Silver machine

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In set theory, Silver machines are devices used for bypassing fine structure theory in proofs of statements holding in L. They were invented as a means of proving global square holds in the constructible universe.

[edit] Preliminaries

An ordinal α 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 α is the unique ordinal for which \models_{L_\gamma} \phi(\alpha^\circ,\beta_1^\circ, \ldots , \beta^\circ_n) where for all α we define \alpha^\circ to be the name for α within Lγ.

A structure \langle X, < , (h_i)_{i<\omega} \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)_{i<\omega} \rangle is an eligible structure then Nλ is defined to be as before but with all occurrences of X replaced with X \cap \lambda.

Let N1,N2 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)})

[edit] Silver machine

A Silver machine is an eligible structure of the form M=\langle On, < , (h_i)_{i<\omega} \rangle which satisfies the following conditions:

Condensation principle. If N \triangleleft M_\lambda then there is an α such that N \cong M_\alpha.

Finiteness principle. For each λ 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 α is *definable from the set X \subseteq On, then \alpha \in M[X]; moreover there is an ordinal \lambda < [sup(X) \cup \alpha]^+, uniformly Σ1 definable from X \cup \{\alpha\}, such that \alpha \in M_\lambda[X].

[edit] References

  • Keith J Devlin (1984). “Chapter IX”, Constructibility. ISBN 0387132589. - Please note that errors have been found in some results in this book concerning Kripke Platek set theory.
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