iterated inductive definitions



Systems of iterated inductive definitions are important formal systems introduced by Feferman (1970), especially of use in proof-theoretic investigations. They are obviously related as well to inductive types in type theory.

Below we record the main definitions and proof-theoretic results about these systems. See also the table at ordinal analysis.


Non-iterated case

Let us begin by the case of a non-iterated inductive definition, as incarnated in the system ID1_1, which axiomatizes the least fixed point of an arithmetically definable positive inductive definition.

A subset II \subseteq \mathbb{N} is called inductively defined if it is the least fixed point of a monotone operator Γ:𝒫()𝒫()\Gamma : \mathcal{P}(\mathbb{N}) \to \mathcal{P}(\mathbb{N}). In order to formalize this situation relative to first-order number theory, we consider operators Γ A\Gamma_A defined by positive operator forms.

The language of ID 1ID_1, ID 1\mathcal{L}_{ID_1}, is obtained from that of first-order number theory, \mathcal{L}_{\mathbb{N}}, by the addition of a set (or predicate) constant I AI_A for every XX-positive formula A(X,x)A(X,x) in [X]\mathcal{L}_{\mathbb{N}}[X] that only contains XX (a new set variable) and xx (a number variable) as free variables. The term XX-positive means that XX only occurs positively in AA. If we take the background logic to be constructive, it is common to require that XX only occurs strictly positively (never to the left of an implication).

We allow ourselves a bit of set-theoretic notation, and we consider formulas F(x)F(x) in a distinguished free variable xx to represent the subclass {xF(x)}\{ x\in\mathbb{N} \mid F(x) \}. We write sFs\in F to mean F(s)F(s), and if FF and GG are two such formulas, we write FGF \subseteq G to mean x.F(x)G(x)\forall x. F(x) \to G(x).

Then ID 1ID_1 contains the axioms of first-order number theory with the induction scheme extended to the new language as well as the axiom

(1)(ID 1) 1A(I A)I A (ID_1)^1\qquad A(I_A) \subseteq I_A

and the scheme

(2)(ID 1) 2A(F)FI AF (ID_1)^2\qquad A(F) \subseteq F \to I_A \subseteq F

where F(x)F(x) ranges over all ID 1\mathcal{L}_{ID_1}-formulas.

Note that (ID 1) 1(ID_1)^1 expresses that I AI_A is closed under the arithmetically definable set operator Γ A(S)={xA(S,x)}\Gamma_A(S) = \{ x\in\mathbb{N} \mid \mathbb{N} \vDash A(S,x) \}, while (ID 1) 2(ID_1)^2 expresses that I AI_A is the least such (at least among sets definable in ID 1\mathcal{L}_{ID_1}).

Thus, I AI_A is meant to be the least pre-fixed-point, and hence the least fixed point of the operator Γ A\Gamma_A.

Iterated inductive definitions

It is possible to define systems of finitely iterated inductive definitions by simply repeating the process of adding a level of inductive definitions and extending the existing schemas to the new language. It’s important that the constants I AI_A for the previous levels may occur negatively in the operators for the later levels. So these systems can be defined in stages by (metamathematical) induction on \mathbb{N}. However, to define the transfinitely iterated variants another approach is needed.

To define the system of ν\nu-times iterated inductive definitions, where ν\nu is an ordinal, let {\prec} be a primitive recursive well-ordering of order type ν\nu. We use Greek letters to denote elements of the field of {\prec}.

The language of ID νID_\nu, ID ν\mathcal{L}_{ID_\nu} is obtained from \mathcal{L}_{\mathbb{N}} by the addition of a binary predicate constant J AJ_A for every XX-positive \mathcal{L}_{\mathbb{N}}[X,Y] formula A(X,Y,μ,x)A(X,Y,\mu,x) that contains at most the shown free variables, where XX is again a unary (set) variable, and YY is a fresh binary predicate variable. We write xJ A μx\in J_A^\mu instead of J A(μ,x)J_A(\mu,x), thinking of xx as a distinguished variable in the latter formula.

The system ID νID_\nu is now obtained from the system of first-order number theory by expanding the induction scheme to the new language and adding the scheme

(3)(TI ν)TI(,F) (TI_\nu)\qquad TI({\prec},F)

expressing transfinite induction along {\prec} for an arbitrary ID ν\mathcal{L}_{ID_\nu} formula FF as well as the axiom

(4)(ID ν) 1μν.A μ(J A μ)J A μ (ID_\nu)^1\qquad \forall\mu\prec\nu. A^\mu(J_A^\mu) \subseteq J_A^\mu

and the scheme

(5)(ID ν) 2μν.A μ(F)FJ A μF (ID_\nu)^2\qquad \forall\mu\prec\nu. A^\mu(F) \subseteq F \to J_A^\mu \subseteq F

where F(x)F(x) is an arbitary ID ν\mathcal{L}_{ID_\nu} formula. In (ID ν) 1(ID_\nu)^1 and (ID ν) 2(ID_\nu)^2 we used the abbreviation A μ(F)A^\mu(F) for the formula A(F,(λγy.γμyJ A γ),μ,x)A(F, (\lambda\gamma y. \gamma\prec\mu \wedge y\in J_A^\gamma), \mu, x), where xx is the distinguished variable. We see that these express that each J A μJ_A^\mu, for μν\mu\prec\nu, is the least fixed point (among definable sets) for the operator Γ A μ(S)={n(,(J A γ) γμ)A(S,(J A γ) γμ,μ,n)}\Gamma_A^\mu(S) = \{ n\in\mathbb{N} \mid (\mathbb{N},(J_A^\gamma)_{\gamma\prec\mu}) \vDash A(S,(J_A^\gamma)_{\gamma\prec\mu},\mu,n) \}. Note how all the previous sets J A γJ_A^\gamma, for γμ\gamma\prec\mu, are used as parameters.

We define ID ν= ξνID ξID_{\prec\nu} = \bigcup_{\xi\prec\nu} ID_\xi.


It is for instance possible to define the constructive ordinal number classes 𝒪 μ\mathcal{O}_\mu for μν\mu\le\nu in ID νID_\nu. Here we describe the simpler example of the constructive tree classes.

The first tree class 𝒯 0\mathcal{T}_0 is a non-iterated inductive definition, which informally states that

  • 0𝒯 00 \in \mathcal{T}_0, and
  • if ee is (the code of) a total recursive function enumerating elements of 𝒯 0\mathcal{T}_0, then e+1𝒯 0e+1 \in \mathcal{T}_0.

Elements of 𝒯 0\mathcal{T}_0 represent \mathbb{N}-branching trees. Formally, 𝒯 0\mathcal{T}_0 is represented by I AI_A for the operator A(X,x)A(X,x) defined by

(6)x=0(x0n.{x1}(n)X). x = 0 \vee (x \ne 0 \wedge \forall n. \{x-1\}(n) \in X).

The higher tree classes 𝒯 μ\mathcal{T}_\mu, for μν\mu\prec\nu, are given by an iterated inductive definition, which informally states that

  • 0𝒯 μ0 \in \mathcal{T}_\mu, and
  • if ee is (the code of) a total recursive function enumerating elements of 𝒯 μ\mathcal{T}_\mu, then 0,e𝒯 μ\langle 0,e\rangle \in \mathcal{T}_\mu, and
  • if γμ\gamma\prec\mu and ee is (the code of) a partial recursive function whose domain includes 𝒯 γ\mathcal{T}_\gamma and whose values thereon are in in 𝒯 μ\mathcal{T}_\mu, then 1+γ,e𝒯 μ\langle 1+\gamma,e\rangle \in \mathcal{T}_\mu.

Now the elements represent infinite trees branching at each internal node over either \mathbb{N} or one of the previous tree classes 𝒯 γ\mathcal{T}_\gamma, for γμ\gamma\prec\mu.

To define these higher tree classes, consider the operator form A(X,Y,μ,x)A(X,Y,\mu,x) defined by the disjunction of the corresponding three clauses:

  • x=0x = 0,
  • e.x=0,en.{e}(n)X\exists e. x = \langle 0,e\rangle \wedge \forall n. \{e\}(n) \in X,
  • γ,e.γμx=1+γ,eyY γ.{e}(y)X\exists \gamma, e. \gamma \prec \mu \wedge x = \langle 1+\gamma,e\rangle \wedge \forall y \in Y^\gamma. \{e\}(y) \in X.

In the last clause we used the class term Y γY^\gamma, thinking of yy in Y(γ,y)Y(\gamma,y) as the distinguished variable.

Proof-theoretic results

The book by Buchholz, Feferman, Pohlers and Sieg (1981) contains most proof-theoretic results concerning these systems. For example, the proof-theoretic ordinal of ID νID_\nu is ψ Ω(ε Ω ν+1)\psi_\Omega(\varepsilon_{\Omega_\nu+1}) for ν\nu a limit ordinal, and that of ID νID_{\prec\nu} is ψ Ω(Ω ν)\psi_\Omega(\Omega_\nu) when ν=ω ρ\nu=\omega^\rho for ρ\rho a limit ordinal.

Girard (1982) gave another analysis of the ID νID_\nu systems using what he calls Π 2 1\Pi^1_2-logic, making heavy use of dilators?.

See ordinal analysis for the relations between some of the ID νID_\nu and ID νID_{\prec\nu} systems and other systems.


  • Buchholz, Feferman, Pohlers and Sieg: Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies, Lecture Notes in Mathematics 897 (1981).

  • Feferman: Formal theories for transfinite iterations of generalized inductive definitions and some subsystems of analysis. In: Intuitionism and Proof Theory (Proc. Conf., Buffalo, N.Y., 1968). 1970.

  • Feferman: The proof theory of classical and constructive inductive definitions, a 40 year saga, 1968-2008. In: Ways of Proof Theory. 2013. pdf

  • Girard: Proof-theoretic investigations of inductive definitions, I. In: Logic and algorithmic (Proc. Conf., Zurich, 1980). 1982.

  • Pohlers: Subsystems of Set Theory and Second Order Number Theory. In: Handbook of Proof Theory. 1998.

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