Why does the second smallest worldly cardinal believe the smallest worldly cardinal is worldly?

Question

It is known that the property of being a worldly cardinal is not absolute (a cardinal $\kappa$ is worldly iff $V_{\kappa} \vDash \textsf{ZFC}$). See here and here for more.

This said, it is the case that the existence of two worldly cardinals has strictly greater consistency strength than the existence of one worldly cardinal, and any proof I can see of this will require that if $\kappa_1 < \kappa_2$ are worldly cardinals, then $V_{\kappa_2}$ must believe $\kappa_1$ is worldly. (Easiest proof: show that $V_{\kappa_2}$ believes that $V_{\kappa_1} \vDash \textsf{ZFC}$.)

I wasn’t sure how to show that this was the case — I was thinking that maybe because $V_{\kappa_2}$ is transitive it must agree with $V$ on whether $V_{\kappa_1}$ models $\textsf{ZFC}$, but I’m not totally clear on/sure of this.

Answer 1

The truth of a first-order sentence $\varphi$ in a structure $\mathfrak{M}$ is absolute between $V$ (= reality) and sufficiently large transitive sets containing $\mathfrak{M}$. In particular, already $V_{\kappa+2}$ correctly computes the full first-order theory of $V_\kappa$ for each infinite ordinal $\kappa$, and even that is overkill. The key point is that truth of a first-order sentence in a structure is witnessed by an appropriate family of Skolem functions, and the relevant properties are all low-complexity.

Since $\kappa_2>\kappa_1+2$, the result follows.

(This is treated in more detail in Barwise's book Admissible sets and structures, if memory serves, but there really are no hidden surprises.)

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EDIT (addressing Francois' comment above): It's also true that worldliness relativizes downwards to (but not necessarily upwards from!) $L$, but this isn't actually related to the above. The issue is "worldliness-in-$L$" is a property of $V_\kappa^L$ (which for reasonably closed $\kappa$ is just $L_\kappa$), not of the genuine $V_\kappa$ ($=V_\kappa^V$ if you like) considered inside $L$. So the key fact is Godel's theorem that if $V_\kappa\models\mathsf{ZFC}$ then $L_\kappa\models\mathsf{ZFC}$. But this actually involves a nontrivial analysis of $L$. And the general failure of upwards absoluteness ruins any hope of an application the other way.

(To see that worldliness need not be absolute upwards from $L$, consider what happens if $\kappa$ is worldly in $L$ but countable in $V$ - which can happen, e.g. by forcing over $L$ with $Col(\omega,\kappa)$.)