Silver machine

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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 $\langle X, < , (h_i)_{i<\omega} \rangle$ is eligible if and only if:

$X \subseteq On$ .
< is the ordering on On restricted to X.
$\forall i, h_i$ is a partial function from $X^{k(i)}$ to X, for some integer k(i).

If $N=\langle X, < , (h_i)_{i<\omega} \rangle$ 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 $M=\langle On, < , (h_i)_{i<\omega} \rangle$ 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 $\lambda < [sup(X) \cup \alpha]^+$ , uniformly $\Sigma_1$ definable from $X \cup \{\alpha\}$ , such that $\alpha \in M_\lambda[X]$ .

References

{{cite book | title=Constructibility | chapter=Chapter IX | author=Keith J Devlin | author-link=Keith Devlin | isbn= 0-387-13258-9 | year = 1984}}

Category:Constructible universe