KEISLER’S THEOREM AND CARDINAL INVARIANTS

Abstract

We consider several variants of Keisler’s isomorphism theorem. We separate these variants by showing implications between them and cardinal invariants hypotheses. We characterize saturation hypotheses that are stronger than Keisler’s theorem with respect to models of size $\aleph _1$ and $\aleph _0$ by $\mathrm {CH}$ and $\operatorname {cov}(\mathsf {meager}) = \mathfrak {c} \land 2^{<\mathfrak {c}} = \mathfrak {c}$ respectively. We prove that Keisler’s theorem for models of size $\aleph _1$ and $\aleph _0$ implies $\mathfrak {b} = \aleph _1$ and $\operatorname {cov}(\mathsf {null}) \le \mathfrak {d}$ respectively. As a consequence, Keisler’s theorem for models of size $\aleph _0$ fails in the random model. We also show that for Keisler’s theorem for models of size $\aleph _1$ to hold it is not necessary that $\operatorname {cov}(\mathsf {meager})$ equals $\mathfrak {c}$.

1 Introduction

The method of ultrapowers is one of the most important ways to construct models. Ultrapowers are models obtained by properly equating the elements of product sets of the models using ultrafilters. We consider the problem of when there exists an ultrafilter $\mathcal {U}$ on $\omega $ such that for two models $\mathcal {A}, \mathcal {B}$ in a countable language $\mathcal {L}$ , the respective ultrapowers $\mathcal {A}^\omega /\mathcal {U}, \mathcal {B}^\omega /\mathcal {U}$ are isomorphic. Since ultrapowers are elementary extensions of original models, if $\mathcal {A}^\omega /\mathcal {U}$ and $\mathcal {B}^\omega /\mathcal {U}$ are isomorphic, then $\mathcal {A}$ and $\mathcal {B}$ must be elementarily equivalent. Keisler showed, under CH, conversely if $\mathcal {A}$ and $\mathcal {B}$ are elementarily equivalent and have size $\le \mathfrak {c}$ , then for every ultrafilter $\mathcal {U}$ over $\omega $ , $\mathcal {A}^\omega /\mathcal {U}$ and $\mathcal {B}^\omega /\mathcal {U}$ are isomorphic. The purpose of this paper is to give necessary conditions and sufficient conditions for when Keisler’s theorem holds in a model where CH does not hold, and to separate the variants of Keisler’s theorem using those conditions.

Convention. All ultrafilters considered in this paper are nonprincipal.

Definition 1.1. Let $\kappa $ be a cardinal.

1. (1) We say $\operatorname {KT}(\kappa )$ holds if for every countable language $\mathcal {L}$ and $\mathcal {L}$ -structures $\mathcal {A}, \mathcal {B}$ of size $\le \kappa $ which are elementarily equivalent, there exists an ultrafilter $\mathcal {U}$ over $\omega $ such that $\mathcal {A}^\omega /\mathcal {U} \simeq \mathcal {B}^\omega /\mathcal {U}$ .

2. (2) We say $\mathrm {SAT}(\kappa )$ holds if there exists an ultrafilter $\mathcal {U}$ over $\omega $ such that for every countable language $\mathcal {L}$ and every sequence of $\mathcal {L}$ -structures $(\mathcal {A}_i)_{i\in \omega }$ with each $\mathcal {A}_i$ of size $\le \kappa $ , $\prod _{i \in \omega } \mathcal {A}_i / \mathcal {U}$ is saturated.

$\mathrm {SAT}(\kappa )$ implies $\operatorname {KT}(\kappa )$ for every $\kappa \le \mathfrak {c}$ by the fact that two saturated structures which are elementarily equivalent and have the same size are isomorphic. Golshani and Shelah [6] proved $\neg \operatorname {KT}(\aleph _2)$ and later we will prove $\neg \mathrm {SAT}(\aleph _2)$ in Theorem 2.2. So this implication $\mathrm {SAT}(\kappa ) \Rightarrow \operatorname {KT}(\kappa )$ holds formally for every $\kappa $ .

Keisler [7] proved $\mathrm {CH} \Rightarrow \mathrm {SAT}(\mathfrak {c})$ . The result $\neg \operatorname {KT}(\aleph _2)$ of Golshani and Shelah implies $\operatorname {KT}(\mathfrak {c}) \Rightarrow \mathrm {CH}$ . So $\mathrm {CH}$ , $\mathrm {SAT}(\mathfrak {c})$ , and $\operatorname {KT}(\mathfrak {c})$ are equivalent. Golshani and Shelah also proved that $\operatorname {cov}(\mathsf {meager}) = \mathfrak {c} \land \operatorname {cf}(\mathfrak {c}) = \aleph _1$ implies $\operatorname {KT}(\aleph _1)$ . Another classical result is the theorem by Ellentuck–Rucker [5] which shows that $\operatorname {MA}(\sigma \text {-centered})$ implies $\mathrm {SAT}(\aleph _0)$ . Moreover, Shelah [9] showed $\operatorname {Con}(\neg \operatorname {KT}(\aleph _0))$ by showing that $\mathfrak {d} < \mathfrak {v}^\forall $ implies $\neg \operatorname {KT}(\aleph _0)$ and that the former is consistent.

In this paper, we prove the implications indicated by thick lines in Figure 1.

Figure 1 Implications; thick arrows indicate our results.

In the rest of this section, we recall basic notions related to cardinal invariants.

Definition 1.2.

1. (1) If $X, Y$ are sets and R is a subset of $X \times Y$ , we call a triple $(X, Y, R)$ a relational system.

2. (2) For a relational system $\mathcal {A} = (X, Y, R)$ , define $\mathcal {A}^\perp = (Y, X, \hat {R})$ , where $\hat {R} = \{ (y, x) \in Y \times X : \neg (x \mathrel {R} y)\})$ .

3. (3) For a relational system $\mathcal {A} = (X, Y, R)$ , define .

4. (4) For relational systems $\mathcal {A} = (X, Y, R), \mathcal {B} = (X', Y', S)$ , we call a pair $(\varphi , \psi )$ a Galois–Tukey morphism from $\mathcal {A}$ to $\mathcal {B}$ if $\varphi \colon X \to X'$ , $\psi \colon Y' \to Y$ and $(\forall x \in X)(\forall y \in Y')(\varphi (x) \mathrel {S} y \implies x \mathrel {R} \psi (y))$ hold.

Fact 1.3 [4, Theorem 4.9]

If there is a Galois–Tukey morphism $(\varphi , \psi )$ from $\mathcal {A}$ to $\mathcal {B}$ , then and hold.

Definition 1.4.

1. (1) For $f, g \in \omega ^\omega $ , define $f <^* g$ iff $(\forall ^\infty n) (f(n) < g(n))$ .

2. (2) For $c \in (\omega +1)^\omega $ and $h \in \omega ^\omega $ , define $\prod c = \prod _{n \in \omega } c(n)$ and $S(c, h) = \prod _{n \in \omega } [c(n)]^{\le h(n)}$ .

3. (3) For $x \in \prod c$ and $\varphi \in S(c, h)$ , define $x \in ^* \varphi $ iff $(\forall ^\infty n)(x(n) \in \varphi (n))$ and define $x \in ^\infty \varphi $ iff $(\exists ^\infty n)(x(n) \in \varphi (n))$ .

Definition 1.5.

1. (1) Define $\mathbf {D} = (\omega ^\omega , \omega ^\omega , <^*)$ , and .

2. (2) For a poset $(P, <)$ , define a relational system $\mathbf {Cof}(P, <)$ by $\mathbf {Cof}(P, <) = (P, P, <)$ . Then we have .

3. (3) For $c \in (\omega +1)^\omega , h \in \omega ^\omega $ , define $\mathbf {Lc}(c, h) = (\prod c, S(c, h), \in ^*)$ , and .

4. (4) Define $\mathbf {wLc}(c, h) = (\prod c, S(c, h), \in ^\infty )$ , , and .

5. (5) For an ideal I on X, define $\mathbf {Cov}(I) = (X, I, \in )$ , , and .

By the definition of the norm , the next lemma is obvious.

Lemma 1.6.

1. (1) $\mathfrak {c}^\forall _{c, h} = \min \{ \left \lvert {S} \right \rvert : S \subseteq S(c, h), (\forall x \in \prod c)(\exists \varphi \in S)(\forall ^\infty n) (x(n) \in \varphi (n)) \}$ .

2. (2) $\mathfrak {c}^\exists _{c, h} = \min \{ \left \lvert {S} \right \rvert : S \subseteq S(c, h), (\forall x \in \prod c)(\exists \varphi \in S)(\exists ^\infty n) (x(n) \in \varphi (n)) \}$ .

3. (3) $\mathfrak {v}^\forall _{c, h} = \min \{ \left \lvert {X} \right \rvert : X \subseteq \prod c, (\forall \varphi \in S(c, h))(\exists x \in X)(\exists ^\infty n) (x(n) \not \in \varphi (n)) \}$ .

4. (4) $\mathfrak {v}^\exists _{c, h} = \min \{ \left \lvert {X} \right \rvert : X \subseteq \prod c, (\forall \varphi \in S(c, h))(\exists x \in X)(\forall ^\infty n) (x(n) \not \in \varphi (n)) \}$ .

Definition 1.7.

1. (1) Define $\mathfrak {v}^\forall = \min \{ \mathfrak {v}^\forall _{c, h} : c, h \in \omega ^\omega , \lim _{n \to \infty } h(n) = \infty \}$ .

2. (2) Define $\mathfrak {c}^\exists = \min \{ \mathfrak {c}^\exists _{c, h} : c, h \in \omega ^\omega , \sum _{n \in \omega } h(n) / c(n) < \infty \}$ .

Fact 1.8 [3, Lemma 3.5 and Theorem 3.12]

Let ${\langle \mathcal {A}_i : i \in \omega \rangle }$ be a sequence of structures in a language $\mathcal {L}$ such that each $\mathcal {A}_i$ has size $\le \mathfrak {c}$ . Let $\mathcal {U}$ be an ultrafilter over $\omega $ . Then the ultraproduct $\prod _{i \in \omega } \mathcal {A}_i / \mathcal {U}$ has size either finite or $\mathfrak {c}$ .

2 $\mathrm {SAT}(\aleph _1)$ and $\operatorname {KT}(\aleph _1)$

In this section, we prove that $\mathrm {SAT}(\aleph _1)$ is equivalent to $\mathrm {CH}$ and that $\operatorname {KT}(\aleph _1)$ implies $\mathfrak {b} = \aleph _1$ .

Theorem 2.1. $\mathrm {SAT}(\aleph _1)$ implies CH.

Proof Assume $\mathrm {SAT}(\aleph _1)$ and $\neg \mathrm {CH}$ . Take an ultrafilter $\mathcal {U}$ over $\omega $ that witnesses $\mathrm {SAT}(\aleph _1)$ . Let $\mathcal {A}_\ast = (\omega _1, <)^\omega / \mathcal {U}$ . For $\alpha < \omega _1$ , put $\alpha _\ast = [{\langle \alpha , \alpha , \alpha , \dots \rangle }]$ . Define a set p of formulas with a free variable x by

$$\begin{align*}p = \{ \ulcorner \alpha_\ast < x \urcorner : \alpha < \omega_1 \}. \end{align*}$$

This p is finitely satisfiable and the number of parameters occurring in p is $\aleph _1 < \mathfrak {c} = \left \lvert {\mathcal {A}_\ast } \right \rvert $ by $\neg \mathrm {CH}$ . Thus, by $\mathrm {SAT}(\aleph _1)$ , we can take $f \colon \omega \to \omega _1$ such that $[f]$ realizes p. Put $\beta = \sup _{n \in \omega } f(n)$ . Now we have $\{ n \in \omega : \beta < f(n) \} \in \mathcal {U}$ and this contradicts the definition of $\beta $ .

Theorem 2.2. $\neg \mathrm {SAT}(\aleph _2)$ holds.

Proof Take an ultrafilter $\mathcal {U}$ over $\omega $ that witnesses $\mathrm {SAT}(\aleph _2)$ . Let $\mathcal {A}_\ast = (\omega _2, <)^\omega / \mathcal {U}$ . For $\alpha < \omega _1$ , put $\alpha _\ast = [{\langle \alpha , \alpha , \alpha , \dots \rangle }]$ . Define a set p of formulas with a free variable x by

$$\begin{align*}p = \{ \ulcorner \alpha_\ast < x < (\omega_1)_\ast \urcorner : \alpha < \omega_1 \}. \end{align*}$$

The remaining argument is the same as Theorem 2.1.

Definition 2.3. Let $\mathfrak {mcf} = \min \{ \operatorname {cf}(\omega ^\omega /\mathcal {U}) : \mathcal {U} \text { an ultrafilter over } \omega \}$ .

The order of $\omega ^\omega /\mathcal {U}$ is the almost domination order modulo $\mathcal {U}$ and $\operatorname {cf}(\omega ^\omega /\mathcal {U})$ is the dominating number of this relation. So it is clear that $\mathfrak {b} \le \mathfrak {mcf} \le \mathfrak {d}$ .

Lemma 2.4 [6, Claim 2.2]

Let $\mathcal {A}$ be a structure in a language $\mathcal {L} = \{<\}$ . Suppose that $a \in \mathcal {A}$ has cofinality $\omega _1$ . Let $\mathcal {U}$ be an ultrafilter over $\omega $ . Then $a_\ast = [{\langle a, a, a, \dots \rangle }]$ has cofinality $\omega _1$ in $\mathcal {A}^\omega / \mathcal {U}$ .

Proof Take an increasing cofinal sequence ${\langle x_\alpha : \alpha < \omega _1\rangle }$ of points in $\mathcal {A}$ below a. Then ${\langle x_\alpha ^* : \alpha < \omega _1\rangle }$ is an increasing cofinal sequence in $\mathcal {A}_\ast $ , where $x_\alpha ^* = [{\langle x_\alpha , x_\alpha , x_\alpha , \dots \rangle }]$ for each $\alpha < \omega _1$ . This can be shown by regularity of $\omega _1$ .

Lemma 2.5 [6, Claim 2.4]

Let $\mathcal {U}$ be an ultrafilter over $\omega $ and $\mathcal {B}_\ast = (\mathbb {Q}, <)^\omega /\mathcal {U}$ . Then for every $a, b \in \mathcal {B}_\ast $ , there is an automorphism on $\mathcal {B}_\ast $ that sends a to b.

Proof Consider the map $F \colon \mathbb {Q}^3 \to \mathbb {Q}$ defined by $F(x, y, z) = x - y + z$ . Then we have

$$\begin{align*}(\forall y, z \in \mathbb{Q}) (\text{the map } x \mapsto F(x, y, z) \text{ is an automorphism on } (\mathbb{Q}, <) \text{ that sends }y\text{ to }z). \end{align*}$$

This statement can be written by a first-order formula in the language $\mathcal {L}' = \{<, F\}$ . Thus the same statement is true in $(\mathbb {Q}, <, F)^\omega /\mathcal {U}$ . The map $F_\ast : \mathcal {B}_\ast ^3 \to \mathcal {B}_\ast $ induced by F satisfies that

$$\begin{align*}&(\forall y, z \in \mathcal{B}_\ast) \\&\quad(\text{the map } x \mapsto F(x, y, z) \text{ is an automorphism on } (\mathcal{B}_\ast, <) \text{ that sends }y\text{ to }z).\\[-35pt] \end{align*}$$

Theorem 2.6. $\operatorname {KT}(\aleph _1)$ implies $\mathfrak {mcf} = \aleph _1$ .

Proof This proof is based on [6, Theorem 2.1]. Assume that $\mathfrak {mcf} \ge \aleph _2$ . We shall show $\neg \operatorname {KT}(\aleph _1)$ .

Let $\mathcal {L} = \{ < \}$ , $\mathcal {A} = (\mathbb {Q}, <)$ and $\mathcal {B} = (\mathbb {Q} + ((\omega _1+1) \times \mathbb {Q}_{\ge 0}), <_{\mathcal {B}})$ . Here $<_{\mathcal {B}}$ is defined by a lexicographical order and a disjoint union order. $\mathcal {A}$ and $\mathcal {B}$ are dense linear ordered sets, so by completeness of DLO, we have $\mathcal {A} \equiv \mathcal {B}$ . Take an ultrafilter $\mathcal {U}$ over $\omega $ . Put $\mathcal {A}_* = \mathcal {A}^\omega / \mathcal {U}, \mathcal {B}_* = \mathcal {B}^\omega / \mathcal {U}$ .

There is a point a in $\mathcal {B}$ such that $\operatorname {cf}(\mathcal {B}_a) = \aleph _1$ , where $\mathcal {B}_a = \{ x \in \mathcal {B} : x < a \}$ . Then $a_* \in \mathcal {B}_*$ has cofinality $\aleph _1$ by Lemma 2.4. Here $a_* = [{\langle a, a, a, \dots \rangle }]$ . On the other hand, we shall show every point in $\mathcal {A}_*$ has cofinality $\ge \mathfrak {mcf}$ . If we do this, since we assumed $\mathfrak {mcf} \ge \aleph _2$ , we will have $\mathcal {A}_* \not \simeq \mathcal {B}_*$ .

By Lemma 2.5, it suffices to consider the point $0_* = [{\langle 0, 0, 0, \dots \rangle }]$ . Since $\mathbb {Q}$ is symmetrical, we consider $\operatorname {cf}((\mathbb {Q}_{>0})^\omega /\mathcal {U}, >_{\mathcal {U}})$ .

Now we construct a Galois–Tukey morphism $(\varphi , \psi ) \colon \mathbf {Cof}(\omega ^\omega /\mathcal {U}) \kern1.2pt{\to}\kern1.2pt \mathbf {Cof}((\mathbb {Q}_{>0})^\omega /\mathcal {U}, >_{\mathcal {U}})$ by

$$ \begin{align*} \varphi &\colon \omega^\omega/\mathcal{U} \to (\mathbb{Q}_{>0})^\omega/\mathcal{U}; [f] \mapsto [{\langle1/(f(n)+1) : n \in \omega\rangle}], \\ \psi &\colon (\mathbb{Q}_{>0})^\omega/\mathcal{U} \to \omega^\omega/\mathcal{U}; [g] \mapsto [{\langle \left\lfloor {1/g(n)-1} \right\rfloor : n \in \omega\rangle}]. \end{align*} $$

So we have $\operatorname {cf}((\mathbb {Q}_{>0})^\omega /\mathcal {U}, >_{\mathcal {U}}) \ge \operatorname {cf}(\omega ^\omega /\mathcal {U}, <_{\mathcal {U}})$ .

Thus we have $\operatorname {cf}((\mathbb {Q}_{>0})^\omega /\mathcal {U}, >_{\mathcal {U}}) \ge \mathfrak {mcf}$ . We are done.

Corollary 2.7. $\operatorname {KT}(\aleph _1)$ implies $\mathfrak {b} = \aleph _1$ .

Proof This follows from Theorem 2.6 and the fact that $\mathfrak {b} \le \mathfrak {mcf}$ .

3 $\mathrm {SAT}(\aleph _0)$ and $\operatorname {KT}(\aleph _0)$

In this section, we first briefly mention consistency of $\operatorname {KT}(\aleph _0) + \neg \operatorname {KT}(\aleph _1)$ . And we prove that $\mathrm {SAT}(\aleph _0)$ is equivalent to $\operatorname {cov}(\mathsf {meager}) = \mathfrak {c} \land 2^{<\mathfrak {c}} = \mathfrak {c}$ .

Fact 3.1 [4, Theorem 7.13]

The statement $\operatorname {cov}(\mathsf {meager}) = \mathfrak {c}$ is equivalent to $\operatorname {MA}(\textrm {countable})$ , that is for every countable poset $\mathbb {P}$ and a family of dense sets $\mathcal {D}$ with $ \left \lvert {\mathcal {D}} \right \rvert < \mathfrak {c}$ there is a filter G of $\mathbb {P}$ that intersects all $D \in \mathcal {D}$ .

Theorem 3.2. $\operatorname {cov}(\mathsf {meager}) = \mathfrak {c}$ implies $\operatorname {KT}(\aleph _0)$ .

Proof [6, Theorem 3.3] shows that $\operatorname {cov}(\mathsf {meager}) = \mathfrak {c} \land \operatorname {cf}(\mathfrak {c})> \aleph _1$ implies $\operatorname {KT}(\aleph _1)$ and the exact same proof works for $\operatorname {KT}(\aleph _0)$ without the assumption $\operatorname {cf}(\mathfrak {c})> \aleph _1$ .

Here we sketch the proof.

Let $\mathcal {L}$ be a countable language and $\mathcal {A}^0$ and $\mathcal {A}^1$ are countable $\mathcal {L}$ -structures which are elementarily equivalent.

Enumerate $(\mathcal {A}^i)^\omega $ for $i=0, 1$ as

$$ \begin{align*} (\mathcal{A}^i)^\omega = \{f^i_\alpha : \alpha < \mathfrak{c} \}. \end{align*} $$

By a back-and-forth method, we construct a sequence of triples ${\langle (\mathcal {U}_\alpha , g^0_\alpha , g^1_\alpha ) : \alpha < \mathfrak {c}\rangle }$ satisfying:

1. (1) $g^0_\alpha \in \mathcal {A}^0$ .

2. (2) $g^1_\alpha \in \mathcal {A}^1$ .

3. (3) $\mathcal {U}_\alpha $ is a filter over $\omega $ generated by $\aleph _0 + \left \lvert {\alpha } \right \rvert $ sets.

4. (4) $(\mathcal {U}_\alpha : \alpha < \mathfrak {c})$ is an increasing continuous sequence.

5. (5) If $\varphi (x_0, \dots , x_{n-1}$ is an $\mathcal {L}$ -formula and $\beta _0, \dots , \beta _n \le \alpha $ , then the set

   $$\begin{align*}\{k \in \omega : \mathcal{M}^0 \models \varphi(g^0_{\beta_0}(k), \dots, g^0_{\beta_{n-1}}(k)) \iff \mathcal{M}^1 \models \varphi(g^1_{\beta_0}(k), \dots, g^1_{\beta_{n-1}}(k)) \}\end{align*}$$

   belongs to $\mathcal {U}_{\alpha +1}$ .

In the construction, when $\alpha $ is even, we put $g^0_\alpha = f^0_\gamma $ where $\gamma $ is the least ordinal $f^0_\gamma \not \in \{g^0_\beta : \beta < \alpha \}$ . And $\mathbb {P}$ is the poset of finite partial functions from $\omega $ to $\mathcal {A}^1$ . Take a generating set $\mathcal {F}$ of $\mathcal {U}_\alpha $ of size $\aleph _0 + \left \lvert {\alpha } \right \rvert $ . Then by using $\operatorname {MA}(\textrm {countable})$ , take a $\mathbb {P}$ -generic filter G with respect to the following family of dense sets of $\mathbb {P}$ :

$$\begin{align*}D_n = \{ p \in \mathbb{P} : n \in \operatorname{dom} p \} \text{ (for }n \in \omega) \end{align*}$$

and

$$ \begin{align*} E_{X,{\langle\varphi_\iota : \iota \in I\rangle},{\langle\gamma^\iota_1,\dots,\gamma^\iota_{n_\iota} : \iota \in I\rangle}} = &\{ p \in \mathbb{P} : (\exists k \in \operatorname{dom}(p) \cap X)(\forall \iota \in I) \\ & \hspace{0.4cm} (M^0 \models \varphi_\iota(g^0_{\gamma^\iota_1}(k), \dots g^0_{\gamma^\iota_{n_\iota}}(k), g^0_\alpha(k)) \Leftrightarrow \\ & \hspace{0.54cm} M^1 \models \varphi_\iota(g^1_{\gamma^\iota_1}(k), \dots g^1_{\gamma^\iota_{n_\iota}}(k), p(k)) \}), \end{align*} $$

where $X \in \mathcal {F}$ , ${\langle \varphi _\iota : \iota \in I\rangle }$ is a finite sequence of $\mathcal {L}$ -formulas and $\gamma ^\iota _1, \dots , \gamma ^\iota _{n_\iota }$ for $\iota \in I$ are ordinals less than $\alpha $ . Then putting $g^1_\alpha = \bigcup G$ satisfies the induction hypothesis.

Then the appropriate construction guarantees that $\mathcal {U} = \bigcup _{\alpha < \mathfrak {c}} \mathcal {U}_\alpha $ is an ultrafilter and that the function

$$\begin{align*}{\langle([g^0_\alpha]_{\mathcal{U}}, [g^1_\alpha]_{\mathcal{U}}) : \alpha < \mathfrak{c}\rangle} \end{align*}$$

is isomorphic from $(M^0)^\omega /\mathcal {U}$ to $(M^1)^\omega /\mathcal {U}$ .

Corollary 3.3. Assume $\operatorname {Con}(\mathrm {ZFC})$ . Then $\operatorname {Con}(\mathrm {ZFC} + \operatorname {KT}(\aleph _0) + \neg \operatorname {KT}(\aleph _1))$ .

Proof $\mathrm {MA} + \neg \mathrm {CH}$ implies $\operatorname {KT}(\aleph _0) \land \neg \operatorname {KT}(\aleph _1)$ by Theorems 2.6 and 3.2.

Fact 3.4 [2, Lemma 2.4.2]

$\operatorname {cov}(\mathsf {meager}) = \mathfrak {v}^\exists _{{\langle \omega : n \in \omega \rangle }, \mathrm {id}}$ . In other words, $\operatorname {cov}(\mathsf {meager}) \ge \kappa $ holds iff $(\forall X \subseteq \omega ^\omega \text { of size } <\kappa ) (\exists S \in \prod _{i \in \omega } [\omega ]^{\le i}) (\forall x \in X) (\exists ^\infty n) (x(n) \in S(n))$ holds.

Theorem 3.5. $\mathrm {SAT}(\aleph _0)$ implies $\operatorname {cov}(\mathsf {meager}) = \mathfrak {c}$ .

Proof Take an ultrafilter $\mathcal {U}$ that witnesses $\mathrm {SAT}(\aleph _0)$ . Fix $X \subseteq \omega ^\omega $ of size $< \mathfrak {c}$ . Define a language $\mathcal {L}$ by $\mathcal {L} = \{\subseteq \}$ and for each $i \in \omega $ , define an $\mathcal {L}$ -structure $\mathcal {A}_i$ by $\mathcal {A}_i = ([\omega ]^{\le i}, \subseteq )$ . For each $x \in \omega ^\omega $ , let $S_x = {\langle \{x(i)\} : i \in \omega \rangle }$ . In the ultraproduct $\mathcal {A}_* = \prod _{i\in \omega } \mathcal {A}_i / \mathcal {U}$ , define a set p of formulas of one free variable S by

$$\begin{align*}p = \{ \ulcorner [S_x] \subseteq S \urcorner : x \in X \}. \end{align*}$$

This p is finitely satisfiable. In order to check this, let $x_0, \dots , x_n$ be finitely many members of X. Define S by $S(m) = \{ x_0(m), \dots , x_n(m) \}$ for $m \ge n$ . We don’t need to care about $S(m)$ for $m < n$ . Then this S satisfies $[S_{x_i}] \subseteq [S]$ for all $i \le n$ . Moreover, the number of parameters of p is $< \mathfrak {c}$ .

So by $\mathrm {SAT}(\aleph _0)$ , we can take $[S] \in \mathcal {A}_*$ that realizes p. Then S fulfills $(\forall x \in X) (\{ n \in \omega : x(n) \in S(n) \} \in \mathcal {U})$ . Thus $(\forall x \in X) (\exists ^\infty n) (x(n) \in S(n))$ .

Theorem 3.6. $\mathrm {SAT}(\aleph _0)$ implies $2^{<\mathfrak {c}} = \mathfrak {c}$ .

Proof Take an ultrafilter $\mathcal {U}$ over $\omega $ that witnesses $\mathrm {SAT}(\aleph _0)$ . Fix $\kappa < \mathfrak {c}$ .

Put $\mathcal {L} = \{\subseteq \}$ and define an $\mathcal {L}$ -structure $\mathcal {A}$ by $\mathcal {A} = ([\omega ]^{<\omega }, \subseteq )$ . Put $\mathcal {A}^* = \mathcal {A}^\omega /\mathcal {U}$ .

Define a map $\iota \colon \omega ^\omega /\mathcal {U} \to \mathcal {A}^*$ by $\iota ([x]) = [{\langle \{x(n)\} : n \in \omega \rangle }]$ . By Fact 1.8, we have $ \left \lvert {\omega ^\omega /\mathcal {U}} \right \rvert = \mathfrak {c}$ . Take a subset F of $\omega ^\omega /\mathcal {U}$ of size $\kappa $ .

For each $X \subseteq F$ , let $p_X$ be a set of formulas with a free variable z defined by

$$\begin{align*}p_X = \{ \ulcorner \iota(y) \subseteq z \urcorner : y \in X \} \cup \{ \ulcorner \iota(y) \not \subseteq z \urcorner : y \in F \smallsetminus X \}. \end{align*}$$

Each $p_X$ is finitely satisfiable. In order to check this, take $[x_0], \dots , [x_n] \in X$ and $[y_0], \dots , [y_m] \in F \smallsetminus X$ . Put $z(i) = \{ x_0(i), \dots , x_n(i) \}$ . Then $\iota ([x_0]), \dots , \iota ([x_n]) \subseteq _{\mathcal {U}} [z]$ . In order to prove $\iota ([y_j]) \not \subseteq _{\mathcal {U}} [z]$ for each $j \le m$ , suppose that $\{ i \in \omega : y_j(i) \in z(i) \} \in \mathcal {U}$ . Then for each $i \in \omega $ , there is a $k_i \le n$ such that $\{ i \in \omega : y_j(i) = x_{k_i}(i)\} \in \mathcal {U}$ . Then there is a $k \le n$ such that $\{ i \in \omega : y_j(i) = x_k(i)\} \in \mathcal {U}$ . This implies $[y_j] = [x_k]$ , which is a contradiction.

By $\mathrm {SAT}(\aleph _0)$ , for each $X \subseteq F$ , take $[z_X] \in \mathcal {A}^*$ that realizes $p_X$ . For $X, Y \subseteq F$ with $X \ne Y$ , we have $[z_X] \ne [z_Y]$ . So $2^\kappa = \left \lvert {\{[z_X] : X \subseteq F\}} \right \rvert \le \left \lvert {\mathcal {A}^*} \right \rvert = \mathfrak {c}$ . Therefore we have proved $2^{<\mathfrak {c}} = \mathfrak {c}$ .

Theorem 3.7. $\operatorname {cov}(\mathsf {meager}) = \mathfrak {c} \land 2^{<\mathfrak {c}} = \mathfrak {c}$ implies $\mathrm {SAT}(\aleph _0)$ .

Proof This proof is based on [5, Theorem 1].

Let ${\langle b_\alpha : \alpha < \mathfrak {c}\rangle }$ be an enumeration of $\omega ^\omega $ . Let ${\langle (\mathcal {L}_\xi , \mathcal {B}_\xi , \Delta _\xi ) : \xi < \mathfrak {c}\rangle }$ be an enumeration of triples $(\mathcal {L}, \mathcal {B}, \Delta )$ such that $\mathcal {L}$ is a countable language, $\mathcal {B} = {\langle \mathcal {A}_i : i \in \omega \rangle }$ is a sequence of $\mathcal {L}$ -structures with universe $\omega $ , and $\Delta $ is a subset of $\operatorname {Fml}(\mathcal {L}^+)$ with $ \left \lvert {\Delta } \right \rvert < \mathfrak {c}$ . Here $\mathcal {L}^+ = \mathcal {L} \cup \{ c_\alpha : \alpha < \mathfrak {c} \}$ where the $c_\alpha $ ’s are new constant symbols and $\operatorname {Fml}(\mathcal {L}^+)$ is the set of all $\mathcal {L}^+$ formulas with one free variable. Here we used the assumption $2^{<\mathfrak {c}} = \mathfrak {c}$ . And ensure each $(\mathcal {L}, \mathcal {B}, \Delta )$ occurs cofinally in this sequence.

For $\mathcal {B}_\xi = {\langle \mathcal {A}^\xi _i : i \in \omega \rangle }$ , put $\mathcal {B}_\xi (i) = (\mathcal {A}^\xi _i, b_0(i), b_1(i), \dots )$ , which is an $\mathcal {L}^+$ -structure.

Let ${\langle X_\xi : \xi < \mathfrak {c}\rangle }$ be an enumeration of $\mathcal {P}(\omega )$ .

We construct a sequence ${\langle F_\xi : \xi < \mathfrak {c}\rangle }$ of filters inductively so that the following properties hold:

1. (1) $F_0$ is the filter consisting of all cofinite subsets of $\omega $ .

2. (2) $F_\xi \subseteq F_{\xi + 1}$ and $F_\xi = \bigcup _{\alpha < \xi } F_\alpha $ for $\xi $ limit.

3. (3) $X_\xi \in F_{\xi +1}$ or $\omega \smallsetminus X_\xi \in F_{\xi +1}$ .

4. (4) $F_\xi $ is generated by $< \mathfrak {c}$ members.

5. (5) If

   (*) $$ \begin{align} \text{for all } \Gamma \subseteq \Delta_\xi \text{ finite, } \{i\in\omega : \Gamma \text{ is satisfiable in } \mathcal{B}_\xi(i) \} \in F_\xi, \end{align} $$

   then there is an $f \in \omega ^\omega $ such that for all $\varphi \in \Delta _\xi $ , $\{ i \in \omega : f(i) \text { satisfies } \varphi \text {in } \mathcal {B}_\xi (i) \} \in F_{\xi +1}$ .

Suppose we have constructed $F_\xi $ . We construct $F_{\xi +1}$ . Let $F_\xi '$ be a generating subset of $F_\xi $ with $ \left \lvert {F_\xi '} \right \rvert < \mathfrak {c}$ . If (*) is false, let $F_{\xi +1}$ be the filter generated by $F_\xi ' \cup \{X_\xi \}$ or $F_\xi ' \cup \{\omega \smallsetminus X_\xi \}$ . Suppose (*).

Put $\mathbb {P} = \operatorname {Fn}(\omega , \omega ) = \{ p : p\text { is a finite partial function from }\omega \text { to }\omega \}$ . For $n \in \omega $ , put

$$\begin{align*}D_n = \{ p \in \mathbb{P} : n \in \operatorname{dom} p \}.\end{align*}$$

For $A \in F_\xi '$ and $\varphi _1, \dots , \varphi _n \in \Delta _\xi $ , put

$$\begin{align*}E_{A,\varphi_1,\dots,\varphi_n} = \{ p \in \mathbb{P} : (\exists i \in \operatorname{dom} p \cap A) (p(i) \text{ satisfies } \varphi_1, \dots, \varphi_n \text{ in } \mathcal{B}_\xi(i)) \}. \end{align*}$$

Each $D_n$ is clearly dense. In order to show that each $E_{A,\varphi _1,\dots ,\varphi _n}$ is dense, take $p \in \mathbb {P}$ . By (*) and the property $A \in F_\xi $ , we can take $i \in A \smallsetminus \operatorname {dom} p$ and $k \in \omega $ such that k satisfies $\varphi _1, \dots , \varphi _n$ in $\mathcal {B}_\xi (i)$ . Put $q = p \cup \{ (i, k) \}$ . This is an extension of p in $E_{A,\varphi _1,\dots ,\varphi _n}$ .

By using $\operatorname {MA}(\textrm {countable})$ , take a generic filter $G \subseteq \mathbb {P}$ with respect to above dense sets. Put $f = \bigcup G$ . Then $F_\xi " := F_\xi ' \cup \{ Y_\varphi : \varphi \in \Delta _\xi \}$ satisfies finite intersection property, where $Y_\varphi = \{ i \in \omega : f(i) \text { satisfies } \varphi \text { in } \mathcal {B}_\xi (i) \}$ . In order to check this, let $A \in F_\xi '$ and $\varphi _1, \dots , \varphi _n \in \Delta _\xi $ . Then by genericity, we can take $p \in G \cap E_{A,\varphi _1,\dots ,\varphi _n}$ . So we can take $i \in \operatorname {dom} p \cap A$ such that $p(i) \text { satisfies } \varphi _1, \dots , \varphi _n \text { in } \mathcal {B}_\xi (i)$ . Then we have $i \in A \cap Y_{\varphi _1} \cap \dots \cap Y_{\varphi _n}$ .

Let $F_{\xi +1}$ be the filter generated by $F_\xi '' \cup \{X_\xi \}$ or $F_\xi '' \cup \{\omega \smallsetminus X_\xi \}$ .

We have constructed ${\langle F_\xi : \xi < \mathfrak {c}\rangle }$ . In order to check that the resulting ultrafilter $F = \bigcup _{\xi < \mathfrak {c}} F_\xi $ witnesses $\mathrm {SAT}(\aleph _0)$ , let $\mathcal {L}$ and $\mathcal {B} = {\langle \mathcal {A}_i : i \in \omega \rangle }$ satisfy the assumption of the theorem. Let $\Delta $ be a subset of $\operatorname {Fml}(\mathcal {L}^+)$ with $ \left \lvert {\Delta } \right \rvert < \mathfrak {c}$ . Assume that for all $\Gamma \subseteq \Delta $ finite, $X_\Gamma := \{i\in \omega : \Gamma \text { is satisfiable in } \mathcal {B}_\xi (i) \} \in F$ . By the regularity of $\mathfrak {c}$ , we have $\alpha < \mathfrak {c}$ such that for all $\Gamma \subseteq \Delta $ finite, $X_\Gamma \in F_\alpha $ . Let $\xi \ge \alpha $ be satisfying $(\mathcal {L}_\xi , \mathcal {B}_\xi , \Delta _\xi ) = (\mathcal {L}, \mathcal {B}, \Delta )$ . Then by (5), there is an $f \in \omega $ such that for all $\varphi \in \Delta $ , $\{ i \in \omega : f(i) \text { satisfies } \varphi \text { in } \mathcal {B}(i) \} \in F$ . Thus $\prod _{i \in \omega } \mathcal {A}_i / F$ is saturated.

4 $\operatorname {KT}(\aleph _0)$ implies $ \mathfrak {c}^\exists \le \mathfrak {d}$

In this section, we will show the following theorem. This proof is based on [9, Theorem 1.1] and [1, Theorem 3.7].

Theorem 4.1. $\operatorname {KT}(\aleph _0)$ implies $\mathfrak {c}^\exists \le \mathfrak {d}$ .

Definition 4.2. Define a language $\mathcal {L}$ by $\mathcal {L} = \{E, U, V\}$ , where E is a binary predicate and $U, V$ are unary predicates. We say an $\mathcal {L}$ -structure $M = ( \left \lvert {M} \right \rvert , E^M, U^M, V^M)$ is a bipartite directed graph if the following conditions hold:

1. (1) $U^M \cup V^M = \left \lvert {M} \right \rvert $ .

2. (2) $U^M \cap V^M = \varnothing $ .

3. (3) $(\forall x, y \in \left \lvert {M} \right \rvert ) (x \mathrel {E^M} y \rightarrow (x \in U^M \text { and } y \in V^M))$ .

Definition 4.3. For $n, k \in \omega $ with $k \le n$ , define a bipartite directed graph $\Delta _{n,k}$ as follows:

1. (1) $U^{\Delta _{n,k}} = \{ 1, 2, 3, \dots , n \}$ .

2. (2) $V^{\Delta _{n,k}} = [\{ 1, 2, 3, \dots , n\}]^{\le k} \smallsetminus \{\varnothing \}$ .

3. (3) For $u \in U^{\Delta _{n,k}}, v \in V^{\Delta _{n,k}}$ , $u \mathrel {E^{\Delta _{n,k}}} v$ iff $u \in v$ .

Definition 4.4. For $n \in \omega $ , let $G_n = \Delta _{n^3, n}$ . Let $\Gamma $ be the disjoint union of $(G_n : n \ge 2)$ .

We define a natural order $\triangleleft $ on $\Gamma $ by $x \triangleleft y$ if $m < n$ for $x \in G_m, y \in G_n$ . Then $\Gamma $ is a bipartite directed graph with an order $\triangleleft $ . Put $\mathcal {L}' = \mathcal {L} \cup \{ \triangleleft \}$ . From now on, we consider $\mathcal {L}'$ -structures which are elementarily equivalent to $\Gamma $ .

Definition 4.5. Let $\Gamma _{\mathrm {NS}}$ be a countable non-standard elementary extension of  $\Gamma $ .

When we say connected components, we mean the connected components when we ignore the orientation of the edges.

Lemma 4.6. Let M be an $\mathcal {L}'$ -structure that is elementarily equivalent to $\Gamma $ . Then the connected components of M are precisely the maximal antichains of M with respect to  $\triangleleft $ .

Proof Suppose that $A \subseteq M$ is connected but not an antichain. Then we can find elements $a_0, \dots , a_n \in M$ such that

$$ \begin{align*} M \models &(a_0 E a_1 \lor a_1 E a_0) \land \dots \land (a_{n-1} E a_n \lor a_n E a_{n-1}) \land \\ &(a_0\text{ and }a_n\text{ are comparable with respect to }\triangleleft). \end{align*} $$

By elementarity, we have $n+1$ many elements in $\Gamma $ that satisfies the same formula. This is a contradiction. So every connected subset in M is an antichain.

Note that any two connected vertexes in $\Gamma $ have a path of length at most $4$ . Thus we have

$$ \begin{align*} \Gamma \models (\forall a, b) &((a\text{ and }b\text{ are incomparable with respect to }\triangleleft) \\ &\rightarrow (\text{there is a path between }a\text{ and }b\text{ with length at most } 4)). \end{align*} $$

By elementarity, the same formula holds in M. So every antichain in M is connected.

Therefore the connected components of M are precisely the maximal antichains of M with respect to $\triangleleft $ .

Therefore, $\triangleleft $ induces an order on the connected components of M and it is denoted also by $\triangleleft $ .

Lemma 4.7. Every infinite connected component C of $\Gamma _{\mathrm {NS}}$ satisfies the following:

$$\begin{align*}(\forall F \subseteq C \cap U \text{ finite}) (\exists v \in C \cap V) (v \text{ has an edge to each point in } F). \end{align*}$$

Proof Let $F = \{ u_1, \dots , u_n \}$ . Observe that

$$ \begin{align*} \Gamma \models (\forall x_1) \dots (\forall x_n) & [ x_1, \dots, x_n \text{ are points in } U \text{ and belong to} \\ & \text {the same connected component and } \\ &\text{ the index of this connected component is } \ge n \\ & \rightarrow (\exists y) [y \text{ belongs to this component, } y \in V \text{ and } x_1, \dots, x_n \mathrel{E} y]]. \end{align*} $$

By elementarity, $\Gamma _{\mathrm {NS}}$ satisfies the same formula.

Lemma 4.8. Let ${\langle \Delta _n : n \in \omega \rangle }$ be a sequence of bipartite directed graphs with $ \left \lvert {U^{\Delta _n}} \right \rvert = \left \lvert {V^{\Delta _n}} \right \rvert = \aleph _0$ . Suppose that for each $n \in \omega $ ,

$$\begin{align*}(\forall F \subseteq U^{\Delta_n} \text{ finite}) (\exists v \in V^{\Delta_n}) (v \text{ has an edge to each point in } F). \end{align*}$$

Then for every ultraproduct $R := \prod _{n \in \omega } \Delta _n / \mathcal {V}$ , we have

$$\begin{align*}(\exists {\langle v_i : i < \mathfrak{d}\rangle} \text{ with each } v_i \in V^R) (\forall u \in U^R) (\exists i < \mathfrak{d}) (u \mathrel{E^R} v_i). \end{align*}$$

Proof We may assume that each $U^{\Delta _n} = \omega $ . Let $\{ f_i : i < \mathfrak {d} \}$ be a cofinal subset of $(\omega ^\omega , <^*)$ . For each $n, m \in \omega $ , take $v_{n, m} \in V^{\Delta _n}$ that is connected with first m points in $U^{\Delta _n}$ . For $i < \mathfrak {d}$ , put

$$\begin{align*}v_i = [{\langle v_{n, f_i(n)} : n \in \omega\rangle}]. \end{align*}$$

Let $[u] \in U^R$ . Consider u as an element of $\omega ^\omega $ . Take $f_i$ that dominates u. Then we have

$$\begin{align*}\{ n \in \omega : u(n) \mathrel{E^{\Delta_n}} v_{n, f_i(n)} \} \in \mathcal{V}. \end{align*}$$

Therefore $[u] \mathrel {E^R} v_i$ .

Lemma 4.9. Let $\mathcal {V}$ be an ultrafilter over $\omega $ and put $Q = (\Gamma _{\mathrm {NS}})^\omega /\mathcal {V}$ . Then there exist cofinally many connected components C with respect to $\triangleleft $ such that

$$\begin{align*}(\exists {\langle v_i : i < \mathfrak{d}\rangle} \text{ with each } v_i \in C \cap V^Q) (\forall u \in C \cap U^Q) (\exists i < \mathfrak{d}) (u \mathrel{E^Q} v_i). \end{align*}$$

Proof Fix a connected component $C_0$ of Q and $[x_0] \in C_0$ . Then for each $n \in \omega $ , there is an infinite component $C_n$ above $x_0(n)$ . Now

$$\begin{align*}C = \{ [x] \in Q : x \in (\Gamma_{\mathrm{NS}})^\omega \text{ and } (\forall n \in \omega)(x(n) \in C_n) \} \end{align*}$$

is a connected component of Q above $C_0$ . Since C can be viewed as $C = \prod _{n \in \omega } C_n / \mathcal {V}$ , the conclusion of the lemma holds for C by Lemmas 4.7 and 4.8.

Lemma 4.10. Let $\kappa < \mathfrak {c}^\exists $ and $\mathcal {U}$ be an ultrafilter over $\omega $ and put $P = \Gamma ^\omega /\mathcal {U}$ . Then for every C in a final segment of connected components of P, we have

$$\begin{align*}(\forall {\langle v_i : i < \kappa\rangle} \text{ with each } v_i \in C \cap V^P) (\exists u \in C \cap U^P) (\forall i < \kappa) (u \not \mathrel{E^P} v_i). \end{align*}$$

Proof Let $f \colon \omega \to \Gamma $ satisfy $f(n) \in G_n$ for all n. Let $C_0$ be the connected component that $[f]$ belongs to. Take a connected component C such that $C_0 \triangleleft C$ and an element $[g] \in C$ . Take a function $h \colon \omega \to \omega $ such that $\{ n \in \omega : g(n) \in G_{h(n)} \} \in \mathcal {U}$ . Then $A := \{ n \in \omega : h(n) \ge n \} \in \mathcal {U}$ . Put $h'(n) = \max \{ h(n), n\}$ .

Take ${\langle [v_i] : i < \kappa \rangle }$ with each $[v_i] \in C \cap V^P$ . Then we have

$$\begin{align*}B_i := \{ n \in \omega : v_i(n) \in G_{h(n)} \cap V^\Gamma \} \in \mathcal{U}. \end{align*}$$

Take $v_i'$ such that $v_i'(n) = v_i(n)$ for $n \in A \cap B_i$ and $v_i'(n) \in [h'(n)^3]^{\le h'(n)}$ for $n\in \omega $ . The assumption $\kappa < \mathfrak {c}^\exists $ and the calculation

$$\begin{align*}\sum_{n \ge 1} \frac{h'(n)}{h'(n)^3} = \sum_{n \ge 1} \frac{1}{h'(n)^2} \le \sum_{n \ge 1} \frac{1}{n^2} < \infty \end{align*}$$

give an $x \in \prod h'$ such that for all $i < \kappa $ , $(\forall ^\infty n) (x(n) \not \in v_i'(n))$ . For each $i < \kappa $ , take $n_i$ such that $(\forall n \ge n_i) (x(n) \not \in v_i'(n))$ .

Take a point $[u] \in C \cap U^P$ such that $u(n) = x(n)$ for all $n \in A$ . Then for all $i < \kappa $ we have

$$\begin{align*}\{ n \in \omega : u(n) \not \mathrel{E^\Gamma} v_i(n) \} \supseteq A \cap B_i \cap [n_i, \omega) \in {\mathcal{U}}. \end{align*}$$

Therefore $[u] \not \mathrel{E^P} [v_i]$ for all $i < \kappa $ .

Assume that $\mathfrak {d} < \mathfrak {c}^\exists $ . Then by Lemmas 4.10 and 4.9, for any two ultrafilters $\mathcal {U}, \mathcal {V}$ over $\omega $ , we have $\Gamma ^\omega /\mathcal {U} \not \simeq (\Gamma _{\mathrm {NS}})^\omega /\mathcal {V}$ . So $\neg \operatorname {KT}(\aleph _0)$ holds. We have proved Theorem 4.1.

Fact 4.11 [8, Lemma 2.3]

$\operatorname {cov}(\mathsf {null}) \le \mathfrak {c}^\exists $ .

Corollary 4.12. In the random model, $\neg \operatorname {KT}(\aleph _0)$ holds.

Proof This corollary holds since $\aleph _1 = \mathfrak {d} < \operatorname {cov}(\mathsf {null}) = \mathfrak {c}$ in the random model.

Remark 4.13. $\mathfrak {v}^\forall \le \mathfrak {c}^\exists $ follows from [8, Lemma 2.6]. So the implication $\operatorname {KT}(\aleph _0) \implies \mathfrak {d} \ge \mathfrak {c}^\exists $ strengthens the implication $\operatorname {KT}(\aleph _0) \implies \mathfrak {d} \ge \mathfrak {v}^\forall $ .

Remark 4.14. In [9], Shelah constructed a creature forcing that forces the following statements:

1. (1) There are a finite language $\mathcal {L}$ and countable $\mathcal {L}$ -structures $\mathcal {A}, \mathcal {B}$ with $\mathcal {A} \equiv \mathcal {B}$ such that for all ultrafilters $\mathcal {U}, \mathcal {V}$ over $\omega $ , we have $\mathcal {A}^\omega /\mathcal {U} \not \simeq \mathcal {B}^\omega /\mathcal {V}$ .

2. (2) There is an ultrafilter $\mathcal {U}$ over $\omega $ such that for every countable language $\mathcal {L}$ and any sequence ${\langle (\mathcal {A}_n, \mathcal {B}_n) : n \in \omega \rangle }$ of pairs of finite $\mathcal {L}$ -structures, if $\prod _{n \in \omega } \mathcal {A}_n/\mathcal {U} \equiv \prod _{n \in \omega } \mathcal {B}_n/\mathcal {U}$ , then these ultraproducts are isomorphic.

Shelah himself pointed out in [9, Remark 2.2] item 2 holds in the random model. On the other hand, we have proved item 1 also holds in the random model. Therefore both of above two statements hold in the random model.

5 $\operatorname {KT}(\aleph _1)$ in forcing extensions

A theorem by Golshani and Shelah [6] states that $\operatorname {cov}(\mathsf {meager}) = \mathfrak {c} \land \operatorname {cf}(\mathfrak {c}) = \aleph _1$ implies $\operatorname {KT}(\aleph _1)$ . In [6], it was also proved that $\operatorname {cf}(\mathfrak {c}) = \aleph _1$ is not necessary for $\operatorname {KT}(\aleph _1)$ . In this section, we prove that $\operatorname {cov}(\mathsf {meager}) = \mathfrak {c}$ is also not necessary for $\operatorname {KT}(\aleph _1)$ .

Theorem 5.1. Let $\lambda> \aleph _1$ be a regular cardinal with $\lambda ^{<\lambda } = \lambda $ . Let ${\langle \mathbb {P}_\alpha , \dot {\mathbb {Q}}_\alpha : \alpha < \omega _1\rangle }$ be a finite support forcing iteration. Suppose that for all $\alpha < \omega _1$ , $\Vdash _\alpha "\dot {\mathbb {Q}}_\alpha \text { is ccc and } \left \lvert {\dot {\mathbb {Q}}_\alpha } \right \rvert \le \lambda "$ . And suppose that for all even $\alpha < \omega _1$ , $\Vdash _\alpha \dot {\mathbb {Q}}_\alpha = \mathbb {C}_\lambda $ . Here $\mathbb {C}_\lambda $ denotes the Cohen forcing adjoining $\lambda $ many Cohen reals. Then, $\Vdash _{\omega _1} \operatorname {KT}(\aleph _1)$ .

Proof This proof is based on [6, Theorem 3.3].

Let G be a $(V, \mathbb {P}_{\omega _1})$ -generic filter.

Let $\mathcal {L}$ be a countable language and $M^0 \equiv M^1$ be two $\mathcal {L}$ -structures of size $\le \aleph _1$ in $V[G]$ . Take sequences ${\langle M^l_i : i < \omega _1\rangle }$ for $l = 0, 1$ that are increasing and continuous such that each $M^l_i$ is countable elementary substructure of $M^l$ and $M^l = \bigcup _{i < \omega _1} M^l_i$ . We can take an increasing sequence ${\langle \alpha _i : i < \omega _1\rangle }$ of even ordinals such that $M^l_i \in V[G_{\alpha _i + 1}]$ for every $l < 2$ and $i < \omega _1$ .

For $i < \omega _1$ and $\beta < \lambda $ , let $c^i_\beta $ be the $\beta $ -th Cohen real added by $\dot {\mathbb {Q}}_{\alpha _i}$ .

Take an enumeration ${\langle X_\gamma : \gamma < \lambda \cdot \omega _1 \rangle }$ of $\mathcal {P}(\omega )$ such that ${\langle X_\gamma : \gamma < \lambda \cdot (i+1)\rangle } \in V[G_{\alpha _i + 1}]$ for every $i < \omega _1$ . We can take such a sequence. The reason for this is that we can take ${\langle \dot {X}_\gamma : \lambda \cdot i \le \gamma < \lambda \cdot (i+1)\rangle }$ as an enumeration of $\mathbb {P}_{\alpha _i+1}$ nice names for subsets of $\omega $ and put $X_\gamma = (\dot {X}_\gamma )^G$ .

For each $l < 2$ , take an enumeration ${\langle f^l_\gamma : \gamma < \lambda \cdot \omega _1\rangle }$ of $(M^l)^\omega $ such that $f^l_{\lambda \cdot i + \beta } \in (M^l_i)^\omega $ for every $i < \omega _1$ and $\beta < \lambda $ and ${\langle f^l_\gamma : \gamma < \lambda \cdot (i+1)\rangle } \in V[G_{\alpha _i + 1}]$ .

For $\lambda ' < \lambda $ , let $G_{\alpha _i, \lambda '}$ denote $G \cap (\mathbb {P}_{\alpha _i} \ast \mathbb {C}_{\lambda '})$ .

Now we construct a sequence of quadruples ${\langle (\mathcal {U}_\gamma , g^0_\gamma , g^1_\gamma , \lambda _\gamma ) : \gamma < \lambda \cdot \omega _1\rangle }$ by induction so that the following properties hold.

1. (1) Each $\mathcal {U}_\gamma $ is a filter over $\omega $ .

2. (2) For every $l < 2$ , $i < \omega _1$ , $\beta < \lambda $ , and $\gamma = \lambda \cdot i + \beta $ , $g^l_{\gamma } \in (M^l_i)^\omega \cap V[G_{\alpha _i, \lambda _\gamma }]$ .

3. (3) For every $l < 2$ and $i < \omega _1$ , ${\langle g^l_\gamma : \gamma < \lambda \cdot (i+1)\rangle } \in V[G_{\alpha _i + 1}]$ .

4. (4) Each $\lambda _\gamma $ is an ordinal below $ \lambda $ . For $\lambda \cdot i \le \gamma \le \gamma ' < \lambda \cdot (i+1)$ , we have $\lambda _\gamma \le \lambda _{\gamma '}$ .

5. (5) For $i < \omega _1$ and $l < 2$ , $\{ g^l_\gamma : \gamma < \lambda \cdot i \} = \{ f^l_\gamma : \gamma < \lambda \cdot i \}$ .

6. (6) If $\lambda \cdot i \le \gamma < \lambda \cdot (i+1)$ , then $\mathcal {U}_\gamma \in V[G_{\alpha _i}, \lambda _\gamma ]$ .

7. (7) If $\gamma < \delta < \lambda \cdot \omega _1$ , then $\mathcal {U}_\gamma \subseteq \mathcal {U}_\delta $ .

8. (8) If $\gamma < \lambda \cdot \omega _1$ is a limit ordinal, then $\mathcal {U}_\gamma = \bigcup _{\delta < \gamma } \mathcal {U}_\delta $ .

9. (9) $X_\gamma \in \mathcal {U}_{\gamma + 1}$ or $\omega \smallsetminus X_\gamma \in \mathcal {U}_{\gamma +1}$ .

10. (10) If $\varphi (x_1, \dots , x_n)$ is an $\mathcal {L}$ -formula, $\gamma = \lambda \cdot i + \beta $ , and $\gamma _1, \dots , \gamma _n \le \gamma $ , then $Y_{\varphi , \gamma _1, \dots , \gamma _n}$ defined below belongs to $\mathcal {U}_{\gamma +1}$ :

    $$\begin{align*}&Y_{\varphi, \gamma_1, \dots, \gamma_n} = \{ k \in \omega : M^0_i \models \varphi(g^0_{\gamma_1}(k), \dots, g^0_{\gamma_n}(k))\\&\quad \Leftrightarrow M^1_i \models \varphi(g^1_{\gamma_1}(k), \dots, g^1_{\gamma_n}(k)) \}. \end{align*}$$

(Construction) First we let $U_0$ be the set of cofinite subsets of $\omega $ .

Suppose that ${\langle \mathcal {U}_\delta : \delta \le \gamma \rangle }$ and ${\langle g^0_\delta , g^1_\delta , \lambda _\delta : \delta < \gamma \rangle }$ are defined. Now we will define $g^0_\gamma , g^1_\gamma , \lambda _\gamma $ and $\mathcal {U}_{\gamma +1}$ . Take i and $\beta $ such that $\gamma = \lambda \cdot i + \beta $ .

Suppose that $\gamma $ is even.

Let $g^0_\gamma = f^0_{\varepsilon _\gamma }$ , where $\varepsilon _\gamma $ is the minimum ordinal such that $f^0_{\varepsilon _\gamma }$ does not belong to $\{g^0_\delta : \delta < \gamma \}$ .

Take $\lambda ' < \lambda $ such that $M^0_i, M^1_i, {\langle g^0_\delta : \delta \le \gamma \rangle }, {\langle g^1_\delta : \delta < \gamma \rangle } \in V[G_{\alpha _i, \lambda '}]$ . Put $\lambda _\gamma = \lambda ' + 1$ . Take a bijection $\pi ^1_i \colon \omega \to M^1_i$ in $V[G_{\alpha _i, \lambda '}]$ . Define $g^1_\gamma $ by $g^1_\gamma = \pi ^1_i \circ c^i_{\lambda '}$ .

Put $\mathcal {Y} = \{ Y_{\varphi , \gamma _1, \dots , \gamma _n} : \varphi (x_1, \dots , x_n) \text { is an } \mathcal {L}\text {-formula and }\gamma _1, \dots , \gamma _n \le \gamma \}$ . Now we show $\mathcal {U}_\gamma \cup \mathcal {Y}$ has the finite intersection property. In order to show it, let $X \in \mathcal {U}_\gamma $ , ${\langle \varphi _\iota : \iota \in I\rangle }$ is a finite sequence of $\mathcal {L}$ -formulas and $\gamma ^\iota _1, \dots , \gamma ^\iota _{n_\iota }$ for $\iota \in I$ are ordinals that are less than $\gamma $ . It suffices to show that the set $D \in V[G_{\alpha _i, \lambda '}]$ defined below is a dense subset of $\mathbb {C}$ :

$$ \begin{align*} &D = \{ p \in \mathbb{C}:\ (\exists k \in \operatorname{dom}(p) \cap X)(\forall \iota \in I) \\ & \qquad M^0_i \models \varphi_\iota(g^0_{\gamma^\iota_1}(k), \dots g^0_{\gamma^\iota_{n_\iota}}(k), g^0_\gamma(k)) \Leftrightarrow M^1_i \models \varphi_\iota(g^1_{\gamma^\iota_1}(k), \dots g^1_{\gamma^\iota_{n_\iota}}(k), \pi^1_i(p(k))) \}. \end{align*} $$

We now prove this. Let $p \in \mathbb {C}$ .

For each $k \in \omega $ and $\iota \in I$ , put

$$\begin{align*}v(k, \iota) = \begin{cases} 1, & \text{if }M^0_i \models \varphi_\iota(g^0_{\gamma^\iota_1}(k), \dots, g^0_{\gamma^\iota_{n_\iota}}(k), g^0_\gamma(k)), \\ 0, & \text{otherwise.} \end{cases} \end{align*}$$

And for each $k \in \omega $ put

$$\begin{align*}v(k) = {\langle v(k, \iota) : \iota \in I\rangle}. \end{align*}$$

Then by finiteness of ${}^I 2$ , for some $v_0 \in {}^I 2$ , we have $\omega \smallsetminus v^{-1}(v_0) \not \in \mathcal {U}_\gamma $ .

For each $\iota \in I$ , put

$$\begin{align*}\varphi_\iota^+(x^\iota_1, \dots, x^\iota_{n_\iota}, y) \equiv \begin{cases} \varphi_\iota(x^\iota_1, \dots, x^\iota_{n_\iota}, y), & \text{if } v_0(\iota) = 1, \\ \neg \varphi_\iota(x^\iota_1, \dots, x^\iota_{n_\iota}, y), & \text{otherwise.} \end{cases} \end{align*}$$

Put

$$\begin{align*}\psi \equiv \exists y \bigwedge_{\iota \in I} \varphi_\iota^+(x^\iota_1, \dots, x^\iota_{n_\iota}, y). \end{align*}$$

Then by the induction hypothesis (10), $Y_{\psi , {\langle \gamma ^\iota _1, \dots \gamma ^\iota _{n_\iota } : \iota \in I\rangle }} \in \mathcal {U}_\gamma $ . So take $k \in X \cap v^{-1}(v_0) \cap Y_{\psi , {\langle \gamma ^\iota _1, \dots \gamma ^\iota _{n_\iota } : \iota \in I\rangle }} \smallsetminus \operatorname {dom}(p)$ .

Since $M^0_i \models \psi ({\langle g^0_{\gamma ^\iota _1}(k), \dots g^0_{\gamma ^\iota _{n_\iota }}(k) : \iota \in I\rangle })$ , we have $M^1_i \models \psi (\langle g^1_{\gamma ^\iota _1}(k), \dots g^1_{\gamma ^\iota _{n_\iota }}(k) : \iota \in I\rangle )$ .

By the definition of $\psi $ , we can take $y \in M^1_i$ such that $M^1_i \models \varphi _\iota ^+(g^1_{\gamma ^\iota _1}(k), \dots , g^1_{\gamma ^\iota _{n_\iota }}(k), y)$ for every $\iota \in I$ . We now put $q = p \cup \{ (k, (\pi ^1_i)^{-1}(y)) \} \in \mathbb {C}$ . This witnesses denseness of D.

Now we define $\mathcal {U}_{\gamma +1}$ as the filter generated by $\mathcal {U}_\gamma \cup \mathcal {Y} \cup \{ X_\gamma \}$ or the filter generated by $\mathcal {U}_\gamma \cup \mathcal {Y} \cup \{ \omega \smallsetminus X_\gamma \}$ .

When $\gamma $ is odd, do the same construction above except for swapping 0 and 1. Since the above construction below $\lambda \cdot (i+1)$ can be performed in $V[G_{\alpha _i + 1}]$ , (3) in the induction hypothesis holds. (End of Construction.)

Now we put $\mathcal {U} = \bigcup _{\gamma < \lambda \cdot \omega _1} \mathcal {U}_\gamma $ , which is an ultrafilter over $\omega $ . Then the function

$$\begin{align*}{\langle([g^0_\gamma]_{\mathcal{U}}, [g^1_\gamma]_{\mathcal{U}}) : \gamma < \lambda \cdot \omega_1\rangle} \end{align*}$$

witnesses $(M^0)^\omega / \mathcal {U} \simeq (M^1)^\omega / \mathcal {U}$ .

Corollary 5.2. $\operatorname {Con}(\mathrm {ZFC}) \rightarrow \operatorname {Con}(\mathrm {ZFC} + \operatorname {cof}(\mathsf {null}) = \aleph _1 < \mathfrak {c} + \operatorname {KT}(\aleph _1))$ .

Proof Let $\mathbb {A}$ denote the amoeba forcing. Let $\lambda> \aleph _1$ be a regular cardinal with $\lambda ^{<\lambda } = \lambda $ . Let ${\langle \mathbb {P}_\alpha , \dot {\mathbb {Q}}_\alpha : \alpha < \omega _1\rangle }$ be a finite support forcing iteration such that for all even $\alpha < \omega _1$ we have $\Vdash _\alpha \dot {\mathbb {Q}}_\alpha = \mathbb {C}_\lambda $ and for all odd $\alpha < \omega _1$ we have $\Vdash _\alpha \dot {\mathbb {Q}}_\alpha = \mathbb {A}$ .

Then $\mathbb {P}_{\omega _1} \Vdash \operatorname {KT}(\aleph _1)$ by Theorem 5.1.

Moreover, we have $\operatorname {cof}(\mathsf {null}) = \aleph _1$ since the amoeba forcing $\mathbb {A}$ adds a null set containing all null sets coded in the ground model (see [2, p. 106]).

6 Open questions

The following three questions remain.

Question 6.1.

1. (1) Does $\operatorname {KT}(\aleph _1)$ imply a stronger hypothesis than $\mathfrak {mcf} = \aleph _1$ ? In particular does $\operatorname {KT}(\aleph _1)$ imply $\operatorname {non}(\mathsf {meager}) = \aleph _1$ ?

2. (2) Does $\operatorname {KT}(\aleph _0)$ imply a stronger hypothesis than $\mathfrak {c}^\exists \le \mathfrak {d}$ ? In particular does $\operatorname {KT}(\aleph _0)$ imply $\operatorname {non}(\mathsf {meager}) \le \operatorname {cov}(\mathsf {meager})$ ?

3. (3) In the Sacks model, does $\operatorname {KT}(\aleph _0)$ hold? (If in this model $\neg \operatorname {KT}(\aleph _0)$ holds, we can separate $\operatorname {KT}(\aleph _0)$ and $\mathfrak {c}^\exists \le \mathfrak {d}$ .)