I wanted to prove the compactness theorem, p 79 Just/Weese:
The (i) <= (ii) direction is not obvious to me. I thought I could prove it by showing not (i) implies not (ii) as follows:
Assume $T$ does not have a model. Then for every $\varphi \in T$, $T \models \varphi$ and by completeness, $T \vdash \varphi$. That is, there is a finite sequence of $\varphi_i$, $i = 1, \dots n$, with $\varphi = \varphi_n$. (Unfortunately, these $\varphi_i$ are not necessarily all in $T$.) Since $T$ doesn't have a model, we also have $T \models \lnot \varphi$ and hence there are $\bar{\varphi}_i$, $i= 1, \dots m$ with $\bar{\varphi}_m = \lnot \varphi$, again, unfortunately these don't have to be in $T$. My proof then ended as follows: Then $\{\varphi, \lnot \varphi\}$ is a finite subset of $T$ that doesn't have a model.
Unfortunately, I can't assume that $\{\varphi, \lnot \varphi\}$ is a subset of $T$. At first I thought a theory automatically contained all provable formulas, too but that's not the case since the book defines a separate set for that on the same page:
Can someone show me how to fix my proof? Thank you.