Jensen hierarchy

In set theory, a mathematical discipline, the Jensen hierarchy or J-hierarchy is a modification of Gödel's constructible hierarchy, L, that circumvents certain technical difficulties that exist in the constructible hierarchy. The J-Hierarchy figures prominently in fine structure theory, a field pioneered by Ronald Jensen, for whom the Jensen hierarchy is named.

Definition

As in the definition of L, let Def(X) be the collection of sets definable with parameters over X:

The constructible hierarchy, is defined by transfinite recursion. In particular, at successor ordinals, .

The difficulty with this construction is that each of the levels is not closed under the formation of unordered pairs; for a given , the set will not be an element of , since it is not a subset of .

However, does have the desirable property of being closed under 0 separation.

Jensen's modified hierarchy retains this property and the slightly weaker condition that , but is also closed under pairing. The key technique is to encode hereditarily definable sets over by codes; then will contain all sets whose codes are in .

Like , is defined recursively. For each ordinal , we define to be a universal predicate for . We encode hereditarily definable sets as , with . Then set and finally, .

Properties

Each sublevel Jα, n is transitive and contains all ordinals less than or equal to αω + n. The sequence of sublevels is strictly increasing in n, since a Σm predicate is also Σn for any n > m. The levels Jα will thus be transitive and strictly increasing as well, and are also closed under pairing, -comprehension and transitive closure. Moreover, they have the property that

as desired.

The levels and sublevels are themselves Σ1 uniformly definable (i.e. the definition of Jα, n in Jβ does not depend on β), and have a uniform Σ1 well-ordering. Finally, the levels of the Jensen hierarchy satisfy a condensation lemma much like the levels of Gödel's original hierarchy.

Rudimentary functions

A rudimentary function is a function that can be obtained from the following operations:

  • F(x1, x2, ...) = xi is rudimentary
  • F(x1, x2, ...) = {xi, xj} is rudimentary
  • F(x1, x2, ...) = xixj is rudimentary
  • Any composition of rudimentary functions is rudimentary
  • zyG(z, x1, x2, ...) is rudimentary

For any set M let rud(M) be the smallest set containing M∪{M} closed under the rudimentary operations. Then the Jensen hierarchy satisfies Jα+1 = rud(Jα).

References

  • Sy Friedman (2000) Fine Structure and Class Forcing, Walter de Gruyter, ISBN 3-11-016777-8
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