Cardinal correct extendible cardinals

There is little argument out there that first-order logic is the best candidate for the formal system underlying mathematical reasoning. No formal system exists entirely outside of the mathematics it tries to formalize, but first-order logic uses a very small fragment of the background set theoretic universe - only that recursion works on the natural numbers. It also has many almost magical properties such as the Compactness theorem and the downward and upward Löwenheim-Skolem theorems. In fact, by Lindström's theorem, first-order logic is unique among the logics satisfying both the Compactness and the downward Löwenheim-Skolem theorems [1]. The other classical logics studied by set theorists, such as infinitary logics or second-order logic, are much too strong to be compact. But there is a more general notion of compactness that they can satisfy, and surprisingly whether these strong logics have the generalized compactness depends on which large cardinals exist in the universe. A cardinal $\kappa$ is a strong compactness cardinal for a logic $\mathcal L$ if every $\lt\kappa$-satisfiable $\mathcal L$-theory is satisfiable. Tarski was the first to make a connection between strong compactness cardinals and large cardinals when he showed that a cardinal $\kappa$ is a strong compactness cardinal for the infinitary logic $\mathbb L_{\kappa,\kappa}$ if and only if $\kappa$ is strongly compact [2]. Magidor showed that the least strong compactness cardinal for second-order logic $\mathbb L^2$ is the least extendible cardinal (least is important here because of the observation that if $\kappa$ is a strong compactness cardinal for $\mathcal L$, then every $\gamma>\kappa$ is also a strong compactness cardinal) [3]. Magidor also showed that a cardinal $\kappa$ is a strong compactness cardinal for the infinitary logic $\mathbb L_{\omega_1,\omega_1}$ or the well-foundedness logic $\mathbb L(Q^{WF})$ (having the quatifier for identifying well-founded definable relations) if and only if $\kappa$ is $\omega_1$-strongly compact (every $\kappa$-complete filter can be extended to a countably complete ultrafilter). The most intractible logic in terms of characterizing its strong compactness cardinals has proven to be the equicardinality logic $\mathbb L(I)$ (having the quantifier for identifying whether two definable collections have the same cardinality). Very little was known until recently about the strong compactness cardinals for $\mathbb L(I)$. In a joint work with Jonathan Osinski, we introduce a new large cardinal notion that is naturally connected to the niceness properties of $\mathbb L(I)$ [4] (see this post).

Goldberg and Thei recently showed that (assuming the existence of a strongly compact cardinal) it is inconsistent to have an elementary embedding $j:V\to M$, where $M$ is an inner model that is correct about the cardinals (see this talk). But let's consider a set version of this inconsistent large cardinal property. We define that a cardinal $\kappa$ is cardinal correct extendible if for every $\alpha>\kappa$, there is an elementary embedding $j:V_\alpha\to M$ with $\text{crit}(j)=\kappa$, $j(\kappa)>\alpha$ and $M$ correct about cardinals. For example, extendible cardinals are clearly cardinal correct extendible. It is not difficult to show that a cardinal correct extendible $\kappa$ is strongly compact [4]. It is at least consistent that the least cardinal correct extendible is above the least supercompact cardinal (this result was pointed out to us by Alejandro Poveda, but we give a generalized argument in [4]). So, in particular, supercompact cardinals are not always cardinal correct extendible. But there are many interesting and seemingly difficult questions about this new large cardinal notion. While the assumption that $j(\kappa)>\alpha$ comes for free in the case of extendible cardinals, we don't know whether this is the case with cardinal correct extendibles. So we make the following two other definitions in case they don't all turn out to be equivalent. We define that a cardinal $\kappa$ is weakly cardinal correct extendible if we remove the assumption that $j(\kappa)>\alpha$. A weakly cardinal correct extendible $\kappa$ is either strongly compact or $j(\kappa)$ is inaccessible and $V_{j(\kappa)}$ is a model with a strongly compact cardinal. If $\kappa$ is weakly cardinal correct extendible, but not cardinal correct extendible, then there must be some ordinal $\gamma$ such that for every $\alpha>\kappa$ and embedding $j:V_\alpha\to M$ with $\text{crit}(j)=\kappa$ and $M$ correct about cardinals, $j^n(\kappa)\lt\gamma$ for every $n\lt\omega$. It is not clear whether this situation is even consistent. Finally, we define that $\kappa$ is cardinal correct extendible pushing up some $\delta\geq\kappa$ if for every $\alpha>\delta$, there is $j:V_\alpha\to M$ with $\text{crit}(j)=\kappa$, $j(\delta)>\alpha$ and $M$ correct about cardinals. In other words, the embeddings may not map $\kappa$ arbitrarily high, but they do map $\delta$ arbitrarily high. This last version of the large cardinal is particularly important in the context of the properites of the logic $\mathbb L(I)$.

Here are some other questions we have about cardinal correct extendibles.

Question: Is every cardinal correct extendible actually extendible? (unlikely)

Question: Are cardinal correct extendibles and extendibles equiconsistent? (possible)

Question: Can the least cardinal correct extendible be the least measurable? (possible)

So how did this large cardinal notion come up in the first place? With Jonathan Osinski, we considered upward Löwenheim-Skolem properties for general logics. The following weak version of the upward Löwenheim-Skolem theorem holds for every logic. The Hanf number of a logic $\mathcal L$ is the least cardinal $\delta$ such that for every $\mathcal L$-sentence $\varphi$ if there is a model $M\models\varphi$ of size at least $\delta$, then there are models of $\varphi$ of arbitrarily large cardinalities. It is a theorem of ${\rm ZFC}$ that every logic has a Hanf number. Galeotti, Khomskii and Väänänen strengthened the notion of the Hanf number by adding the requirement that the arbitarily large models of $\varphi$ had $M$ as a (not necessarily elementary) substructure and called the stronger notion the upward Löwenheim-Skolem number [5]. They showed that if $\mathbb L^2$ had an upward Löwenheim-Skolem number, then there was a partially extendible cardinal, thus showing that unlike the Hanf number, the existence of the upward Löwenheim-Skolem number could have large cardinal strength. We also considered the full generalized version of the upward Löwenheim-Skolem theorem by defining that a strong upward Löwenheim-Skolem number of a logic $\mathcal L$ is the least cardinal $\delta$ such that if $M$ is a model of size at least $\delta$, then it has elementary superstructures of arbitrarily large cardinalities. We were able to characterize the upward Löwenheim-Skolem and the strong upward Löwenheim-Skolem numbers of several classical logics such as second-order logic, well-foundedness logic, infinitary logics, and sort logics. The hardest turned out, not suprisingly, to be once again the equicardinality logic. We showed that if there is a cardinal correct extendible cardinal $\kappa$ pushing up some $\delta\geq\kappa$, then the strong upward Löwenheim-Skolem number for $\mathbb L(I)$ exists and is bounded by $\delta$. We also showed that if the upward Löwenheim-Skolem number for $\mathbb L(I)$ exists, then there is a cardinal $\kappa$ and some $\delta\geq\kappa$ such that $\kappa$ is cardinal correct extendible pushing up $\delta$. Finally, we showed that a strong compactness cardinal for $\mathbb L(I)$ is a mixture of cardinal correct extendible and $\delta$-strongly compact.

Theorem: For every $\gamma > \delta$ there is $\alpha \geq \gamma$, a transitive set $M$ and an elementary embedding $j: V_\alpha \rightarrow M$ such that $M$ is cardinal correct, $\text{crit}(j) \leq \delta$ and there exists $d \in M$ such that $j''\gamma \subseteq d$ and $M \models |d| \lt j(\delta)$.

References

1. C. C. Chang and H. J. Keisler, Model theory, Third., vol. 73. North-Holland Publishing Co., Amsterdam, 1990, p. xvi+650.
2. A. Tarski, “Some problems and results relevant to the foundations of set theory,” in Logic, Methodology and Philosophy of Science, Proceedings of the 1960 International Congress, 1962.
3. M. Magidor, “On the role of supercompact and extendible cardinals in logic,” Israel Journal of Mathematics, vol. 10, pp. 147–157, 1971.
4. V. Gitman and J. Osinski, “Upward Löwenheim-Skolem numbers for abstract logics,” Manuscript, 2023.
5. L. Galeotti, Y. Khomskii, and J. Väänänen, “Bounded Symbiosis and Upwards Reflection,” Manuscript.