# Models of infinite cardinal and compactness

I'm stuck with this problem:

$L$ is first-order language with identity and $L_q$ a language obtained by adding to $L$ the quantifier $Q$. If $P$ is a formula and $x$ a variable, $QxP$ is a formula of $L_q$. If $A$ is a $L_q$-structure, then $A\vDash QxP$ if and only if $\#\left(\{a \in |A| : A\vDash P[a]\}\right)>\aleph_0$.

I have to show that there exists $T\subset\operatorname{For}(L_q)$ of cardinal superior to $\aleph_0$ such that T does not have a model and every finite subset has a model

Any ideas??

thanks!

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And the quantifier $Q$ is...? – Asaf Karagila Jan 22 '13 at 19:22
Use negation and Q to say the universe is countable. Then for uncountably many constant symbols, say they're all distinct. – Andreas Blass Jan 22 '13 at 19:52
@AndreasBlass: you should post this as an answer... – tomasz Jan 23 '13 at 15:45

## 1 Answer

To get this off the unanswered list:

Use negation and Q to say the universe is countable. Then for uncountably many constant symbols, say they're all distinct. – Andreas Blass Jan 22 at 19:52

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