Rudimentary Function

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'':

: $\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 ''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

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

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.R. B. Jensen
The Fine Structure of the Constructible Hierarchy
(1972), p.247. Accessed 13 January 2023.

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).

References

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

Constructible universe