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:

${\displaystyle Def(X):=\{\{y\in X\mid \Phi (y,z_{1},...,z_{n}){\text{ is true in }}(X,\in )\}\mid \Phi {\text{ is a first order formula}},z_{1},...,z_{n}\in X\}}$

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

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

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

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

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

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, ${\displaystyle \Delta _{0}}$-comprehension and transitive closure. Moreover, they have the property that

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

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 Gödel'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