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](http://jdh.hamkins.org/worldly-cardinals-are-not-always-downwards-absolute/) and [here](https://mathoverflow.net/questions/130019/forcing-mildly-over-a-worldly-cardinal) 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.



Why does the second smallest worldly cardinal believe the smallest worldly cardinal is worldly?
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 $Col(\omega,\kappa)$.)


