Any injective map from a finite set to itself is surjective. Ax's Theorem extends this to algebraic varieties and regular maps. Gromov invented sofic groups as a way to extend to this result to cellular automata and other settings. We'll re-prove his results via sofic entropy theory. Similarly we use sofic mean dimension to prove non-embeddability results in topological dynamics and $l^p$ dimension to prove non-embeddability results in representation theory.