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.
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
does have the desirable property of being closed under Σ0 separation.
[1] 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
will contain all sets whose codes are in
is defined recursively.
We encode hereditarily definable sets as
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,
-comprehension and transitive closure.
Moreover, they have the property that as desired.
(Or a bit more generally,
[2]) 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.
, there is a Skolem function for that relation that is itself definable by a
[3] 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:[2] 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).
[2] Jensen defines
is defined similarly.
One of the main results of fine structure theory is that
ω γ
is (in the terminology of α-recursion theory)
[2] Lerman defines the
In a Jensen-style characterization,