Toward a Theory of Second-Order Consequence

Notre Dame Journal of Formal Logic 40 (3):315-325 (1999)
Abstract
There is little doubt that a second-order axiomatization of Zermelo-Fraenkel set theory plus the axiom of choice (ZFC) is desirable. One advantage of such an axiomatization is that it permits us to express the principles underlying the first-order schemata of separation and replacement. Another is its almost-categoricity: M is a model of second-order ZFC if and only if it is isomorphic to a model of the form Vκ, ∈ ∩ (Vκ × Vκ) , for κ a strongly inaccessible ordinal.
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Gabriel Uzquiano (2006). The Price of Universality. Philosophical Studies 129 (1):137 - 169.

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