(First posted in [StackExchange](https://math.stackexchange.com/q/1912968/365891))


I am interested in the Löwenheim numbers associated with quantifiers that are prevalent in ordinary language use, such as "more" and "most". 


Definition: The *Löwenheim number* of a logic 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 Härtig 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 Väänänen 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 Väänänen, 2011](http://www.worldscientific.com/doi/abs/10.1142/S0219061311001018)).


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 interesting results?



Lowenheim numbers for ordinary language quantifiers
$\def\L#1{\mathcal L(#1)}\let\fii\varphi\let\eq\leftrightarrow\DeclareMathOperator\I{I}\DeclareMathOperator\most{Most}\DeclareMathOperator\more{More}$I don’t know what other quantifiers from ordinary language are there in the wild, but the three mentioned in the question have the same Löwenheim number:



> 
> **Theorem:** $\ell(\L\I)=\ell(\L\most)=\ell(\L\more)$.
> 
> 
> 


As already noted in the question itself, the inequalities
$$\ell(\L\I)\le\ell(\L\more),\qquad\ell(\L\most)\le\ell(\L\more)$$
follow immediately from the fact that the quantifiers $\I x$ and $\most x$ are definable from $\more$ using
$$\begin{align\*}
\I x\,(\fii x,\psi x)&\eq\neg\bigl(\more x\,(\fii x,\psi x)\lor\more x\,(\psi x,\fii x)\bigr),\\
\most x\,(\fii x,\psi x)&\eq\more x\,(\fii x\land\psi x,\fii x\land\neg\psi x).
\end{align\*}$$


For the
$$\ell(\L\more)\le\ell(\L\most)$$
inequality, note that $\more$ is *almost* definable from $\most$: if $(\fii x)^M$ or $(\psi x)^M$ is infinite, then
$$M\models\more x\,(\fii x,\psi x)\eq\most x\,(\fii x\lor\psi x,\neg\psi x).$$
Rather than trying to figure out if we can fix this definition to work for finite sets as well, let me use a different approach: we expand the models in such a way that we can define disjoint union.


Let $\alpha$ be an $\L\more$-sentence. Choose a fresh binary function symbol $p(x,y)$ and constants $a,b$. Let $\theta$ be the FO sentence stating that $p$ is a pairing function, and $a\ne b$. Let $\alpha^\*$ be the sentence $\alpha$ in which each subformula of the form
$$\more x\,(\fii x,\psi x)$$
is replaced with
$$\most z\,(\fii'z\lor\psi'z,\fii'z),$$
where
$$\begin{align\*}
\fii'z&\eq\exists x\,(z=p(x,a)\land\fii x),\\
\psi'z&\eq\exists x\,(z=p(x,b)\land\psi x).
\end{align\*}$$
Clearly,


* any infinite model in the signature of $\alpha$ can be expanded to a model of $\theta$, and
* if $M\models\theta$, then $M\models\alpha\eq\alpha^\*$.


Thus, for any infinite $\kappa$,



> 
> $\alpha$ has a model of cardinality $\kappa$ if and only if the $\L\most$-sentence $\theta\land\alpha^\*$ has a model of cardinality $\kappa$.
> 
> 
> 


Since $\alpha$ was an arbitrary $\L\more$-sentence, $\ell(\L\more)\le\ell(\L\most)$ follows.


For the remaining
$$\ell(\L\more)\le\ell(\L\I)$$
inequality, we can use a similar expansion approach. Let $\alpha$ be an $\L\more$-sentence, and let $<$ be a fresh binary relation symbol. We define $\L\I$-sentences $\alpha^\#$ and $\xi$ as follows:


$\alpha^\#$ is $\alpha$ with each subformula
$$\more x\,(\fii x,\psi x)$$
replaced with
$$\exists v\,(\I x\,(\psi x,x<v)\land\forall u\,(\I x\,(\fii x,x<u)\to v<u)),$$
that is: there is an initial segment of $<$ of the same size as $(\psi x)^M$ whose no initial segment has the same size as $(\fii x)^M$.


Let $\xi$ be the conjunction of the formulas


* $<$ is a (strict) linear order,
* $\forall\vec y\,\exists u\,\I x\,(\fii(x,\vec y),x<u)$,


for every subformula $\fii(x,\vec y)$ of $\alpha^\#$ with all variables shown (we only need subformulas involved in the translations of $\more$ as above, of course); in words, every definable set in sight has the same size as an initial segment. Again, it is easy to see that


* every infinite model $M$ in the signature of $\alpha$ has an expansion to a model of $\xi$: indeed, if $|M|=\lambda$, then any well order $<$ on the domain of $M$ of type at least $\lambda+1$ will do;
* if $M$ is a model of $\xi$, then $M\models\alpha\eq\alpha^\#$.


Consequently,



> 
> $\alpha$ has a model of cardinality $\kappa$ if and only if the $\L\I$-sentence $\xi\land\alpha^\#$ has a model of cardinality $\kappa$
> 
> 
> 


for infinite $\kappa$.


