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.
Let us begin by the case of a non-iterated inductive definition, as incarnated in the system ID, which axiomatizes the least fixed point of an arithmetically definable positive inductive definition.
A subset is called inductively defined if it is the least fixed point of a monotone operator . In order to formalize this situation relative to first-order number theory, we consider operators defined by positive operator forms.
The language of , , is obtained from that of first-order number theory, , by the addition of a set (or predicate) constant for every -positive formula in that only contains (a new set variable) and (a number variable) as free variables. The term -positive means that only occurs positively in . If we take the background logic to be constructive, it is common to require that only occurs strictly positively (never to the left of an implication).
We allow ourselves a bit of set-theoretic notation, and we consider formulas in a distinguished free variable to represent the subclass . We write to mean , and if and are two such formulas, we write to mean .
Then contains the axioms of first-order number theory with the induction scheme extended to the new language as well as the axiom
and the scheme
where ranges over all -formulas.
Note that expresses that is closed under the arithmetically definable set operator , while expresses that is the least such (at least among sets definable in ).
Thus, is meant to be the least pre-fixed-point, and hence the least fixed point of the operator .
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 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 . However, to define the transfinitely iterated variants another approach is needed.
To define the system of -times iterated inductive definitions, where is an ordinal, let be a primitive recursive well-ordering of order type . We use Greek letters to denote elements of the field of .
The language of , is obtained from by the addition of a binary predicate constant for every -positive [X,Y] formula that contains at most the shown free variables, where is again a unary (set) variable, and is a fresh binary predicate variable. We write instead of , thinking of as a distinguished variable in the latter formula.
The system is now obtained from the system of first-order number theory by expanding the induction scheme to the new language and adding the scheme
expressing transfinite induction along for an arbitrary formula as well as the axiom
and the scheme
where is an arbitary formula. In and we used the abbreviation for the formula , where is the distinguished variable. We see that these express that each , for , is the least fixed point (among definable sets) for the operator . Note how all the previous sets , for , are used as parameters.
We define .
It is for instance possible to define the constructive ordinal number classes for in . Here we describe the simpler example of the constructive tree classes.
The first tree class is a non-iterated inductive definition, which informally states that
Elements of represent -branching trees. Formally, is represented by for the operator defined by
The higher tree classes , for , are given by an iterated inductive definition, which informally states that
Now the elements represent infinite trees branching at each internal node over either or one of the previous tree classes , for .
To define these higher tree classes, consider the operator form defined by the disjunction of the corresponding three clauses:
In the last clause we used the class term , thinking of in as the distinguished variable.
The book by Buchholz, Feferman, Pohlers and Sieg (1981) contains most proof-theoretic results concerning these systems. For example, the proof-theoretic ordinal of is for a limit ordinal, and that of is when for a limit ordinal.
Girard (1982) gave another analysis of the systems using what he calls -logic, making heavy use of dilators?.
See ordinal analysis for the relations between some of the and 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.
Last revised on October 26, 2016 at 10:58:48. See the history of this page for a list of all contributions to it.