Originally defined in [[Magidor, Malitz - Compact extesions of L(Q) (part 1a)]] (MM1) **Def** Magidor-Malitz quantifiers: for an ordinal $\alpha$ and integer $n$ we set $ \begin{gather*} \mathcal{M}\vDash Q_{\alpha}^{\mathrm{MM},n}x_{1} \dots x_{n} \varphi \left(x_{1} ,\dots, x_{n},\vec{b}\right)\iff\\ \exists X \subseteq M\left(\left|X\right|\geq\aleph_{\alpha}\land\forall a_{1},\dots, a_{n}\in M^{n}:\mathcal{M}\vDash \varphi \left(a_{1},\dots, a_{n},\vec{b}\right)\right) \end{gather*}$^def For every $1\leq n<\omega$ and cardinal $\kappa$ let $\mathcal{L}^{n}_{\kappa}=\mathcal{L}_{\omega \omega}(Q^{n}_{\kappa})$, $\mathcal{L}^{<\omega}_{\kappa}=\mathcal{L}_{\omega \omega}(Q_{\kappa}^{n}\mid n<\omega)$ ### Observation This is the same as saying that the coloring $c:M^n\to \{ 0,1 \}$ defined by $c(a_{1},\dots,a_{n})=0 \iff \mathcal{M}\vDash \varphi \left(a_{1},\dots, a_{n},\vec{b}\right)$ has a homogeneous set for color $0$ of size $\geq \aleph_{\alpha}$ . # Facts ## Basic facts (MM1) - There is a prenex normal form. - $Q^{2}$ doen't distribute over disjunction. - Lowenheim-Skolem-Tarski theorem down to $\kappa$. ## Compactness and incompactness MM1: - *Incompactness bound:* If $\lambda=\mathrm{cf}(\kappa)$ then $L(Q_{\kappa}^{2})$ is not $\lambda$-compact - *Compactness with weakly compact*: Let $\mu$ be weakly compact, $\Sigma\subseteq \mathcal{L}(Q^{n}\mid n<\omega)$ of cardinality lt;\mu$. If every finite subset of $\Sigma$ has a model in the $\mu$-interpretation, then $\Sigma$ has a model in the $\kappa$ interpretation for every regular $\kappa>|\Sigma|+\omega$. - In particular, if $\kappa$ is a [[Weakly compact cardinal]] then $\mathcal{L}^{<\omega}_{\kappa}$ is lt;\kappa$ compact. - $\diamondsuit(\aleph_{1})$ implies that $\mathcal{L}_{\omega_{1}}^{<\omega}$ is $\aleph_{0}$-compact. Shelah: - It is consistent that $\mathcal{L}_{\omega \omega}(Q^{2}_{1})$ is not $\aleph_{0}$-compact. ## Relative expressive strength Recall ![[Generalized logics#^e77292]] - Clearly $\mathcal{L}_{\omega \omega}(Q_{1})\leq\mathcal{L}_{\omega \omega}(Q^{2}_{1})$. - $\mathcal{L}_{\omega \omega}(Q_{1})<\mathcal{L}_{\omega \omega}(Q^{2}_{1})$ Let $\varphi(E)$ say that $E$ is an equivalence relation with only uncountable equivalence classes. The formula $Q_{1}^{2}xy(E(x,y)\to x=y)$ says that there are uncountably many classes. But models of $\varphi(E)$ with countably many or uncountably many classes are $\mathcal{L}_{\omega \omega}(Q_{1})$-elementarily equivalent, as can be shown by EF-games. (MM1): - For every cardinal $\kappa>\omega$ and every $n<\omega$, $\mathcal{L}_{\kappa}^{n}<\mathcal{L}_{\kappa}^{n+1}$. [[Garavaglia - Relative strength of Malitz quantifiers.]] - Shelah: this is true even when considering $\leq_{PC}$ # Elementary equivalence There are EF games. See [[Badger - An Ehrenfeucht game for the multivariable quantifiers of Malitz and some applications.]] - Suppose $\mathfrak{A}$ and $\mathfrak{B}$ are $\kappa$-dense linear orders without endpoints of power $\kappa$. Then $\mathfrak{A}\equiv_{\underline{L}}\mathfrak{B}$. - # Questions - Is the theory of Souslin pseudo-trees complete (assuming it is consistent)? - Is the theory of non-Souslin Aronszajn pseudo-trees complete? - In particular - can special trees be characterised only using the $\leq$ symbol? Idea from Menachem: try to specialize a tree with a forcing that preserves the logic. E.g with Knaster property. Check the specializing poset of Abraham and Shelah. We do know: } - The class of $\omega_1$-like trees is PC in $\Lu$. - The class of $\omega_1$-like Aronszajn trees in PC in $\LM$. - The class of $\omega_1$-like Souslin trees is EC in $\LM$. can we show that the first two are strict? Perhaps take a non-Souslin Aronszajn tree, and inductively replace each node with many copies. Does this preserve its $\Lu / \LM$ theory? **Conjecture** The theory of $\omega$-splitting wide $\omega_{1}$-like trees with least element such that for every $x$, the tree $x\uparrow$ is not Souslin, is complete. Maybe there needs to be some more information regarding initial segments. Perhaps that every element has a successor is enough. Idea: Take two such trees $T_{1},T_{2}$. In existential moves we use a first order strategy. For $Q$ moves, if I chooses $X \subseteq T_{1}$ uncountable, we consider $X \cap a\uparrow$ and $X \cap a \downarrow$ for each $a$ already chosen. If $X \cap a \uparrow$ contains incompatible elements, when choosing $Y$ we make sure that if $b$ corresponded to $a$ then $Y\cap b\uparrow$ is an antichain. This can be done by assumption. Otherwise, $X\cap a\uparrow$ is countable (as there are no uncountable branches) and we duplicate it. Need lemma saying we can always duplicate chains. Also to take care of $X\cap a\downarrow$. # Bibliography - [[Magidor, Malitz - Compact extesions of L(Q) (part 1a)]] 1977 - [[Magidor, Malitz - Compactness and transfer for a fragment of L²]] 1977 - [[Badger - An Ehrenfeucht game for the multivariable quantifiers of Malitz and some applications.]] 1977 - [[Garavaglia - Relative strength of Malitz quantifiers.]] 1978 - [[Morgenstern - On amalgamations of languages with magidor-malitz quantifiers]] 1979 - [[Rubin, Shelah - On the expressibility hierarchy of magidor-malitz quantifiers]] 1983 - [[Rapp - ON THE EXPRESSIVE POWER OF THE LOGICS L(Qαn1,…,nm)]] 1984 - [[Abraham, Shelah - A Δ₂² well-order of the reals and incompactness of L (QMM)]] 1991 - [[Hayut - Magidor–Malitz reflection]] 2017 - [[Barwise, Feferman - Model-theoretic logics]] IV.5 pg. 153