if $\kappa$ is a cardinal such that $V_\kappa \models \sf ZFC$, then $\kappa$ is called a worldly cardinal and this is not necessarily downward absolute [Hamkins], i.e. there is a model $V$ of $\sf ZFC$ which satisfies the existence of a worldly cardinal $\kappa$, and there is a subset $W$ of $V$ that is a transitive inner model of $\sf ZFC$ and yet doesn't satisfy $\kappa$ being a worldly cardinal.

Would that phenomena repeat itself upwardly beyond inaccessibility? I mean for example the worldly cardinak $\kappa$ such that $V_\kappa \models (\sf ZFC+ \text {there is an inaccessible})$ would that similarily be not downward absolute for that theory?


1 Answer 1


It depends on the large cardinal property.

For example, inaccessibility is downwards-absolute: if $V\models$ "$\kappa$ is inaccessible" and $W$ is an inner model of $V$ then $W\models$ "$\kappa$ is inaccessible." This is an immediate consequence of the fact that inaccessibility is $\Pi_1$-expressible. Many other large cardinal properties are also $\Pi_1$-expressible (e.g. weak compactness) and so again are downwards-absolute.

On the other hand, measurability is not downwards-absolute, and in fact there is a canonical way that this fails: regardless of $V$'s stance on the matter, $L\models$ "There are no measurable cardinals." In particular, no cardinal can be "downwards-absolutely measurable." This is Scott's theorem.

Broadly speaking, "small" large cardinal properties (below a measurable) tend to be downwards-absolute while "big" large cardinal properties tend not to be. However, as worldliness demonstrates, that heuristic doesn't hold universally.

  • $\begingroup$ Yes! Very nice! But I also want an output on what happens at the small large cardinals, are there some of them that are larger than the first inaccessible and yet not downward absolute, the example I gave is the first worldly cardinal $\kappa$ such that $V_\kappa \models \sf (ZFC + \text {there is an inaccessible })$, now I suspect that this cardinal is "SINGULAR", and so may not be downwardly absolute i.e. there is transitive inner model of ZFC + there is an inaccessible that doesn't see it as worldly cardinal "of the theory ZFC + there is an inaccessible". Is that correct? $\endgroup$ Jul 5, 2022 at 17:58
  • $\begingroup$ to put it in full: there is a model $V$ of (ZFC+ there is an inaccessible), that satisfy the existence of a cardinal $\kappa$ such that $V_\kappa \models \sf (ZFC + \text {there is an inaccessible})$, and there is a subset $W$ of $V$ that is a transitive inner model of $\sf ZFC + \text{ there is an inaccessible}$ such that $W$ doesn't satisfy existence of a worldly cardinal $\kappa$ such that $V_\kappa \models \sf (ZFC + \text {there is an inaccessible})$. You see that matters are all relative to ZFC+ existence of inaccessible. $\endgroup$ Jul 5, 2022 at 18:09
  • $\begingroup$ In nutshell what I'm trying to figure out is whether that even among the small large cardinals (larger than the first inaccessible) there are worldly cardinals that fail to be seen as such downwardly, and there may even be a proper class of those among the SMALL large cardinals. $\endgroup$ Jul 5, 2022 at 18:15
  • $\begingroup$ Nice answer! Weak compactness is not downwards-absolute though. In "Saturated Ideals", Kunen kills and then resurrects weak compactness of a cardinal. $\endgroup$ Jul 6, 2022 at 17:58

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