# Does the logical system with the “there exist uncountably many”-quantifier satisfy a variant of upwards Löwenheim-Skolem

Let $$\mathcal L_Q$$ be the logical system that includes first order logic together with the quantifier $$Q$$ which is defined as follows:

For an interpretation $$\mathfrak I=(\mathfrak A, \beta)=((A,\mathfrak a), \beta)$$,

$$\mathfrak I\models Qx\varphi$$ iff there exist uncountably many $$a\in A$$ so that $$\mathfrak I \frac a x\models \varphi$$.

My question is the following:

Let $$\Phi$$ be a set of formulas satisfied by a model of cardinality $$\geq \aleph_1$$, is $$\Phi$$ satisfiable by a model of cardinality $$>\kappa$$ for any cardinality $$\kappa$$?

If the answer to the above is negative, does the weaker version where $$\kappa=\mathfrak c$$ have a positive answer?

• Have you tried anything so far? Some ideas to get started: 1. Does Lindström's theorem apply? 2. Can we use Henkin models or Herbrand models to prove this? – Johannes Kloos Oct 21 at 20:55
• @JohannesKloos Neither of those ideas seems relevant. Lindstrom's theorem shows that there can't be a compact logic with the downwards Lowenheim-Skolem property, but $\mathcal{L}_Q$ lacks the compactness property and the OP is asking about the upwards Lowenheim-Skolem property. Henkin models have to do with second-order logic, I don't see how they'd be relevant to $\mathcal{L}_Q$. And Herbrand models are as small as possible - they're exactly the opposite of what the OP would be looking for. – Noah Schweber Oct 22 at 0:41

No, it doesn't.

Consider $$\omega_1$$, thought of as a linear order. This satisfies - in addition to the usual axioms of linear order - the axioms "The universe is uncountable" and "For each $$x$$, the set $$\{y: y is countable." But every uncountable linear order with the countable predecessor property has cardinality $$\aleph_1$$.

What about $$\aleph_2$$?

Well, we can use a variant of the same trick! Consider the structure defined as follows:

• We start with $$\omega_2$$ as a linear order.

• Now we attach to each $$x\in\omega_2$$ a copy of $$\omega_1$$ as a linear order, together with a surjection from that copy to the initial segment $$\{y: y of $$\omega_2$$. We now have a two-sorted structure, one of the sorts being the $$\omega_2$$-part and the other sort being the disjoint union of a bunch of $$\omega_1$$s, and some stuff connecting them appropriately.

Any $$\mathcal{L}_Q$$-elementarily equivalent structure must have, in place of the $$\omega_2$$-part, a linear order with the $$\le\aleph_1$$-predecessor property by the argument above for $$\omega_1$$. But this rules out models of cardinality $$\aleph_3$$ or above.

Similarly we can whip up a theory with models of size $$\aleph_4$$ but no models of size $$\ge\aleph_5$$, and so on. And in fact we don't need theories: all of this can be accomplished by individual $$\mathcal{L}_Q$$-sentences.

The first point where things become a bit slippery is $$\aleph_\omega$$; if I recall correctly we can continue well past this, but I don't immediately see the details.