Jensen hierarchy

In set theory, a mathematical discipline, 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\in X\mid \Phi (y,z_{1},...,z_{n}){\text{ is true in }}(X,\in )\}\mid \Phi {\text{ is a first order formula}},z_{1},...,z_{n}\in X\}

The constructible hierarchy, $L$ is defined by transfinite recursion. In particular, at successor ordinals, $L_{\alpha +1}=Def(L_{\alpha })$ .

The difficulty with this construction is that each of the levels is not closed under the formation of unordered pairs; for a given $x,y\in L_{\alpha +1}\setminus L_{\alpha }$ , the set $\{x,y\}$ will not be an element of $L_{\alpha +1}$ , since it is not a subset of $L_{\alpha }$ .

However, $L_{\alpha }$ does have the desirable property of being closed under ∑₀ separation.

Jensen's modified hierarchy retains this property and the slightly weaker condition that $J_{\alpha +1}\cap {\textrm {P}}(J_{\alpha })={\textrm {Def}}(J_{\alpha })$ , but is also closed under pairing. The key technique is to encode hereditarily definable sets over $J_\alpha$ by codes; then $J_{\alpha +1}$ will contain all sets whose codes are in $J_\alpha$ .

Like $L_{\alpha }$ , $J_\alpha$ is defined recursively. For each ordinal $\alpha$ , we define $W_{n}^{\alpha }$ to be a universal $\Sigma _{n}$ predicate for $J_\alpha$ . We encode hereditarily definable sets as $X_{\alpha }(n+1,e)=\{X_{\alpha }(n,f)\mid W_{n+1}^{\alpha }(e,f)\}$ , with $X_{\alpha }(0,e)=e$ . Then set $J_{\alpha ,n}:=\{X_{\alpha }(n,e)\mid e\in J_{\alpha }\}$ and finally, $J_{\alpha +1}:=\bigcup _{n\in \omega }J_{\alpha ,n}$ .

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

J_{\alpha +1}\cap {\text{Pow}}(J_{\alpha })={\text{Def}}(J_{\alpha }),

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.

Rudimentary functions

A rudimentary function is a function that can be obtained from the following operations:

F(x₁, x₂, ...) = x_i is rudimentary
F(x₁, x₂, ...) = {x_i, x_j} is rudimentary
F(x₁, x₂, ...) = x_i − x_j is rudimentary
Any composition of rudimentary functions is rudimentary
∪_z∈yG(z, x₁, x₂, ...) is rudimentary

For any set M let rud(M) be the smallest set containing M∪{M} closed under the rudimentary operations. Then the Jensen hierarchy satisfies J_α+1 = rud(J_α).

References

Sy Friedman (2000) Fine Structure and Class Forcing, Walter de Gruyter, ISBN 3-11-016777-8

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