Silver Machine
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High Quality Content by WIKIPEDIA articles! In set theory, Silver machines are devices used for bypassing the use of fine structure in proofs of statements holding in L. They were invented as a means of proving global square holds in the constructible universe. An ordinal is definable from a class of ordinals X if and only if there is a formula phi(mu_0,mu_1, ldots ,mu_n) and exists beta_1, ldots , beta_n,gamma in X such that is the unique ordinal for which models_{L_gamma} phi(alpha^circ,beta_1^circ, ldots , beta^circ_n) where for all we define alpha^circ to be the name for within L . A struc...
High Quality Content by WIKIPEDIA articles! In set theory, Silver machines are devices used for bypassing the use of fine structure in proofs of statements holding in L. They were invented as a means of proving global square holds in the constructible universe. An ordinal is definable from a class of ordinals X if and only if there is a formula phi(mu_0,mu_1, ldots ,mu_n) and exists beta_1, ldots , beta_n,gamma in X such that is the unique ordinal for which models_{L_gamma} phi(alpha^circ,beta_1^circ, ldots , beta^circ_n) where for all we define alpha^circ to be the name for within L . A structure langle X, , (h_i)_{iomega} rangle is eligible if and only if: 1. X subseteq On. 2. is the ordering on On restricted to X. 3. forall i, h_i is a partial function from Xk(i) to X, for some integer k(i). If N=langle X, , (h_i)_{iomega} rangle is an eligible structure then N is defined to be as before but with all occurrences of X replaced with X cap lambda.
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