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.
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Definition
As in the definition of L, let Def(X) be the collection of sets definable with parameters over X:
Def(X) = { {y | y ε X and Φ(y, z1, ..., zn) is true in (X, ε)} | Φ is a first order formula and z1, ..., zn are elements of X}.The constructible hierarchy, L is defined by transfinite recursion. In particular, at successor ordinals, Lα+1 = Def(Lα).
The difficulty with this construction is that each of the levels is not closed under the formation of unordered pairs; for a given x, y ε Lα+1 − Lα, the set {x,y} will not be an element of Lα+1, since it is not a subset of Lα.
However, Lα does have the desirable property of being closed under Σ0 separation.
Jensen's modified hierarchy retains this property and the slightly weaker condition that
Like Lα, Jα is defined recursively. For each ordinal α, we define
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, Delta-0 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 Godel'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(M) be the smallest set containing M∪{M} closed under the rudimentary operations. Then the Jensen hierarchy satisfies Jα+1 = rud(Jα).