Compactness Theorem / Set made of formulas of infinite size

Question

Could someone give me an example of an infinite countable set, where formulas contained in it are under the form of a conjunction or disjunction of infinite size, for which the compactness theorem doesn't hold?

Thanks a lot, David

Answer 1

The compactness theorem say "If your finitely consistent, then you are consistent". Let $ L=\{c_i :i\in \omega\}$ or the language containing countably many constant symbols (with equality). Consider the sentence in $L_{\omega_1\omega}$ which says:

$$\varphi \equiv (\forall x)\bigvee_{i\in \omega}(x = c_i) $$

Along with the collection of sentences $\Phi = \{ (\exists x)(x \neq c_i): i \in \omega\}$

Notice that every finite subcollection of $\Phi \cup \varphi$ is consistent (just choose $x$ to be some constant symbol that is not present in $\Phi_0 \subset \Phi$.

However, $\Phi \cup \varphi$ is inconsistent since $\Phi$ says "x is none of the constant symbols" and $\varphi$ says "every x is a constant symbol".