Inner models and large cardinals

§0. The ordinal numbers were Georg Cantor’s deepest contribution to mathematics. After the natural numbers 0, 1, . . . , n, . . . comes the first infinite ordinal number ù, followed by ù + 1, ù + 2, . . . , ù + ù, . . . and so forth. ù is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {í | í < α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ù is identified with the first infinite cardinal א0, similarly for the first uncountable ordinal number ù1 and the first uncountable cardinal number א1, etc. 2 We thus arrive at the following picture: