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Definition

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

Def ( X ) := { { y ∈ X ∣ Φ ( y , z 1 , . . . , z n )  is true in  ( X , ∈ ) } ∣ Φ  is a first order formula , z 1 , . . . , z n ∈ 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, L {\displaystyle L} is defined by transfinite recursion. In particular, at successor ordinals, L α + 1 = Def ( L α ) {\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 x , y ∈ L α + 1 ∖ L α {\displaystyle x,y\in L_{\alpha +1}\setminus L_{\alpha }} , the set { x , y } {\displaystyle \{x,y\}} will not be an element of L α + 1 {\displaystyle L_{\alpha +1}} , since it is not a subset of L α {\displaystyle L_{\alpha }} .

However, L α {\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 J α + 1 ∩ P ( J α ) = Def ( J α ) {\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 J α {\displaystyle J_{\alpha }} by codes; then J α + 1 {\displaystyle J_{\alpha +1}} will contain all sets whose codes are in J α {\displaystyle J_{\alpha }} .

Like L α {\displaystyle L_{\alpha }} , J α {\displaystyle J_{\alpha }} is defined recursively. For each ordinal α {\displaystyle \alpha } , we define W n α {\displaystyle W_{n}^{\alpha }} to be a universal Σ n {\displaystyle \Sigma _{n}} predicate for J α {\displaystyle J_{\alpha }} . We encode hereditarily definable sets as X α ( n + 1 , e ) = { X α ( n , f ) ∣ W n + 1 α ( e , f ) } {\displaystyle X_{\alpha }(n+1,e)=\{X_{\alpha }(n,f)\mid W_{n+1}^{\alpha }(e,f)\}} , with X α ( 0 , e ) = e {\displaystyle X_{\alpha }(0,e)=e} . Then set J α , n := { X α ( n , e ) ∣ e ∈ J α } {\displaystyle J_{\alpha ,n}:=\{X_{\alpha }(n,e)\mid e\in J_{\alpha }\}} and finally, J α + 1 := ⋃ n ∈ ω J α , n {\displaystyle J_{\alpha +1}:=\bigcup _{n\in \omega }J_{\alpha ,n}} .

Properties

Each sublevel Jα, 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 Σm predicate is also Σn for any n > m. The levels Jα will thus be transitive and strictly ⊆-increasing as well, and are also closed under pairing, Δ 0 {\displaystyle \Delta _{0}} -comprehension and transitive closure. Moreover, they have the property that

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

as desired. (Or a bit more generally, L ω + α = J 1 + α ∩ V ω + α {\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 Jα, n in Jβ 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 α {\displaystyle J_{\alpha }} , considering any Σ n {\displaystyle \Sigma _{n}} relation on J α {\displaystyle J_{\alpha }} , there is a Skolem function for that relation that is itself definable by a Σ n {\displaystyle \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:⁴

• F(x1, x2, ...) = xi is rudimentary (see projection function)
• F(x1, x2, ...) = {xi, xj} is rudimentary
• F(x1, x2, ...) = xi − xj 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 Jα+1 = rud(Jα).⁵

Projecta

Jensen defines ρ α n {\displaystyle \rho _{\alpha }^{n}} , the Σ n {\displaystyle \Sigma _{n}} projectum of α {\displaystyle \alpha } , as the largest β ≤ α {\displaystyle \beta \leq \alpha } such that ( J β , A ) {\displaystyle (J_{\beta },A)} is amenable for all A ∈ Σ n ( J α ) ∩ P ( J β ) {\displaystyle A\in \Sigma _{n}(J_{\alpha })\cap {\mathcal {P}}(J_{\beta })} , and the Δ n {\displaystyle \Delta _{n}} projectum of α {\displaystyle \alpha } is defined similarly. One of the main results of fine structure theory is that ρ α n {\displaystyle \rho _{\alpha }^{n}} is also the largest γ {\displaystyle \gamma } such that not every Σ n ( J α ) {\displaystyle \Sigma _{n}(J_{\alpha })} subset of ω γ {\displaystyle \omega \gamma } is (in the terminology of α-recursion theory) α {\displaystyle \alpha } -finite.⁶

Lerman defines the S n {\displaystyle S_{n}} projectum of α {\displaystyle \alpha } to be the largest γ {\displaystyle \gamma } such that not every S n {\displaystyle S_{n}} subset of β {\displaystyle \beta } is α {\displaystyle \alpha } -finite, where a set is S n {\displaystyle S_{n}} if it is the image of a function f ( x ) {\displaystyle f(x)} expressible as lim y 1 lim y 2 … lim y n g ( x , y 1 , y 2 , … , y n ) {\displaystyle \lim _{y_{1}}\lim _{y_{2}}\ldots \lim _{y_{n}}g(x,y_{1},y_{2},\ldots ,y_{n})} where g {\displaystyle g} is α {\displaystyle \alpha } -recursive. In a Jensen-style characterization, S 3 {\displaystyle S_{3}} projectum of α {\displaystyle \alpha } is the largest β ≤ α {\displaystyle \beta \leq \alpha } such that there is an S 3 {\displaystyle S_{3}} epimorphism from β {\displaystyle \beta } onto α {\displaystyle \alpha } . There exists an ordinal α {\displaystyle \alpha } whose Δ 3 {\displaystyle \Delta _{3}} projectum is ω {\displaystyle \omega } , but whose S n {\displaystyle S_{n}} projectum is α {\displaystyle \alpha } for all natural n {\displaystyle n} . ⁷

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

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. https://core.ac.uk/download/pdf/30905237.pdf ↩

3. R. B. Jensen, The Fine Structure of the Constructible Hierarchy (1972), p.247. Accessed 13 January 2023. https://www.math.cmu.edu/~laiken/papers/FineStructure.pdf ↩

4. K. Devlin, An introduction to the fine structure of the constructible hierarchy (1974). Accessed 2022-02-26. https://core.ac.uk/download/pdf/30905237.pdf ↩

5. K. Devlin, An introduction to the fine structure of the constructible hierarchy (1974). Accessed 2022-02-26. https://core.ac.uk/download/pdf/30905237.pdf ↩

6. K. Devlin, An introduction to the fine structure of the constructible hierarchy (1974). Accessed 2022-02-26. https://core.ac.uk/download/pdf/30905237.pdf ↩

7. 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 ↩