# Facts on elementary submodels

In the paper of "Aspero, Larson, Moore - Forcing Axioms and the CH" three facts are stated as well-known. As i have not read them before, they are not that obvious to me. Maybe good references to these facts would help or maybe somebody would be so kind and explain them briefly.

The three facts mentioned are (for $\theta, \lhd, \vec{\kappa}, \vec{U}$ as below):

1. If $M \prec (H(\theta),\in, \lhd)$ and $I \subseteq \kappa_2 \in M$, then $\operatorname{cl}(M,I) \prec (H(\theta),\in, \lhd)$.
2. Let $i<3, M \prec H(\theta)$ s.t. $U_i, \kappa_2 \in M$. If $\eta \in \bigcap (M_\cap U_i)$, then $\operatorname{cl}(M, \lbrace \eta \rbrace) \cap \kappa_i$ is an end-extension of $M \cap \kappa_i$.
3. Under the conditions of Fact 2, let $I \subseteq \kappa_i$ and $\mu \in M$ regular with $\mu > \kappa_i$. Then $\sup(\operatorname{cl}(M,I) \cap \mu) = \sup(M \cap \mu)$.

For a sufficiently large cardinal $\theta$ for $P$ let $\lhd$ be a well-ordering of $H(\theta)$. For an increasing sequence $\kappa_i (i<3)$ of cardinals $> \omega_2$ and $\ M,I \subset H(\theta)$ with $I \subseteq \kappa_2 \in M$ let $\operatorname{cl}(M,I)$ denote the set of values $g(\eta_0, \dots, \eta_{n-1})$, where $g$ is a function in $M$ and $\operatorname{dom}(g) = {\kappa_2}^{<\omega}$, and $\lbrace \eta_0, \dots, \eta_{n-1} \rbrace \subseteq I$ finite.

Fixing $\theta, \lhd, \vec{\kappa}, \vec{U}$ (where $U_i$ are normal ultrafilters on each $\kappa_i$), given $i<3$ and an $M \prec (H(\theta), \in , \lhd)$ with $\vert M \vert < \kappa_i$, we say $\lbrace M_\xi \rbrace_{\xi < \kappa_i}$ is the iteration of $M$ relative to $U_i$ in case $\lbrace M_\xi \rbrace_{\xi < \kappa_i}$ is the unique $\subseteq$-continuous sequence s.t. $M_0=M$ and, $\forall \xi < \kappa_i, \ M_{\xi+1} = \operatorname{cl}(M_\xi, \lbrace \eta_\xi \rbrace)$, where $\eta^i_\xi = \min(\bigcap (U_i \cap M_\xi))$.

• What does $I\subseteq\in M$ mean? Both subset and element of? – Asaf Karagila Aug 28 '15 at 12:14
• Oh thanks, that was a typo. I corrected it. Of course just as in the definition below. – os23 Aug 28 '15 at 13:58
• In part 2, what do you mean by $\eta \in \bigcap (M \cap U_i)$? Is it supposed to read $\eta \in U_i \setminus M$? – Stefan Mesken Aug 31 '15 at 20:40
• If you are still interested in answers to your questions, let me now by commenting on this comment. – Stefan Mesken Sep 5 '15 at 14:39

1. We will use the Tarski Vaught test: Let $\phi$ be a $\{\in, \lhd\}$-formula (notice that I consider $\in$ and $\lhd$ both as formal symbols and sets, in a slight abuse of notation) and let $p \in \operatorname{cl}(M,I)$ be a parameter s.t. $$(H(\theta); \in, \lhd) \models \exists x \phi(x,p)$$ (We only allowed for one parameter in $\phi$, but letting $p = (p_1, \ldots, p_n)$ reduces the general case to the above modulo some basic set theory.)
By the definition of $\operatorname{cl}(M,I)$, there is a function $g \in M$, $g \colon \kappa_2^{< \omega} \to M$ and $\overline \mu \in I^{< \omega}$ s.t. $p = g(\overline \mu)$. We have to show that there is some $x \in \operatorname{cl}(M,I)$ with $(H(\theta); \in, \lhd) \models \phi(x, g(\overline \mu))$.
Towards this goal, let $$f \colon \kappa_2^{< \omega} \to H(\theta), \overline \nu \mapsto \min_\lhd \{ x \in H(\theta) \mid (H(\theta); \in, \lhd) \models \phi(x, g(\overline \nu) \}$$ where $\min_\lhd$ is the minimum with respect to $\lhd$ and $\min_\lhd \emptyset := \emptyset$. $f$ is definable in $(H(\theta); \in, \lhd)$ from $\kappa_2$ and $g$. As $\kappa_2, g \in M$, we have (by Tarski Vaught) $f \in M$ and thus $f(\overline \nu) \in \operatorname{cl}(M,I)$. By the definition of $f$ $$(H(\theta); \in, \lhd) \models \phi(f(\overline \nu), g(\overline \nu)).$$ Refering to the Tarski Vaught test one last time, this yields $\operatorname{cl}(M,I) \prec (H(\theta); \in, \lhd)$.