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