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'':

: $\textrm(X) := \$

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

The difficulty with this construction is that each of the levels is not closed under the formation of unordered pairs; for a given $x, y \in L_ \setminus L_\alpha$, the set $\$ will not be an element of $L_$, 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 $J_ \cap \mathcal P(J_) = \textrm(J_)$, but is also closed under pairing. The key technique is to encode hereditarily definable sets over $J_\alpha$ by codes; then $J_$ will contain all sets whose codes are in $J_\alpha$.

Like $L_\alpha$, $J_\alpha$ is defined recursively. For each ordinal $\alpha$, we define $W^_n$ to be a universal $\Sigma_n$ predicate for $J_\alpha$. We encode hereditarily definable sets as $X_(n+1, e) = \$, with $X_(0, e) = e$. Then set $J_ := \$ and finally, $J_ := \bigcup_ J_$.

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

: $J_ \cap \mathcal P(J_\alpha) = \text(J_\alpha),$

as desired. (Or a bit more generally, $L_=J_\cap V_$.K. Devlin
An introduction to the fine structure of the constructible hierarchy
(1974). Accessed 2022-02-26.)

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''(''x''₁, ''x''₂, ...) = ''x''_''i'' is rudimentary (see projection function)
*''F''(''x''₁, ''x''₂, ...) = is rudimentary
*''F''(''x''₁, ''x''₂, ...) = ''x''_''i'' − ''x''_''j'' is rudimentary
*Any composition of rudimentary functions is rudimentary
*∪_{''z''∈''y''}''G''(''z'', ''x''₁, ''x''₂, ...) is rudimentary, where G is a rudimentary function

For any set ''M'' let rud(''M'') be the smallest set containing ''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 $\beta\leq\alpha$ such that $(J_\beta,A)$ is amenable for all $A\in\Sigma_n(J_\alpha)\cap\mathcal P(J_\beta)$, 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 $\lim_\lim_\ldots\lim_g(x,y_1,y_2,\ldots,y_n)$ where $g$ is $\alpha$-recursive. In a Jensen-style characterization, $S_3$ projectum of $\alpha$ is the largest $\beta\leq\alpha$ 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 $n$. S. G. Simpson, "Short course on admissible recursion theory". Appearing in Studies in Logic and the Foundations of Mathematics vol. 94, ''Generalized Recursion Theory II'' (1978), pp.355--390

References

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

Constructible universe