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

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

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

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

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

h i 1 ( x 1 , , x k ( i ) ) h i 2 ( x 1 , , x k ( i ) ) {\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 M = O n , < , ( h i ) i < ω {\displaystyle M=\langle On,<,(h_{i})_{i<\omega }\rangle } which satisfies the following conditions:

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

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

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

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

References

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