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.

References

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