- 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 infine structure theory , a field pioneered byRonald 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 isnot closed under the formation of unordered pairs; for a given x, y εLα+1, the set {x,y} need not be an element of Lα+1.
References
* Sy. D. Freeman (2000) "Fine Structure and Class Forcing", Walter de Gruyter , ISBN 3110167778
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