I've reading about consistent theories and I had a question. As union of arbitrary functions does not necessarily give a function, I wonder if union of consistent theories is consistent. My intuition says that it is not true at all, so, I tried to construct a counterexample but before it, look at the following reasoning:
Take the sentence $\varphi: \forall x \forall y(xRy→yRx)$. I realized that $\langle \mathbb{N}, = \rangle$ is a model for $\Sigma=\{ \varphi \}$, meaning that $\Sigma$ is satisfiable and then, $\Sigma$ must be consistent.
Is this reasoning true at all? ¡Thanks in advance!