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:

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 J_{\alpha+1} \cap \textrm{Pow}(J_{\alpha}) = \textrm{Def}(J_{\alpha}), but is also closed under pairing. The key technique is to encode hereditarily definable sets over Jα by codes; then Jα+1 will contain all sets whose codes are in Jα.

Like Lα, Jα is defined recursively. For each ordinal α, we define  W^{\alpha}_n to be a universal Σn predicate for Jα. We encode hereditarily definable sets as X_{\alpha}(n+1, e) = \{X(n, f) \mid W^{\alpha}_{n+1}(e, f)\}, with X_{\alpha}(0, e) = e. Then set Jα, n to be {X(n, e) | e in Jα}. Finally, Jα+1 = \bigcup_{n \in \omega} J_{\alpha, n}.

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

J_{\alpha+1} \cap \text{Pow}(J_\alpha) = \text{Def}(J_\alpha),

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).

References

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