Constructible hierarchy: Difference between revisions
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== Definition ==
Say a subset \(X\) of \(Y\) is definable if there are some \(z_0, z_1, \cdots, z_n \in Y\) and some formula \(\varphi\) in the language of set theory so that the elements of \(X\) are precisely the \(x\) so that \(Y\) satisfies \(\varphi(x, z_0, z_1, \cdots, z_n)\). For example, under the von Neumann definition of ordinal, the set of even numbers, the set of odd numbers, the set of prime numbers, the set of perfect squares greater than 17, and so on, are all definable. Using elementary cardinal arithmetic,
Like with the von Neumann hierarchy, the constructible hierarchy is built up in stages, denoted \(L_\alpha\).<ref>K. J. Devlin, "[https://core.ac.uk/download/pdf/30905237.pdf An introduction to the fine structure of the constructible hierarchy]" (1974)</ref>
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