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.

Definition

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

$\text{Def}(X):=\{\{y\in X\mid \Phi (y,z_1,\mathrm{...},z_n)\text{ is true in }(X,\in )\}\mid \Phi \text{ is a first order formula},z_1,\mathrm{...},z_n\in X\}$

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

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 Σ₀ separation.

Jensen's modification of the L hierarchy retains this property and the slightly weaker condition that $J_{\alpha +1}\cap 𝒫(J_{\alpha })=\text{Def}(J_{\alpha })$, 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 ωα + 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

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

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

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 $J_{\alpha }$, considering any ${\Sigma }_n$ relation on $J_{\alpha }$, there is a Skolem function for that relation that is itself definable by a ${\Sigma }_n$ formula.

Rudimentary functions

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:

• 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).

Projecta

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

Lerman defines the $S_n$ projectum of $\alpha$ to be the largest $\gamma$ such that not every $S_n$ subset of $\beta$ is $\alpha$-finite, where a set is $S_n$ if it is the image of a function $f(x)$ expressible as $\underset{y_1}{\lim }\underset{y_2}{\lim }\dots \underset{y_n}{\lim }g(x,y_1,y_2,\dots ,y_n)$ where $g$ is $\alpha$-recursive. In a Jensen-style characterization, $S_3$ projectum of $\alpha$ is the largest $\beta \le \alpha$ such that there is an $S_3$ epimorphism from $\beta$ onto $\alpha$. There exists an ordinal $\alpha$ whose ${\Delta }_3$ projectum is $\omega$, but whose $S_n$ projectum is $\alpha$ for all natural $n$.

References

• Sy Friedman (2000) Fine Structure and Class Forcing, Walter de Gruyter,