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. Rudimentary functions describe a method for iterating through the Jensen hierarchy.

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

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

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

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 Σ₀ separation.^[1]

Jensen's modification of the L hierarchy retains this property and the slightly weaker condition that ${\displaystyle J_{\alpha +1}\cap {\mathcal {P}}(J_{\alpha })={\textrm {Def}}(J_{\alpha })}$, 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, ${\displaystyle J_{\alpha +1}:=\bigcup _{n\in \omega }J_{\alpha ,n}}$.

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

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

as desired. (Or a bit more generally, ${\displaystyle L_{\omega +\alpha }=J_{1+\alpha }\cap V_{\omega +\alpha }}$.^[2])

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. Also, the levels of the Jensen hierarchy satisfy a condensation lemma much like the levels of Gödel's original hierarchy.

For any ${\displaystyle J_{\alpha }}$, considering any ${\displaystyle \Sigma _{n}}$ relation on ${\displaystyle J_{\alpha }}$, there is a Skolem function for that relation that is itself definable by a ${\displaystyle \Sigma _{n}}$ formula.^[3]

A rudimentary function is a Vⁿ→V function (i.e. a finitary function accepting sets as arguments) that can be obtained from the following operations:^[2]

• F(x₁, x₂, ...) = x_i is rudimentary (see projection function)
• 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, where G is a rudimentary function

For any set M let rud(M) be the smallest set containing M∪{M} closed under the rudimentary functions. Then the Jensen hierarchy satisfies J_α+1 = rud(J_α).^[2]

Jensen defines ${\displaystyle \rho _{\alpha }^{n}}$, the ${\displaystyle \Sigma _{n}}$ projectum of ${\displaystyle \alpha }$, as the largest ${\displaystyle \beta \leq \alpha }$ such that ${\displaystyle (J_{\beta },A)}$ is amenable for all ${\displaystyle A\in \Sigma _{n}(J_{\alpha })\cap {\mathcal {P}}(J_{\beta })}$, and the ${\displaystyle \Delta _{n}}$ projectum of ${\displaystyle \alpha }$ is defined similarly. One of the main results of fine structure theory is that ${\displaystyle \rho _{\alpha }^{n}}$ is also the largest ${\displaystyle \gamma }$ such that not every ${\displaystyle \Sigma _{n}(J_{\alpha })}$ subset of ${\displaystyle \omega \gamma }$ is (in the terminology of α-recursion theory) ${\displaystyle \alpha }$-finite.^[2]

Lerman defines the ${\displaystyle S_{n}}$ projectum of ${\displaystyle \alpha }$ to be the largest ${\displaystyle \gamma }$ such that not every ${\displaystyle S_{n}}$ subset of ${\displaystyle \beta }$ is ${\displaystyle \alpha }$-finite, where a set is ${\displaystyle S_{n}}$ if it is the image of a function ${\displaystyle f(x)}$ expressible as ${\displaystyle \lim _{y_{1}}\lim _{y_{2}}\ldots \lim _{y_{n}}g(x,y_{1},y_{2},\ldots ,y_{n})}$ where ${\displaystyle g}$ is ${\displaystyle \alpha }$-recursive. In a Jensen-style characterization, ${\displaystyle S_{3}}$ projectum of ${\displaystyle \alpha }$ is the largest ${\displaystyle \beta \leq \alpha }$ such that there is an ${\displaystyle S_{3}}$ epimorphism from ${\displaystyle \beta }$ onto ${\displaystyle \alpha }$. There exists an ordinal ${\displaystyle \alpha }$ whose ${\displaystyle \Delta _{3}}$ projectum is ${\displaystyle \omega }$, but whose ${\displaystyle S_{n}}$ projectum is ${\displaystyle \alpha }$ for all natural ${\displaystyle n}$. ^[4]

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