Jensen hierarchy

In set theory, 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 | yεX and Φ(y,z₁,...,z_n) is true in (X,ε)} | Φ is a first order formula and z₁,...,z_n are elements of X}.

The constructible hierarchy, L is defined by transfinite recursion. In particular, at successor ordinals, L_α+1 = Def (L_α).

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 L_α+1 - L_α, the set {x,y} will not be an element of L_α+1, since it is not a subset of L_α.

However, L_α does have the desirable property of being closed under Delta-0 Comprehension.

Jensen's modified hierarchy retains this property and the slightly weaker condition that $J_{\alpha +1}\cap Pow(J_{\alpha })=Def(J_{\alpha })$, but is also closed under pairing. The key technique is to encode hereditarily definable sets over J_α by codes; then J_α+1 will contain all sets whose codes are in J_α.

Like L_α, J_α is defined recursively. For each ordinal α, we define $W_{n}^{\alpha }$ to be a universal Sigma-n predicate for J_α. We encode hereditarily sets as $X_{\alpha }(n+1,e)=\{X(n,f)|W_{n+1}^{\alpha }(e,f)\}$, with $X_{\alpha }(0,e)=e$. Then set J_{α, n} to be {X(n, e) | e in J_α}. Finally, J_α+1 = $\bigcup _{n\in \omega }J_{\alpha ,n}$.

Properties

Each sublevel J_{α, 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 Σ_m predicate is also Σ_n for any n > m. The levels J_α 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 $J_{\alpha +1}\cap Pow(J_{\alpha })=Def(J_{\alpha })$, as desired.

The levels and sublevels are themselves Σ₁ uniformly definable [ie the definition of J_{α, n} in J_β 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.

References

• Sy. D. Freeman (2000) Fine Structure and Class Forcing, Walter de Gruyter , ISBN 3110167778