Constructible hierarchy: Difference between revisions
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== Definition == |
== Definition == |
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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, |
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, there are \(\max(\aleph_0, |Y|) = |Y|\) definable subsets of an infinite set \(Y\), and thus "almost all" subsets of an infinite set are not definable. The parameters \(\vec{z}\) are of importance when \(Y\) is uncountable, to ensure that there are more than \(\aleph_0\) definable subsets of \(Y\), but they do not have any effect when \(Y\supseteq\mathbb N\) is countable, since all elements of the natural numbers are definable. |
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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> |
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> |