What are the smallest possible theories?

Question

Im wondering how we could define a general form for the smallest possible theory in some formal language. In other words, if we have the formal language of first order logic, what is the smallest set of sentences which is a valid theory, i.e. if $\phi$ is a sentence, and $T$ the set of sentences comprising the theory, $\phi \in T \longleftrightarrow T \vdash \phi$, what is the smallest $T$? Im particularly interested in what the smallest theory in first order arithmetic would be. Im thinking that it might be 0=0, or any single sentence which is derivable from the empty set of hypotheses, $T=\phi=\{ \phi | \vdash \phi \}, $however im not certain at all this this is correct and would appreciate a clarification. Thanks guys!

Answer 1

Even in the language of equality there are infinitely many sentences $\phi$ such that $\vdash \phi$ (for instance, $\forall x (x=x)$ and similar). So any theory $T$ such that $\phi \in T$ whenever $T \vdash \phi$ must have infinite size.

If you mean smallest in the sense of containment rather than cardinality, then $T = \{\phi\ |\ \vdash \phi\}$ is in fact what you are looking for (since such $\phi$ are theorems of every theory).