{\textrm{"}} b$ such that $\varphi (c) \in e \operatorname{\textrm{"}} b$ and $\eta (e^{-1}(\varphi (c)) = 0$ . Since this is true for $\eta $ and a, we can define a function sending $c \in e \operatorname{\textrm{"}} a$ to $\varphi \in p \cap e \operatorname{\textrm{"}} a$ such that $\varphi (c) \in e \operatorname{\textrm{"}} a$ and $M_{\eta } \models \neg \varphi (c)$ . Thus, any $b \subseteq a$ which is closed under this function, omits the subtype $p \cap e\operatorname{\textrm{"}} b$ .

Let b be a cofinal branch though the tree $\mathcal T$ , and assume that b meets L cofinally. Since b is a cofinal branch, it defines a complete theory extending T and thus a model of T. Let us call this model $M_b$ . We want to verify that the type p is omitted. Indeed, let $z \in M_b$ . Let $x\in P_\kappa \lambda $ contain the ordinal in which the constant for z is enumerated. Let $y\supseteq x$ such that there is $\eta ' \in L \cap \ \mathcal {T}_y$ , $\eta ' = b \restriction y$ . Since $\eta '$ represents a model that omits a sub-type of p and contains the constant z, there must be a formula $\varphi \in p \cap e\operatorname{\textrm{"}} y$ such that $\beta = e(\neg \varphi (z)) \in \operatorname {\textrm{dom}} \eta '$ and $\eta '(\beta ) = 1$ . Thus, z does not realize p.

Let us finally show $(1) \implies (2)$ . Let M be a transitive model of size $\mu $ containing T and L, and closed under $<\kappa $ -sequences. By Theorem 3.3 and the hypothesis, there is an elementary embedding $j\colon M \to N$ , with critical point $\kappa $ , $j\operatorname{\textrm{"}} \mu \in N$ . Let $D \subseteq P_\kappa \mu $ be a club such that for all $x \in D$ , $L \cap \mathcal {T}_x \neq \emptyset $ and belongs to M. Then $j\operatorname{\textrm{"}} \mu \in j(D)$ and in particular, there is some $\tilde {\eta } \in j(\mathcal {T})_{j\operatorname{\textrm{"}} \mu }\cap j(L)$ .

Let b be the following branch:

$$\begin{align*}b(x) = j^{-1} \operatorname{\textrm{"}} (\tilde{\eta} \restriction j\operatorname{\textrm{"}} x) = \{(\zeta, \epsilon) \mid b(j(\zeta)) = \epsilon\}.\end{align*}$$

Buck in M, let $E_\eta $ be the club, as in Definition 4.2(3). Let us apply j to the function $\eta \mapsto E_\eta $ and let $\tilde {E}$ be the obtained club. So $\tilde {E}$ is a club on $P_\kappa j\operatorname{\textrm{"}}\mu $ (since $\kappa = |j\operatorname{\textrm{"}} \mu \cap j(\kappa )|$ ). For every $z \in P_\kappa j\operatorname{\textrm{"}}\mu $ , $z = j(w)$ for some $w \in P_\kappa \mu $ , so $D = j^{-1} \operatorname{\textrm{"}} (j(E)_{\tilde {\eta }})$ is a club in $P_\kappa \mu $ . For all $x \in D$ , $b \restriction x \in L$ (as $j(x) \in E_\eta $ and $j(b \restriction x) = \eta \restriction j(x) \in j(L)$ ), as wanted.⊣

Proof. Let us consider a name for a tree $\dot {T}$ and a ladder system $\dot {L}$ on $P_\kappa \lambda $ of the generic extension. By the $\kappa $ -c.c. of $\mathbb {M}(\kappa )$ , the set $\left (P_\kappa \lambda \right )^V$ is unbounded in $\left (P_\kappa \lambda \right )^{V[G]}$ . Moreover, one can easily code all names for elements in $P_\kappa \lambda $ , $\dot {T}$ and $\dot {L}$ into a transitive structure M of size $\lambda $ . We will assume that M satisfies some portion of $\textrm{ZFC}$ , and in particular it satisfies choice and the basic theory of forcing (including the forcing theorem for $\Sigma _n$ formulas, where n is sufficiently large).

By Lemma 2.8, there is an elementary embedding

$$\begin{align*}j\colon M \to N,\end{align*}$$

such that $j \operatorname{\textrm{"}} M \in N$ . We would like to lift this embedding to an elementary embedding from $M[G]$ to $N[H]$ , where G is a V-generic filter for $\mathbb {M}(\kappa )$ and H is an N-generic filter for $j(\mathbb {M}(\kappa ))$ . We cannot construct H in $V[G]$ , so in order to construct H we force with $j(\mathbb {M}(\kappa )) / \mathbb {M}(\kappa )$ over $V[G]$ .

Indeed, it is obvious that $\mathbb {M}(\kappa ) = j(\mathbb {M}(\kappa )) \restriction \kappa $ . Moreover, since for every $p \in \mathbb {M}(\kappa )$ , $j(p) = p$ , we conclude that for a generic filter $H \subseteq j(\mathbb {M}(\kappa ))$ , letting $G = H \restriction \kappa $ , the embedding j can be extended to an elementary embedding $j^\star \colon M[G] \to N[H]$ .

As in Theorem 4.4, by taking an element $\eta \in j(\dot {T})^{H}_{j\operatorname{\textrm{"}} \lambda } \cap j(\dot {L})^H$ , we obtain a branch through $\dot {T}^G$ ,

$$\begin{align*}b = \{ j^{-1}(\eta \restriction j\operatorname{\textrm{"}} x) \mid x \in \left(P_\kappa \lambda\right)^{V[G]}\}.\end{align*}$$

We would like to show that b belongs to $V[G]$ and that it meets $\dot {L}^G$ on a club.

The forcing $j(\mathbb {M}(\kappa )) / G$ cannot add new branches to a $P_\kappa \lambda $ trees (see, for example, [Reference Weiß27], or Claim 5.12 ahead). Thus, $b\in V[G]$ . Moreover, in $N[H] \subseteq V[H]$ , there is a club in $P_{\omega _1}\lambda $ in which b intersects L, since $\operatorname {\textrm{cf}} \kappa = \omega _1$ in the generic extension. We would like to claim that the same holds in $V[G]$ . Assume otherwise and let us consider

$$\begin{align*}S = \{x \in P_{\omega_1} \lambda \mid b \restriction x \notin L\} \in V[G].\end{align*}$$

In $N[H] \subseteq V[H]$ , S is non-stationary. But the forcing $j(\mathbb {M}(\kappa )) / G$ is proper in $V[G]$ since it is a projection of a product of a $\sigma $ -closed forcing and a c.c.c. forcing.⊣

Notation 5.6. If $S \subseteq \mathbb {S}(\alpha )$ is a generic filter, then:

• • $T_\alpha = \bigcup \{t \mid \exists \langle t, \ell , b, f\rangle \in S\}$ is a binary $\alpha $ -tree,

• • For each $x \in T_\alpha $ , let $B_\alpha (x) = \bigcup \{b(x) \mid \exists \langle t, \ell , b, f\rangle \in S,\, x \in \operatorname {\textrm{dom}} b\} \in {}^\alpha 2$ is a cofinal branch at $T_\alpha $ .

• • $L_\alpha = \bigcup \{\ell \mid \exists \langle t, \ell , b, f\rangle \in S\}$ is a ladder system on $T_\alpha $ .

• • $F_\alpha = \bigcup \{f \mid \exists \langle t, \ell , b, f\rangle \in S\}$ .

When $\alpha $ is clear from the context, we will omit it.

Claim 5.7. $\mathbb {S}(\alpha )$ is $\sigma $ -closed, $\alpha $ -strategically closed, and of size $2^{<\alpha }$ .

Proof. Let $\langle p_\xi \mid \xi < \epsilon \rangle $ be the game played so far, $\epsilon < \alpha $ . We denote by $p_\xi = \langle t_\xi , \ell _\xi , b_\xi , f_\xi \rangle $ , and we let $\delta _\xi = \max \operatorname {\textrm{dom}} \ell _\xi $ and $\gamma _\xi + 1$ be the height of the tree $t_\xi $ .

At successor steps, player Even does not move. At limit steps $\epsilon $ , let us define $t_\epsilon $ . If player Odd did not move co-boundedly below $\epsilon $ , then player Even does need to do anything.

Otherwise, the conditions $p_\xi $ are strictly decreasing on an unbounded subset of $\epsilon $ . Let us construct the condition which player Even would play. First, let us define $t_\epsilon $ . This is a tree of height $(\sup _{\xi < \epsilon } \gamma _\xi ) + 1$ . Let $\tilde {t} = \bigcup _{\xi < \epsilon } t_\xi $ . For each $x \in \tilde {t}$ , let $B(x)$ be $\bigcup _{\xi _\star < \xi < \epsilon } b_\xi (x)$ , where $\xi _\star $ is the level of x. We define

$$\begin{align*}t_{\epsilon} = \tilde{t} \cup \{B(x) \mid x \in \tilde{t}\}.\end{align*}$$

We let $b_\epsilon (x) = B(x)$ for $x \in \tilde {t}$ and $B(x) = x$ for nodes x in the top level of $t_{\epsilon }$ .

Let $\tilde {\ell } = \bigcup _{\xi < \epsilon } \ell _\xi $ . If $\sup \delta _\xi < \sup \gamma _\xi = \gamma _\epsilon $ , we let $\ell _\epsilon = \tilde {\ell }$ . Otherwise, we need to define $\ell (\gamma _\epsilon )$ . For $\operatorname {\textrm{cf}} \epsilon = \omega $ , we can define $\ell (\gamma _\epsilon ) = \emptyset $ , and $f_\epsilon = \bigcup _{\xi < \epsilon } f_\xi \cup \{(x, -1)\}$ for some x with $\sup x = \gamma _\epsilon $ . If $\operatorname {\textrm{cf}} \epsilon> \omega $ , we pick an arbitrary $x \in t_{\gamma _\epsilon }$ , and let

$$\begin{align*}E_x = \{y \in P_{\omega_1} \gamma_\epsilon \mid \sup y \in \{\gamma_\xi \mid \xi < \epsilon\} \}.\end{align*}$$

We set $\ell (\gamma _\epsilon ) = \{x\} \cup \{x \restriction z \mid z \in E_x\}$ .

We need to verify that the definition works. Note that the only non-trivial requirement is the empty intersection of $\operatorname {\textrm{range}} f$ and $\ell (\beta )$ for all $\beta \in \operatorname {\textrm{dom}} f$ . The requirement holds automatically for all $\beta \notin \{\gamma _\xi \mid \xi \leq \epsilon \}$ . For $\beta = \gamma _\xi $ , if $y \in \ell (\beta )$ then $\sup \operatorname {\textrm{dom}} y \in \{\gamma _\xi \mid \xi \leq \epsilon \}$ , but for each such y, if $y \in \operatorname {\textrm{dom}} f$ , then $f(y) = -1$ .

Since the strategy is trivial at finite steps, the forcing is $\sigma $ -closed.⊣

Claim 5.8. Let $\alpha $ be a regular cardinal, $\alpha \geq \omega _2$ . In the generic extension by $\mathbb {S}(\alpha )$ there is no branch of the generic tree $T_\alpha $ that meets the generic ladder system $L_\alpha $ on an $\omega _1$ -club.

Proof. Let $\dot {b}$ be a name for some a branch and let $\dot {C}$ be a name for a club. Let p be a condition in $\mathbb {S}(\alpha )$ . We want to find a condition $q \leq p$ such that $q \Vdash \check {a} \in \dot {C}$ , $\dot {b} \restriction a = \check x$ , and $\check x \notin \dot {L}$ .

Work inside some countable model M such that $p, \dot {b}, \dot {C}, \mathbb {S}(\alpha ) \in M$ , and let $\delta = \sup (M \cap \alpha )$ . By taking an $\omega $ -sequence of extensions of p inside M, we obtain an M-generic filter G. By the $\sigma $ -closure of the forcing, there are many conditions q such that $G = \{q' \in M \mid q' \geq q\}$ . Any such condition is a lower bound for the filter G.

Since G is M-generic, for every $\zeta \in M$ , the value of $\dot {b}(\zeta )$ is determined by some condition in G. Therefore, there is some $x \colon M \cap \alpha \to 2$ such that $(\dot {b} \cap M)^G = x$ . Note that for each condition q as above, $q \Vdash \dot {b} \restriction (M \cap \alpha ) = \check {x}$ , and in particular for some y, $q \Vdash \check y \in \dot {T}_{\alpha }$ and $y \restriction (M \cap \alpha ) = x$ .

Since $\operatorname {\textrm{cf}} \delta = \omega $ , we can pick $q = (t^q, \ell ^q, b^q, f^q)$ to be a lower bound of the conditions in G, such that $f^q(M \cap \delta ) = x$ , and $\ell ^q(\delta ) = \emptyset $ . This is possible, since the height of $t^q$ is at least $\delta + 1$ , $y \in t^q_{\delta + 1}$ , and $x = y \restriction (M \cap \delta )$ .

Since q is M-generic and $\dot {C}\in M$ is forced to a club, $q \Vdash M \cap \alpha \in \dot {C}$ . Finally, $q \Vdash \dot {b} \restriction (\check M \cap \check \alpha ) = \check {x} \notin \dot {L}$ .⊣

Claim 5.10. $\mathbb {S}(\alpha ) \ast \dot {\mathbb {T}}(\alpha )$ contains an $\alpha $ -closed dense subset.

Proof. Let D be the set of all conditions $\langle (t,\ell ,b,f), q\rangle $ such that the height of t is $\gamma + 1$ , $\max \operatorname {\textrm{dom}} \ell = \max q = \gamma $ . The set D is dense and $\alpha $ -closed.⊣

Proof. Since our forcing notion is of the form $\mathbb {M}'(\kappa )\ast \dot {\mathbb {S}(\kappa )}$ , by Claim 5.8, the generic tree and ladder system which is introduced by $\mathbb {S}(\kappa )$ would witness the failure of $\operatorname {\textrm{LSCP}}(P_{\omega _2}\omega _2, \omega _1\text {-club})$ .

Let us turn now to showing that $\operatorname {\textrm{LSCP}}(P_{\omega _2}\lambda , \omega _1\text {-cofinal})$ holds. Let $\dot {\mathcal T}$ be a name for a $P_{\omega _2}\lambda $ -tree and let $\dot {\mathcal L}$ be a name for a ladder system on $\dot {\mathcal T}$ .

As in the proof of the previous case, we start with a transitive model M, which contains all relevant information and obtain from Lemma 2.8 a transitive model N and an elementary embedding, $j\colon M \to N$ with $j\operatorname{\textrm{"}} \lambda \in M$ , $\lambda < j(\kappa )$ . This time, we would like to require more closure from N, so we will assume that M satisfies the further requirements of Lemma 2.9, and conclude that we can pick N to be closed under $<\lambda $ -sequences.

Let $G \ast S\subseteq \mathbb {M}'(\kappa ) \ast \mathbb {S}(\kappa )$ be a generic filter. We would like to find a generic $G' \ast S' \subseteq j(\mathbb {M}'(\kappa ) \ast \mathbb {S}(\kappa ))$ and lift the embedding to an embedding $\tilde {j} \colon M[G][S] \to N[G'][S']$ .

Since the f-part which is introduced in the forcing $\mathbb {S}(\kappa )$ is a non-reflecting stationary set, there is no hope to lift this embedding without a forcing component that would add a club disjoint from it. So, $\tilde {j}$ exists only in a generic extension of $V[G][S]$ .

First, let us show that

$$\begin{align*}j(\mathbb{M}'(\kappa)) \cong \mathbb{M}'(\kappa) \ast \dot{\mathbb{S}}(\kappa) \ast \dot{\mathbb{T}}(\kappa) \ast \dot{\mathbb{Q}}.\end{align*}$$

Indeed, the map sending $(a, m, s, t) \in j(\mathbb {M}'(\kappa ))$ to $(a \restriction \kappa , m \restriction \kappa , s \restriction \kappa + 1, t \restriction \kappa + 1)$ is a projection. Since $V_\kappa \subseteq N$ , it is easy to verify that $\dot {\mathbb {S}}(\kappa ), \dot {\mathbb {T}}(\kappa )$ , and the forcing $\mathbb {M}'(\kappa )$ are computed in the same way in M and in N, and therefore this map projects $j(\mathbb {M}'(\kappa ))$ onto $\mathbb {M}'(\kappa ) \ast \dot {\mathbb {S}}(\kappa ) \ast \dot {\mathbb {T}}(\kappa )$ . Let $T \subseteq \mathbb {T}^{G \ast S}(\kappa )$ be a $V[G][S]$ -generic filter, and let $\mathbb {Q}$ be the quotient forcing:

$$\begin{align*}\mathbb{Q} := j(\mathbb{M}'(\kappa)) /\left(\mathbb{M}'(\kappa) \ast \dot{\mathbb{S}}(\kappa) \ast \dot{\mathbb{T}}(\kappa)\right) = j(\mathbb{M}'(\kappa)) / (G \ast S \ast T).\end{align*}$$

Let $C = \bigcup T$ be the generic club introduced by $\mathbb {T}(\kappa )$ . In order to lift j, we must find a generic filter $G' \subseteq j(\mathbb {M}'(\kappa ))$ and $S' \subseteq j(\mathbb {S}(\kappa ))$ such that for every $p \in G \ast S$ , $j(p) \in G' \ast S'$ . By the structure of the conditions in $\mathbb {M}'(\kappa )$ , this implies that $G' \restriction \kappa = G$ , and for every $s \in S$ , $j(s)$ is in the generic $S'$ for $j(\mathbb {S}(\kappa ))$ . As usual, we choose $G'$ such that $G' \restriction \kappa + 1 = G \ast S \ast T$ and $G' / (G \ast S \ast T)$ is a generic filter for $\dot {\mathbb {Q}}^{G' \restriction \kappa + 1}$ over $M[G' \restriction \kappa + 1]$ .

We would like to find a master condition—a condition in $j(\mathbb {S}(\kappa ))$ , m such that for all condition $s \in \mathbb {S}(\kappa )$ that appear in the generic filter S, $m \leq j(s)$ . This would be sufficient as all conditions in the generic filter G are unmoved by j.

Let $T_\kappa , B_\kappa , L_\kappa , F_\kappa $ be the generic tree, branches, ladder system, and function introduced by $\mathbb {S}(\kappa )$ , respectively, as defined in Notation 5.6 (do not confuse the generic tree $T_\kappa $ with the generic filter for the forcing $\mathbb {T}(\kappa )$ , T).

Take $t_m = T_\kappa \cup \operatorname {\textrm{range}} B_\kappa \in 2^{\leq \kappa }$ . So, $t_m$ is a tree of height $\kappa + 1$ . Let $\ell _m$ extend the generic ladder system $L_\kappa $ by adding one element in the level $\kappa $ . Since $\kappa $ is forced to have uncountable cofinality in the generic extension by $j(\mathbb {M}'(\kappa ))$ , $\ell _m(\kappa )$ is obtained by picking one arbitrary element $\eta $ from the $\kappa $ -th level of the tree and using the generic club C that was introduced by $\mathbb {T}(\kappa )$ : the club $E_\eta $ consists of all $x \in P_{\omega _1} \kappa $ such that $\sup x \in C$ .

Let $b_m = B_\kappa $ , the collection of all generic branches. More precisely, for every $x \in T_\kappa $ , we define $b_m(x)$ to be the node in $t_m$ which lie on top of the cofinal branch $B_\kappa (x)$ , and $b_m(x) = x$ for $x \in t_m \cap {}^\kappa 2$ . Let $f_m = F_\kappa $ . The generic club C witnesses the domain of F to be non-stationary. Moreover, since C does not intersect $\{\sup x \mid x \in \operatorname {\textrm{dom}} F\}$ , we conclude that $\ell _m(\kappa )$ is disjoint from F.

Finally, we take a generic $S'$ such that $m\in S'$ . By the above discussion, in $V[G'][S']$ , the embedding j lifts. Let us denote by $j^\star \colon M[G][S] \to N[G'][S']$ the lifted embedding.

As in the proof of Theorem 4.4, we obtain a branch b by considering the value of the ladder system at $j\operatorname{\textrm{"}} \lambda $ : The element $j \operatorname{\textrm{"}} \lambda $ is a member of the club which is included in the domain on $j^*(\mathcal L)$ . We take $\eta \in j^*(\mathcal L)(j \operatorname{\textrm{"}} \lambda )$ , and define

$$\begin{align*}b = \{j^{-1}(\eta \restriction j\operatorname{\textrm{"}} z) \mid z \in P_{\omega_2} \lambda\}.\end{align*}$$

We claim that $b\in V[G]$ .

Finally, let us show that the set

$$\begin{align*}B = \{x \in P_{\omega_1} \lambda \mid b \restriction x \in \bigcup \operatorname{\textrm{range}} \ell\}\end{align*}$$

is unbounded. Indeed, this set is even stationary as in $N[G'][S']$ (in which $\operatorname {\textrm{cf}} \kappa> \omega $ ) this set contains a club.⊣

Proof. We work with full supercompact embeddings. Let $j \colon V \to M$ be a $\lambda $ -supercompact embedding. As in the proof of Lemma 5.11, we can lift it to an elementary embedding $j^* \colon V[G] \to M[H]$ .

Let us consider now a $P_{\omega _2} \lambda $ -tree $\mathcal {T}$ with a list d. Let us consider the branch b which is generated by $j^*(d)(j \operatorname{\textrm{"}} \lambda ) \in M[H]$ . By the arguments of Lemma 5.11, this branch appears already in $V[G]$ . We need to show that it is ineffable. Working in $V[G]$ , let $B = \{x \in P_{\omega _2} \lambda \mid b(x) = d(x)\}$ . If B is non-stationary in $V[G]$ , then there is a club D, avoiding it. Let us consider $j^*(D)$ . $j\operatorname{\textrm{"}} \lambda = \bigcup _{x\in D} j^*(x) \in j^*(D)$ . Therefore, $j \operatorname{\textrm{"}} \lambda \notin j^*(B)$ , but this is absurd, as

$$\begin{align*}j^*(b)(j\operatorname{\textrm{"}} \lambda) = \bigcup_{x \in P_{\omega_2}\lambda} j^*(b(x)) = j^*(d)(j\operatorname{\textrm{"}} \lambda).\\[-42pt] \end{align*}$$

⊣

Question 6.1. Is it consistent that $\operatorname {\textrm{LSCP}}(P_{\omega _2}\lambda , \omega _1$ -cofinal $)$ holds, but the seemly stronger property $\operatorname {\textrm{LSCP}}(P_{\omega _2}\omega _2, \omega _1$ -stationary $)$ fails, namely there is an $\omega _2$ -tree with a ladder system such that no branch meets that ladder system on a stationary set in $P_{\omega _1}\lambda $ ?

Question 6.2. Does the $\operatorname {\textrm{LSCP}}(P_{\omega _2}\lambda , \omega _1\text {-clubs})$ imply the Ineffable Tree Property at $P_{\omega _2}\lambda '$ for some $\lambda ' < \lambda $ ?