I am interested in the Lowenheim numbers associated with quantifiers that are prevalent in ordinary language use, such as "more" and "most".
Definition: The Lowenheim number of L, $\ell$(L), is the least cardinal $\mu$ such that any satisfiable sentence in L has a model of cardinality less or equal to $\mu$ if such exists. Otherwise, $\ell$(L)=$\infty$.
For a logic L and a quantifier $Q$, let L($Q$) be the logic L with $Q$ added to its language. Let $\mathcal{L}$ denote standard first order logic.
So, for instance: Let $Q_\alpha$ be the unary monadic quantifier ``there are at least $\aleph_\alpha$ many''. We have $\ell$($\mathcal{L}(Q_\alpha)$)=$\aleph_\alpha$ for each ordinal $\alpha$.
Another example: Let the Hartig quantifier $I$ be the binary monadic quantifier stating equal cardinality of sets: $I$ is a binary monadic quantifier such that $M\models Ix(\varphi x,\psi x)$ iff $|(\varphi x)^M|=|(\psi x)^M|$. $\ell(\mathcal{L}(I))$ is very high, and is independent of ZFC. $\ell(\mathcal{L}(I))$ is a fixed point of the function $\alpha\mapsto\aleph_\alpha$, and further, Magidor and Vaananen showed that it is consistent with ZFC both that $\ell(\mathcal{L}(I))$ is under the first weakly inaccessible cardinal and that it is above the measurable cardinal (Magidor and Vaananen, 2011).
Are there known results for "most" and "more"? I could only find the following bounds:
Let $Most$ be the binary monadic quantifier such that $M\models Most\ x (\varphi x, \psi x)$ iff $|(\varphi x)^M\backslash (\psi x)^M|<|(\varphi x)^M\cap(\psi x)^M|$. Then I've found the following lower bound: $\ell(\mathcal{L}(Most))\geq\aleph_\omega$.
Let $More$ be the binary monadic quantifier such that $M\models More\ x (\varphi x, \psi x)$ iff $|(\varphi x)^M|>|(\psi x)^M|$. Then we have both $\ell(\mathcal{L}(More))\geq\ell(\mathcal{L}(I))$ and $\ell(\mathcal{L}(More))\geq\ell(\mathcal{L}(Most))$ (due to considerations of expressive power).
Are there better results? And are there other quantifiers from ordinary language that have better results?
