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| In [[set theory]], a mathematical discipline, the '''Jensen hierarchy''' or '''J-hierarchy''' is a modification of [[Kurt Gödel|Gödel]]'s [[constructible universe|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.
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| ==Definition==
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| As in the definition of ''L'', let Def(''X'') be the collection of sets definable with parameters over ''X'':
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| : Def(''X'') = { {''y'' | ''y'' ε ''X'' and Φ(''y'', ''z''<sub>1</sub>, ..., ''z''<sub>''n''</sub>) is true in (''X'', ε)} | Φ is a first order formula and ''z''<sub>1</sub>, ..., ''z''<sub>''n''</sub> are elements of ''X''}.
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| The constructible hierarchy, L is defined by [[transfinite recursion]]. In particular, at successor ordinals, ''L''<sub>α+1</sub> = Def(''L''<sub>α</sub>).
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| The difficulty with this construction is that each of the levels is
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| not closed under the formation of [[Axiom of pairing|unordered pairs]]; for a given x, y ε
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| ''L''<sub>α+1</sub> − ''L''<sub>α</sub>, the set {''x'',''y''} will not be an element of ''L''<sub>α+1</sub>, since it is not a subset of ''L''<sub>α</sub>.
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| However, ''L''<sub>α</sub> does have the desirable property of being closed under [[Lévy hierarchy|Σ<sub>0</sub>]] [[Axiom schema of separation|separation]].
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| Jensen's modified hierarchy retains this property and the slightly weaker condition that <math>J_{\alpha+1} \cap \textrm{Pow}(J_{\alpha}) = \textrm{Def}(J_{\alpha})</math>, but is also closed under pairing. The key technique is to encode hereditarily definable sets over ''J''<sub>α</sub> by codes; then ''J''<sub>α+1</sub> will contain all sets whose codes are in ''J''<sub>α</sub>.
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| Like ''L''<sub>α</sub>, ''J''<sub>α</sub> is [[recursive definition|defined recursively]]. For each ordinal α, we define <math> W^{\alpha}_n</math> to be a [[universal predicate|universal]] Σ<sub>n</sub> predicate for ''J''<sub>α</sub>. We encode hereditarily definable sets as <math>X_{\alpha}(n+1, e) = \{X(n, f) \mid W^{\alpha}_{n+1}(e, f)\}</math>, with <math>X_{\alpha}(0, e) = e</math>. Then set ''J''<sub>α, n</sub> to be {X(n, e) | e in J<sub>α</sub>}. Finally, ''J''<sub>α+1</sub> = <math>\bigcup_{n \in \omega} J_{\alpha, n}</math>.
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| ==Properties==
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| Each sublevel ''J''<sub>''α'', ''n''</sub> 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<sub>''m''</sub> predicate is also Σ<sub>''n''</sub> for any ''n'' > ''m''. The levels ''J''<sub>''α''</sub> 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
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| : <math>J_{\alpha+1} \cap \text{Pow}(J_\alpha) = \text{Def}(J_\alpha),</math>
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| as desired.
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| The levels and sublevels are themselves Σ<sub>1</sub> uniformly definable [i.e. the definition of ''J''<sub>''α'', ''n''</sub> in ''J''<sub>''β''</sub> does not depend on ''β''], and have a uniform Σ<sub>1</sub> well-ordering. Finally, the levels of the Jensen hierarchy satisfy a [[condensation lemma]] much like the levels of Godel's original hierarchy.
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| ==References== | |
| * [[Sy Friedman]] (2000) ''Fine Structure and Class Forcing'', Walter de Gruyter, ISBN 3-11-016777-8
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| [[Category:Constructible universe]]
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