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| {{about|the device|the [[Hawkwind]] song|Silver Machine}}
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| :''Not to be confused with [[Silver Machines]].'' | |
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| In [[set theory]], '''Silver machines''' are devices used for bypassing the use of [[fine structure theory (set theory)|fine structure]] in proofs of [[statements true in L|statements holding in L]]. They were invented by set theorist [[Jack Silver]] as a means of proving [[global square]] holds in the [[constructible universe]].
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| ==Preliminaries==
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| An [[Ordinal number|ordinal]] <math>\alpha</math> is ''*definable'' from a class of ordinals X if and only if there is a formula <math>\phi(\mu_0,\mu_1, \ldots ,\mu_n)</math> and <math>\exists \beta_1, \ldots , \beta_n,\gamma \in X </math> such that <math>\alpha</math> is the unique ordinal for which <math>\models_{L_\gamma} \phi(\alpha^\circ,\beta_1^\circ, \ldots , \beta^\circ_n)</math> where for all <math>\alpha</math> we define <math>\alpha^\circ</math> to be the name for <math>\alpha</math> within <math>L_\gamma</math>.
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| A structure <math>\langle X, < , (h_i)_{i<\omega} \rangle</math> is ''eligible'' if and only if:
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| # <math>X \subseteq On</math>.
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| # < is the ordering on On restricted to X.
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| # <math>\forall i, h_i</math> is a partial function from <math>X^{k(i)}</math> to X, for some integer k(i).
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| If <math>N=\langle X, < , (h_i)_{i<\omega} \rangle</math> is an eligible structure then <math>N_\lambda</math> is defined to be as before but with all occurrences of X replaced with <math>X \cap \lambda</math>.
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| Let <math>N^1, N^2</math> be two eligible structures which have the same function k. Then we say <math>N^1 \triangleleft N^2</math> if <math>\forall i \in \omega</math> and <math>\forall x_1, \ldots , x_{k(i)} \in X^1</math> we have:
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| <math>h_i^1(x_1, \ldots , x_{k(i)}) \cong h_i^2(x_1, \ldots , x_{k(i)})</math>
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| ==Silver machine==
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| A Silver machine is an eligible structure of the form <math>M=\langle On, < , (h_i)_{i<\omega} \rangle</math> which satisfies the following conditions:
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| ''Condensation principle.'' If <math>N \triangleleft M_\lambda</math> then there is an <math>\alpha</math> such that <math>N \cong M_\alpha</math>.
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| ''Finiteness principle.'' For each <math>\lambda</math> there is a finite set <math>H \subseteq \lambda</math> such that for any set <math>A \subseteq \lambda +1</math> we have
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| : <math>M_{\lambda+1}[A] \subseteq M_\lambda[(A \cap \lambda) \cup H] \cup \{\lambda\}</math>
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| ''Skolem property.'' If <math>\alpha</math> is *definable from the set <math>X \subseteq On</math>, then <math>\alpha \in M[X]</math>; moreover there is an ordinal <math>\lambda < [sup(X) \cup \alpha]^+</math>, uniformly <math>\Sigma_1</math> definable from <math>X \cup \{\alpha\}</math>, such that <math>\alpha \in M_\lambda[X]</math>.
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| ==References==
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| *{{cite book | title=Constructibility | chapter=Chapter IX | author=[[Keith Devlin|Keith J Devlin]] | id=<small>ISBN 0-387-13258-9</small> | year = 1984}} - Please note that errors have been found in some results in this book concerning Kripke Platek set theory.
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| [[Category:Constructible universe]]
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