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. 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:


 * $$\textrm{Def}(X) := \{ \{y \in X \mid \Phi(y,z_1,...,z_n) \text{ is true in } (X,\in)\} \mid \Phi \text{ is a first order formula}, z_1, ..., z_n\in X\}$$

The constructible hierarchy, $$L$$ is defined by transfinite recursion. In particular, at successor ordinals, $$L_{\alpha+1} = \textrm{Def}(L_\alpha)$$.

The difficulty with this construction is that each of the levels is not closed under the formation of unordered pairs; for a given $$x, y \in L_{\alpha+1} \setminus L_\alpha$$, the set $$\{x,y\}$$ will not be an element of $$L_{\alpha+1}$$, 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 $$J_{\alpha+1} \cap \mathcal P(J_{\alpha}) = \textrm{Def}(J_{\alpha})$$, but is also closed under pairing. The key technique is to encode hereditarily definable sets over $$J_\alpha$$ by codes; then $$J_{\alpha+1}$$ will contain all sets whose codes are in $$J_\alpha$$.

Like $$L_\alpha$$, $$J_\alpha$$ is defined recursively. For each ordinal $$\alpha$$, we define $$W^{\alpha}_n$$ to be a universal $$\Sigma_n$$ predicate for $$J_\alpha$$. We encode hereditarily definable sets as $$X_{\alpha}(n+1, e) = \{X_\alpha(n, f) \mid W^{\alpha}_{n+1}(e, f)\}$$, with $$X_{\alpha}(0, e) = e$$. Then set $$J_{\alpha,n} := \{X_\alpha(n, e) \mid e \in J_\alpha\}$$ and finally, $$J_{\alpha+1} := \bigcup_{n \in \omega} J_{\alpha, n}$$.

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


 * $$J_{\alpha+1} \cap \mathcal P(J_\alpha) = \text{Def}(J_\alpha),$$

as desired. (Or a bit more generally, $$L_{\omega+\alpha}=J_{1+\alpha}\cap V_{\omega+\alpha}$$. )

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:
 * F(x1, x2, ...) = xi is rudimentary (see projection function)
 * F(x1, x2, ...) = {xi, xj} is rudimentary
 * F(x1, x2, ...) = xi − xj is rudimentary
 * Any composition of rudimentary functions is rudimentary
 * ∪z∈yG(z, x1, x2, ...) is rudimentary, where G is a rudimentary function

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 $$\rho_\alpha^n$$, the $$\Sigma_n$$ projectum of $$\alpha$$, as the largest $$\beta\leq\alpha$$ such that $$(J_\beta,A)$$ is amenable for all $$A\in\Sigma_n(J_\alpha)\cap\mathcal P(J_\beta)$$, and the $$\Delta_n$$ projectum of $$\alpha$$ is defined similarly. One of the main results of fine structure theory is that $$\rho_\alpha^n$$ is also the largest $$\gamma$$ such that not every $$\Sigma_n(J_\alpha)$$ subset of $$\omega\gamma$$ 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 $$\lim_{y_1}\lim_{y_2}\ldots\lim_{y_n}g(x,y_1,y_2,\ldots,y_n)$$ where $$g$$ is $$\alpha$$-recursive. In a Jensen-style characterization, $$S_3$$ projectum of $$\alpha$$ is the largest $$\beta\leq\alpha$$ 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 $$n$$.