Silver machine

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.

This article is about the kind of mathematical object. For the Hawkwind song, see Silver Machine. For the Vapors song, see Silver Machines.

Preliminaries

An ordinal ${\displaystyle \alpha }$ is *definable from a class of ordinals X if and only if there is a formula ${\displaystyle \phi (\mu _{0},\mu _{1},\ldots ,\mu _{n})}$ and ${\displaystyle \exists \beta _{1},\ldots ,\beta _{n},\gamma \in X}$ such that ${\displaystyle \alpha }$ is the unique ordinal for which ${\displaystyle \models _{L_{\gamma }}\phi (\alpha ^{\circ },\beta _{1}^{\circ },\ldots ,\beta _{n}^{\circ })}$ where for all ${\displaystyle \alpha }$ we define ${\displaystyle \alpha ^{\circ }}$ to be the name for ${\displaystyle \alpha }$ within ${\displaystyle L_{\gamma }}$.

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

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

If ${\displaystyle N=\langle X,<,(h_{i})_{i<\omega }\rangle }$ is an eligible structure then ${\displaystyle N_{\lambda }}$ is defined to be as before but with all occurrences of X replaced with ${\displaystyle X\cap \lambda }$.

Let ${\displaystyle N^{1},N^{2}}$ be two eligible structures which have the same function k. Then we say ${\displaystyle N^{1}\triangleleft N^{2}}$ if ${\displaystyle \forall i\in \omega }$ and ${\displaystyle \forall x_{1},\ldots ,x_{k(i)}\in X^{1}}$ we have:

${\displaystyle h_{i}^{1}(x_{1},\ldots ,x_{k(i)})\cong h_{i}^{2}(x_{1},\ldots ,x_{k(i)})}$

Silver machine

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

Condensation principle. If ${\displaystyle N\triangleleft M_{\lambda }}$ then there is an ${\displaystyle \alpha }$ such that ${\displaystyle N\cong M_{\alpha }}$.

Finiteness principle. For each ${\displaystyle \lambda }$ there is a finite set ${\displaystyle H\subseteq \lambda }$ such that for any set ${\displaystyle A\subseteq \lambda +1}$ we have

${\displaystyle M_{\lambda +1}[A]\subseteq M_{\lambda }[(A\cap \lambda )\cup H]\cup \{\lambda \}}$

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

References

• Keith J Devlin (1984). "Chapter IX". Constructibility. ISBN 0-387-13258-9.