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[edit]

An ordinal is *definable from a class of ordinals X if and only if there is a formula and such that is the unique ordinal for which where for all we define to be the name for within .

A structure is eligible if and only if:

  1. .
  2. < is the ordering on On restricted to X.
  3. is a partial function from to X, for some integer k(i).

If is an eligible structure then is defined to be as before but with all occurrences of X replaced with .

Let be two eligible structures which have the same function k. Then we say if and we have:

Silver machine[edit]

A Silver machine is an eligible structure of the form which satisfies the following conditions:

Condensation principle. If then there is an such that .

Finiteness principle. For each there is a finite set such that for any set we have

Skolem property. If is *definable from the set , then ; moreover there is an ordinal , uniformly definable from , such that .

References[edit]