Silver machine

"Silver Machine" is a 1972 song by the UK rock group Hawkwind. It was originally released as a single on 9 June 1972, reaching number three on the UK singles chart. The single was re-issued in 1976, again in 1978 reaching number 34 on the UK singles charts, and once again in 1983 reaching number 67 on the UK singles charts. The original mix has been re-released on the remasters version of In Search of Space.

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 by set theorist Jack Silver as a means of proving global square holds in the constructible universe. ==Preliminaries== An ordinal \alpha 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 ordinals \beta_1, \ldots , \beta_n,\gamma \in X such that \alpha is the unique ordinal for which \models_{L_\gamma} \phi(\alpha^\circ,\beta_1^\circ, \ldots , \beta^\circ_n) where for all \alpha we define \alpha^\circ to be the name for \alpha within L_\gamma . A structure \langle X, is eligible if and only if: X \subseteq On . \forall i, h_i is a partial function from X^{k(i)} to X, for some integer k(i). If N=\langle X, is an eligible structure then N_\lambda is defined to be as before but with all occurrences of X replaced with X \cap \lambda . Let N^1, N^2 be two eligible structures which have the same function k. Then we say N^1 \triangleleft N^2 if \forall i \in \omega and \forall x_1, \ldots , x_{k(i)} \in X^1 we have: h_i^1(x_1, \ldots , x_{k(i)}) \cong h_i^2(x_1, \ldots , x_{k(i)}) ==Silver machine== A Silver machine is an eligible structure of the form M=\langle On, which satisfies the following conditions: Condensation principle. If N \triangleleft M_\lambda then there is an \alpha such that N \cong M_\alpha . Finiteness principle. For each \lambda there is a finite set H \subseteq \lambda such that for any set A \subseteq \lambda +1 we have M_{\lambda+1}[A] \subseteq M_\lambda[(A \cap \lambda) \cup H] \cup \{\lambda\} Skolem property. If \alpha is *definable from the set X \subseteq On , then \alpha \in M[X] ; moreover there is an ordinal \lambda , uniformly \Sigma_1 definable from X \cup \{\alpha\} , such that \alpha \in M_\lambda[X] . ==References==

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