Proof about $T_M$ is consistent for every Model $M$

Question

$T_M$ is consistent for every Model $M$. Where $T_M = \{ A \in FO(S) \mid M \vDash A\}$ and $S$ is a Signature.

So i tried proving.

Suppose $T_M$ is inconsistent. Then we have $M \vDash A$ and $M \vDash \lnot A$ for a particular A.

This would mean that $A$ and $\lnot A$ would be true at the same time. Which would lead to a contradiction.

Is my proof correct? Any suggestions ?

Answer 1

If the definition of consistency to be used is the semantical one:

there doesnt exist a sentence $A$ such that $M \vDash A$ and $M \vDash ¬A$,

the result is straightforward, as you suggested.

By the basic definition of semantics: $M \vDash ¬A$ iff $M \nvDash A$ (i.e. not $M \vDash A$).

Thus, in no model $M$ we can have $A$ and $¬A$ both true.