Considered as an alternative set theory axiom schema, it is called the Axiom (schema) of (set) induction.
It states, all properties , that if for every set , the truth of follows from the truth of for all elements of , then this property holds for all sets. In symbols:
Comparison with natural number inductionEdit
The above can be compared with -induction over the natural numbers for number properties . This may be expressed as
or, using the symbol for the tautologically true statement, ,
Fix a predicate and define a new predicate equivalent to , except with an argument offset by one and equal to for . With this we can also get a statement equivalent to induction for , but without a conjunction. By abuse of notation, letting " " denote , an instance of the -induction schema may thus be expressed as
This now mirrors an instance of the Set-Induction schema.
Conversely, Set-Induction may also be treated in a way that treats the bottom case explicitly.
With classical tautologies such as and , an instance of the -induction principle can be translated to the following statement:
This expresses that, for any property , either there is any (first) number for which does not hold, despite holding for the preceding case, or - if there is no such failure case - is true for all numbers.
Accordingly, in classical ZF, an instance of the set-induction can be translated to the following statement, clarifying what form of counter-example prevents a set-property to hold for all sets:
This expresses that, for any property , either there a set for which does not hold while being true for all elements of , or holds for all sets. For any property, if one can prove that implies , then the failure case is ruled out and the formula states that the disjunct must hold.
In the context of the constructive set theory CZF, adopting the Axiom of regularity would imply the law of excluded middle and also set-induction. But then the resulting theory would be standard ZF. However, conversely, the set-induction implies neither of the two. In other words, with a constructive logic framework, set-induction as stated above is strictly weaker than regularity.