Relative interpretations In Kunen's Set theory (2011), he says that there is a finitistic proof that $ $Con$(ZF^-)\implies $Con$(ZF)$. He also mentions elsewhere that if $\Theta$ is at least as strong as finitistic reasoning and we have a finitistic proof of $ $Con$(\Lambda)\implies $Con$(\Gamma)$ then: $\Theta \vdash $Con$(\Lambda)\implies $Con$(\Gamma)$ Now my question is when he says there is a finitistic proof of $ $Con$(ZF^-)\implies $Con$(ZF)$, does he mean that $\Theta \vdash $Con$(ZF^-)\implies $Con$(ZF)$ for any $\Theta$ at least as strong as finitistic reasoning (like $BST^-$)? My confusion is this: Understanding an Easy Relative Consistency Proof Which leads me to believe that the $\Theta$ needs to be stronger than $BST^-$ in this particular case. If this is the case, then what's the reason Kunen is not more clear on this point, ie why doesn't he explicitly say: "We have shown that $ZF^- \vdash $Con$(ZF^-)\implies $Con$(ZF)$" Any help is appreciated, thanks!

Look at the third and fourth paragraphs of the answer you linked to. There Andres Caicedo gives a syntactic argument, which looks finitistic to me. What exactly leads you to believe that you need something stronger than finitistic reasoning for this argument?