In set theory and logic, Buchholz's ID hierarchy is a hierarchy of subsystems of first-order arithmetic. The systems/theories are referred to as "the formal theories of ý-times iterated inductive definitions". ID<sub>ý</sub> extends PA by ý iterated least fixed points of monotone operators.
The formal theory ID<sub>ÃÂ</sub> (and ID<sub>ý</sub> in general) is an extension of Peano Arithmetic, formulated in the language L<sub>ID</sub>, by the following axioms:
The theory ID<sub>ý</sub> with ý â àis defined as:
A set is called inductively defined if for some monotonic operator , , where denotes the least fixed point of . The language of ID<sub>1</sub>, , is obtained from that of first-order number theory, , by the addition of a set (or predicate) constant I<sub>A</sub> for every X-positive formula A(X, x) in L<sub>N</sub>[X] that only contains X (a new set variable) and x (a number variable) as free variables. The term X-positive means that X only occurs positively in A (X is never on the left of an implication). We allow ourselves a bit of set-theoretic notation:
Then ID<sub>1</sub> contains the axioms of first-order number theory (PA) with the induction scheme extended to the new language as well as these axioms:
Where ranges over all formulas.
Note that expresses that ÃÂ is closed under the arithmetically definable set operator , while ÃÂ expresses that ÃÂ is the least such (at least among sets definable in ).
Thus, ÃÂ is meant to be the least pre-fixed-point, and hence the least fixed point of the operator .
To define the system of ý-times iterated inductive definitions, where ýàis an ordinal, let àbe a primitive recursive well-ordering of order type ý. We use Greek letters to denote elements of the field of . The language of ID<sub>ý</sub>, is obtained from by the addition of a binary predicate constant J<sub>A</sub> for every X-positive formula that contains at most the shown free variables, where X is again a unary (set) variable, and Y is a fresh binary predicate variable. We write instead of , thinking of x as a distinguished variable in the latter formula.
The system ID<sub>ý</sub>àis now obtained from the system of first-order number theory (PA) by expanding the induction scheme to the new language and adding the scheme expressing transfinite induction along for an arbitrary àformula àas well as the axioms:
where ÃÂ is an arbitrary ÃÂ formula. In ÃÂ and ÃÂ we used the abbreviation ÃÂ for the formula , where ÃÂ 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 then define .
- is a weakened version of . In the system of , a set is instead called inductively defined if for some monotonic operator , is a fixed point of , rather than the least fixed point. This subtle difference makes the system significantly weaker: , while .
is weakened even further. In , not only does it use fixed points rather than least fixed points, and has induction only for positive formulas. This once again subtle difference makes the system even weaker: , while .
is the weakest of all variants of , based on W-types. The amount of weakening compared to regular iterated inductive definitions is identical to removing bar induction given a certain subsystem of second-order arithmetic. , while .
is an "unfolding" strengthening of . It is not exactly a first-order arithmetic system, but captures what one can get by predicative reasoning based on ý-times iterated generalized inductive definitions. The amount of increase in strength is identical to the increase from to : , while .