Silver machine explained

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

\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,} _{1} \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 .

1. < 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 _{λ} is defined to be as before but with all occurrences of X replaced with X \cap λ .

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 ^{1 (x} _{1 ,} \ldots , x _{k(i)} ) \cong h ^{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 _{λ} then there is an \alpha such that N \cong M _{\alpha} .

Finiteness principle. For each

λ there is a finite set H \subseteq λ such that for any set A \subseteq λ +1 we haveM _{λ+1} [A] \subseteq M _{λ[(A} \cap λ ) \cup H] \cup \{ λ \}

Skolem property. If

\alpha is *definable from the set X \subseteq On , then \alpha \in M[X] ; moreover there is an ordinal λ < [sup(X) \cup \alpha] ^{+} , uniformly \Sigma _{1} definable from X \cup \{\alpha\} , such that \alpha \in M _{λ[X]} .

References

• Book: Constructibility . Chapter IX . Keith J Devlin . Keith Devlin . 0-387-13258-9 . 1984.