How strong is "ZFC is 'very consistent'"?

Question

Suppose we add a predicate "$x$ is very consistent" to $\textsf{ZFC}$, and we add the following axioms:

1. $\textsf{ZFC}$ is very consistent
2. If some theory $T$ is very consistent, then it is consistent
3. If some theory $T$ is very consistent, then $T + T\text{ is consistent}$ is also very consistent
4. Given a set $\mathcal{T}$ of very consistent theories, $\textsf{ZFC} + (\forall T \in \mathcal{T}. T \text{ is consistent})$ is very consistent

How strong is this theory? Is there some large cardinal axiom which proves its consistency?

Answer 1

Here's a quick argument that any reasonable way of making your notion precise gets something no stronger (and I suspect vastly weaker) than the theory $T=$ $\mathsf{ZFC}$ + "There is a worldly cardinal."

Within $T$, we'll interpret "very consistent" as "has a transitive model." Conditions 1-3 are trivial (once we restrict condition 3 to theories extending $\mathsf{ZFC}$, say, to address the issue Alex Kruckman pointed out); in particular, note that every transitive model is an $\omega$-model and so does not believe false inconsistency statements.

For 4, letting $\kappa$ be the smallest inaccessible cardinal note (reasoning within $T$) that $V_\kappa$ is itself a transitive model of $\mathsf{ZFC}$ which correctly sees transitive models of all set theories which have transitive models: if $A$ is a transitive model of set theory, then by downward Lowenheim-Skolem + Mostowski collapse $A$ has a countable transitive model, and any such model is a fortiori an element of $V_\kappa$.