Jensen hierarchy explained

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:

rm{Def}(X) : = \{ \{y \in X \mid \Phi(y,z _{1,...,z} _{n)} istruein (X,\in)\} \mid \Phi isafirstorderformula , z _{1,} ..., z _{n\in} X\}

The constructible hierarchy,

L is defined by transfinite recursion. In particular, at successor ordinals, L _{\alpha+1} = rm{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.^[1]

Jensen's modification of the L hierarchy retains this property and the slightly weaker condition that

J _{\alpha+1} \cap l P(J _{\alpha} ) = rm{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 ^{\alpha} _{n} 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 ^{\alpha} _{n +1} (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} : = c up _{n} J _{\alpha,} .

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 thatJ _{\alpha+1} \cap l P(J _{\alpha)} = Def (J _{\alpha),}

as desired. (Or a bit more generally,

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

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.^[3]

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, ...) = 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∪ closed under the rudimentary functions. Then the Jensen hierarchy satisfies J_α+1 = rud(J_α).

Projecta

Jensen defines

\rho ^{n} _{\alpha} , 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)\caplP(J} _{\beta)} , and the \Delta _{n} projectum of \alpha is defined similarly. One of the main results of fine structure theory is that \rho ^{n} _{\alpha} 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 _{y _{1}} \lim _{y _{2}} \ldots\lim _{y _{n}} 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 . ^[4]

References

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

Notes and References

1. Wolfram Pohlers, Proof Theory: The First Step Into Impredicativity (2009) (p.247)
2. K. Devlin, An introduction to the fine structure of the constructible hierarchy (1974). Accessed 2022-02-26.
3. R. B. Jensen, The Fine Structure of the Constructible Hierarchy (1972), p.247. Accessed 13 January 2023.
4. 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