The terms "local" and "global" when applied to large cardinal axioms seem to have a well understood intuitive meaning, although a formalized definition of them in (a metalanguage for)ZFC might be quite unwieldy. Given a large cardinal axiom, set theorists can immediately classify it as local or global. Loosely speaking, it is local iff you do not have to "look at" sets of arbitrarily high rank in the set theoretic hierarchy to determine whether the axiom does or does not hold. I have often wondered whether any global large cardinal can ever be "smaller" than some local large cardinal. I have never seen a statement that this is not possible. But is it? To be more specific: Is there any example of a local large cardinal axiom L, such that a sentence which states that the smallest supercompact cardinal is smaller than the smallest cardinal satisfying Lis not known to be inconsistent with ZFC? We always assume, of course, that the conjunction of both axioms holds.

3$\begingroup$ You'd have to have a "local" axiom which is nonetheless more complex than usual, namely more than $\Sigma_2$. $\endgroup$– Monroe EskewJun 15, 2014 at 18:58

$\begingroup$ Not related to your more specific question, but maybe still interesting: the existence of a strongly compact cardinal has strictly greater consistency strength than the existence of a Woodin cardinal; and in my view, the first is "local" and the second is "global." Of course, the relevant question is whether the least Woodin can be smaller than the least strongly comapct, which I don't know the answer to. $\endgroup$– Noah SchweberJun 15, 2014 at 18:58

$\begingroup$ At the same time, I think the global/local distinction might be misleading: measurability sounds like a local definition, but maybe the "right" way to think of it is as providing a global object. $\endgroup$– Noah SchweberJun 15, 2014 at 18:59

$\begingroup$ @NoahS To answer your question about least Woodin, Magidor proved from Con(ZFC+supercompact) that there can be a supercompact with no strong compacts below. But supercompact is always a limit of Woodins. $\endgroup$– Monroe EskewJun 15, 2014 at 19:06
1 Answer
I don't agree that it is difficult to formalize the local/global distinction, and indeed, I think that there is a largely agreedupon technical meaning for these notions.
Specifically, a property is locally verifiable if it can be verified inside any sufficiently large rank initial segment $V_\theta$ of the universe. So $\varphi(x)$ is such a property if it is equivalent to $\exists\theta\ V_\theta\models\psi(x)$, for some assertion $\psi$ (of any complexity). It is an excellent exercise to prove that the local properties are precisely the $\Sigma_2$ expressible assertions. To be truly local, both the property and its negation should be locally verifiable, which would make the property $\Delta_2$.
Update: I have posted further discussion of this issue on my blog at Local properties in set theory, including a proof of the equivalence I mentioned above.
Examples of local large cardinal properties would include inaccessible, Mahlo, weakly compact, Ramsey, measurable, Woodin, superstrong, almost huge, huge, rankintorank and many others. In particular, some local large cardinal properties are extremely high in the large cardinal hierarchy.
Global properties, in contrast, are not verifiable inside any particular $V_\theta$, and most of the commonly considered global large cardinal properties have complexity $\Pi_2$ or $\Pi_3$. For example, $\kappa$ is supercompact if and only if for every $\lambda$ it is $\lambda$supercompact. For any particular $\lambda$, the assertion that $\kappa$ is $\lambda$supercompact is local, since it can be verified inside $V_{\lambda+3}$ or so, and so it is $\Sigma_2$. The universal quantifier in front would seem to make supercompactness a $\Pi_3$ property, but actually it is $\Pi_2$, since the failure of the supercompactness of $\kappa$ is locally verifiable, as $\kappa$ fails to be supercompact just in case this is true inside some tall enough $V_\theta$ (thanks to Kostas for his comment on my blog about this, and he says it is mentioned also in Kanamori Ch. 5 after exercise 22.8). Other global large cardinal properties include: uplifting, strongly uplifting, unfoldable, strongly unfoldable, tall, strong, strongly compact, supercompact and many others.
Some of the global properties occur very low in the large cardinal hierarchy. For example, the uplifting cardinals are weaker than Mahlo in consistency strength, yet they are not local properties. So there are numerous instances where a global property has weaker consistency strength than a local property.
Meanwhile, however, every supercompact cardinal (and every strong cardinal) is $\Sigma_2$reflecting, and therefore if there is any typeL large cardinal above $\kappa$, then there would have to be many below $\kappa$. So the least supercompact cardinal can never be less than the least $L$cardinal, if $L$ is a local property.

1$\begingroup$ More obviously, I would imagine that things of the form "there exists a proper class of ..." are almost always global. $\;$ $\endgroup$– user5810Jun 16, 2014 at 3:22

$\begingroup$ Yes, that's right. Asserting a proper class of inaccessible cardinals (the universe axiom) is $\Pi_3$, as is asserting a proper class of any other local large cardinal. Asserting a proper class of supercompact cardinals appears to be $\Pi_5$. $\endgroup$ Jun 16, 2014 at 10:35

$\begingroup$ @ Joel: Many thanks for your answer. It seems then that the least cardinal satisfying any global property can never be less than the least cardinal satisfying any local property. $\endgroup$ Jun 16, 2014 at 19:48

$\begingroup$ You're welcome. But I think you've overstated things a bit. What I would agree with is that no $\Sigma_2$reflecting cardinal can be the least cardinal satisfying a local property, since the existence of such a cardinal would reflect below it. But not every global large cardinal is necessarily $\Sigma_2$reflecting. For example, unfoldable cardinals need not be reflecting, and in a recent paper, Gitik, Shanker, Cody and I proved that the least unfoldable cardinal can be the least weakly compact cardinal. See jdh.hamkins.org/leastweaklycompact. $\endgroup$ Jun 16, 2014 at 19:52

$\begingroup$ Incidentally, I think I'll make a blog post about the fact that $\Sigma_2$ is the same as the concept of local I mention in my answer, since it keeps coming up. Check out my web page in a few weeks. $\endgroup$ Jun 16, 2014 at 19:53