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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.~~		Andrew Simcox is the name his parents gave him and he completely enjoys this title. Her family lives in Alaska but her spouse desires them to transfer. Since he was eighteen he's been operating as tarot card readings - [http://gcjcteam.org/index.php?mid=etc_video&document_srl=696611&sort_index=regdate&order_type=desc website link] - an info officer but he plans on altering it. What me and [http://c045.danah.co.kr/home/index.php?document_srl=1356970&mid=qna love psychics] my family members love is to climb but I'm considering on starting some thing new.<br><br>Visit my webpage; best psychics ([https://www-ocl.gist.ac.kr/work/xe/?document_srl=605236 updated blog post])

	~~==Definition==~~
	~~As in the definition of ''L'', let Def(''X'') be the collection of sets definable with parameters over ''X''~~:

	~~: 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''}~~.

	~~The constructible hierarchy, L is defined by [[transfinite recursion]]. In particular, at successor ordinals, ''L''<sub>α+1</sub>~~ = ~~Def(''L''<sub>α</sub>).~~

	~~The difficulty with this construction is that each of the levels is~~
	~~not closed under the formation of [[Axiom of pairing\|unordered pairs]]; for a given x, y ε~~
	~~''L''<sub>α+1</sub>~~&~~nbsp;~~&~~minus;~~&~~nbsp;''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>~~.

	~~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]].~~

	~~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>~~.

	~~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>~~.

	~~==Properties==~~
	~~Each sublevel ''J''~~<~~sub~~>~~''α'', ''n''~~<~~/sub~~> ~~is transitive and contains all ordinals less than or equal to ''αω''&nbsp~~;~~+ ''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~~

	~~: <math>J_{\alpha+1} \cap \text{Pow}(J_\alpha) = \text{Def}(J_\alpha),</math>~~

	~~as desired.~~

	~~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.~~

	=~~=References==~~
	* [[Sy Friedman]~~] (2000~~) ~~''Fine Structure and Class Forcing'', Walter de Gruyter, ISBN 3-11-016777-8~~

	~~[[Category:Constructible universe]]~~

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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.

==Definition==
As in the definition of ''L'', let Def(''X'') be the collection of sets definable with parameters over ''X'':

: Def(''X'') = { {''y'' | ''y'' ε ''X'' and Φ(''y'', ''z''1, ..., ''z''''n'') is true in (''X'', ε)} | Φ is a first order formula and ''z''1, ..., ''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 [[Axiom of pairing|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 [[Lévy hierarchy|Σ0]] [[Axiom schema of separation|separation]].

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''α by codes; then ''J''α+1 will contain all sets whose codes are in ''J''α.

Like ''L''α, ''J''α is [[recursive definition|defined recursively]]. For each ordinal α, we define <math> W^{\alpha}_n</math> to be a [[universal predicate|universal]] Σn predicate for ''J''α. 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''α, n to be {X(n, e) | e in Jα}. Finally, ''J''α+1 = <math>\bigcup_{n \in \omega} J_{\alpha, n}</math>.

==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

: <math>J_{\alpha+1} \cap \text{Pow}(J_\alpha) = \text{Def}(J_\alpha),</math>

as desired.

The levels and sublevels are themselves Σ1 uniformly definable [i.e. 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 Friedman]] (2000) ''Fine Structure and Class Forcing'', Walter de Gruyter, ISBN 3-11-016777-8

[[Category:Constructible universe]]

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