Inductive set

This article is about the notion in descriptive set theory. For the use in foundations of mathematics, see axiom of infinity.

Bourbaki also defines an inductive set to be a partially ordered set that satisfies the hypothesis of Zorn's lemma when nonempty.

In descriptive set theory, an inductive set of real numbers (or more generally, an inductive subset of a Polish space) is one that can be defined as the least fixed point of a monotone operation definable by a positive Σ¹_n formula, for some natural number n, together with a real parameter.

Inductive set can be defined as follows: A set J of real numbers is called an inductive set if it obeys the following : the number 1 belongs to J, if d is an element of J then d+1 is also an element of J.

The inductive sets form a boldface pointclass; that is, they are closed under continuous preimages. In the Wadge hierarchy, they lie above the projective sets and below the sets in L(R). Assuming sufficient determinacy, the class of inductive sets has the scale property and thus the prewellordering property.

References

Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.

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