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. Rudimentary functions describe a method for iterating through the Jensen hierarchy.

Definition

As in the definition of L, let Def(X) be the collection of sets definable with parameters over X: The constructible hierarchy, L 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 {x,y} will not be an element of , since it is not a subset of L_\alpha. However, L_\alpha does have the desirable property of being closed under Σ0 separation. Jensen's modification of the L 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 J_\alpha by codes; then will contain all sets whose codes are in J_\alpha. Like L_\alpha, J_\alpha is defined recursively. For each ordinal \alpha, we define to be a universal \Sigma_n predicate for J_\alpha. 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 ωα + 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, \Delta_0-comprehension and transitive closure. Moreover, they have the property that as desired. (Or a bit more generally, . ) 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. Also, the levels of the Jensen hierarchy satisfy a condensation lemma much like the levels of Gödel's original hierarchy. For any J_\alpha, considering any \Sigma_n relation on J_\alpha, there is a Skolem function for that relation that is itself definable by a \Sigma_n formula.

Rudimentary functions

A rudimentary function is a Vn→V function (i.e. a finitary function accepting sets as arguments) that can be obtained from the following operations: For any set M let rud(M) be the smallest set containing M∪{M} closed under the rudimentary functions. Then the Jensen hierarchy satisfies Jα+1 = rud(Jα).

Projecta

Jensen defines, the \Sigma_n projectum of \alpha, as the largest such that (J_\beta,A) is amenable for all , and the \Delta_n projectum of \alpha is defined similarly. One of the main results of fine structure theory is that is also the largest \gamma such that not every subset of is (in the terminology of α-recursion theory) \alpha-finite. Lerman defines the S_n projectum of \alpha to be the largest \gamma such that not every S_n subset of \beta is \alpha-finite, where a set is S_n if it is the image of a function f(x) expressible as where g is \alpha-recursive. In a Jensen-style characterization, S_3 projectum of \alpha is the largest such that there is an S_3 epimorphism from \beta onto \alpha. There exists an ordinal \alpha whose \Delta_3 projectum is \omega, but whose S_n projectum is \alpha for all natural n.