Large Cardinals As Principles of Structural Reflection
LARGE CARDINALS AS PRINCIPLES OF STRUCTURAL REFLECTION JOAN BAGARIA Abstract. After discussing the limitations inherent to all set- theoretic reflection principles akin to those studied by A. L´evy et. al. in the 1960’s, we introduce new principles of reflection based on the general notion of Structural Reflection and argue that they are in strong agreement with the conception of reflec- tion implicit in Cantor’s original idea of the unknowability of the Absolute, which was subsequently developed in the works of Ack- ermann, L´evy, G¨odel, Reinhardt, and others. We then present a comprehensive survey of results showing that different forms of the new principles of Structural Reflection are equivalent to well-known large cardinals axioms covering all regions of the large-cardinal hi- erarchy, thereby justifying the naturalness of the latter. In the framework of Zermelo-Fraenkel (ZF) set theory1 the universe V of all sets is usually represented as a stratified cumulative hierarchy V V of sets indexed by the ordinal numbers. Namely, = Sα∈OR α, where V0 = ∅ Vα+1 = P(Vα), namely the set of all subsets of Vα, and Vλ = Sα<λ Vα, if λ is a limit ordinal. This view of the set-theoretic universe justifies a posteriori the ZF axioms. For not only all ZF axioms are true in V , but they are also necessary to build V . Indeed, the axioms of Extensionality, pairing, Union, Power-set, and Separation are used to define the set-theoretic operation given by G(x)= [{P(y): ∃z(hz,yi ∈ x)}. The axiom of Replacement is then needed to prove by transfinite recur- arXiv:2107.01580v1 [math.LO] 4 Jul 2021 sion that the operation V on the ordinals given by V (α)= G(V ↾ α) is well-defined and unique.
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