Would a theory with model $\{\}$ be consistent?

Question

I don't know much about model theory, but it was suggested in another math forum that a theory with model $\{ \}$ would be consistent. Is this correct?

Answer 1

If $T$ is inconsistent then it proves any statement of first order logic and so it proves also $\exists x \exists y. \neg (x=y) $ so the empty set is not a model.

Answer 2

This is really up to convention.

In most cases it is convenient to disallow the empty set from being a model of anything. It helps simplifying a lot of statements, for example: "Every finite partial order has a maximal element" is false if we allow the empty set to be a model of the theory of partial orders; another example would be the inference rule $\forall x\varphi\rightarrow\exists x\varphi$ (which in turns implies that the empty set is never a model of a consistent theory, since $\forall x(x=x)\rightarrow\exists x(x=x)$, so in particular there is some $x$ in the universe).

On the other hand, in cases like ordinals and the likes, it is convenient to have the empty set as a partial order, as it simplifies a lot of things when dealing with ordinals. And certainly there are other examples of this nature elsewhere.

Answer 3

If an axiom/inference set has any model at all, then it is as consistent as model theory. That includes having a model with an empty universe.

This comes from the fact that if $A$ is consistent, and $A \vdash B$, then $B$ is consistent. Here $A$ is your possibly empty model + model theory, and $B$ is your axioms/inferences.