Model theoretic characterizations of large cardinals revisited

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W. Boney, S. Dimopoulos, V. Gitman, and M. Magidor, “Model theoretic characterizations of large cardinals revisited,” Manuscript.

The compactness of strong logics and set theory have been intertwined since Tarski [1] defined (weakly and strongly) compact cardinals in terms of the properties of the infinitary logic $\mathbb L_{\kappa, \omega}$. This established a strong connection between abstract model theory and the theory of large cardinals, which has also become apparent by the recent breakthroughs in the theory of Abstract Elementary Classes-a purely model theoretic framework-where certain important results depend on the existence of large cardinal axioms.

The interaction between the two fields is also strengthened by the first author's article [2], which establishes new characterizations of established large cardinal notions, expressed in model theoretic terms as compactness properties. This paper is a sequel to [2] and characterizes more large cardinals this way, namely Woodin, various virtual large cardinals and subtle cardinals. Notably, this has led us to generalise or define new concepts in abstract model theory, that may be useful outside the scope of the current exposition.

One of the main philosophical open questions about the large cardinal hierarchy is to explain the fact that it appears to be linear. Hence, apart from the intrinsic interest, we believe that the framework of compactness principles that we invoke offers a new insight into this problem.

We give a model-theoretic characterisation of Woodin cardinals by introducing a notion of Henkin models for arbitrary abstract logics. We characterise various virtual large cardinals, by introducing the notion of a pseudo-model for a theory. Finally, we characterise (a class version of) subtle cardinals as a natural weakening of Vopěnka's principle by showing that if ${\rm Ord}$ is subtle, then every abstract logic has a stationary class of weak compactness cardinals.

The conversations leading to this paper began at the conference "Accessible categories and their connections" organized by Andrew Brooke-Taylor, and we would like to thank Brooke-Taylor for organizing a fascinating meeting.

References

  1. 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.
  2. W. Boney, “Model theoretic characterizations of large cardinals,” Israel J. Math., vol. 236, no. 1, pp. 133–181, 2020. Available at: https://doi.org/10.1007/s11856-020-1971-6