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,...,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 Delta-0 Comprehension.
Jensen's modified hierarchy retains this property and the slightly weaker condition that , but is also closed under pairing.
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References
- Sy. D. Freeman (2000) Fine Structure and Class Forcing, Walter de Gruyter , ISBN 3110167778