:''Bourbaki also defines an inductive set to be a partially ordered set that satisfies the hypothesis of
Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...
when nonempty.''
In
descriptive set theory
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" set (mathematics), subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has a ...
, an inductive set of
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s (or more generally, an inductive
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of a
Polish space
In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that h ...
) is one that can be defined as the least fixed point of a monotone operation definable by a positive Σ
1''n'' formula, for some natural number ''n'', together with a real parameter.
The inductive sets form a
boldface pointclass
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a ''point'' is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by ...
; that is, they are closed under
continuous preimage
In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y.
More generally, evaluating f at each ...
s. In the
Wadge hierarchy In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. W ...
, they lie above the
projective set
In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is \boldsymbol^1_n for some positive integer n. Here A is
* \boldsymbol^1_1 if A is analytic
* \boldsymbol^1_n if the complement of A, X\se ...
s and below the sets in
L(R)
In set theory, L(R) (pronounced ''L of R'') is the smallest transitive class, transitive inner model of Zermelo–Fraenkel set theory, ZF containing all the ordinal number, ordinals and all the real number, reals.
Construction
L(R) can be constru ...
. Assuming sufficient
determinacy
Determinacy is a subfield of game theory and set theory that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "dete ...
, the class of inductive sets has the
scale property and thus the
prewellordering property.
The term can have a number of different meanings:
*Russell's definition, an inductive set is a nonempty partially ordered set in which every element has a successor. An example is the set of natural numbers N, where 0 is the first element, and the others are produced by adding 1 successively.
*Roitman considers the same construction in a more concrete form: the elements are sets, the empty set
among them, and the successor of every element
is the set
. In particular, every inductive set contains the sequence
.
*For many other authors (e.g., ''Bourbaki)'', an inductive set is a partially ordered set in which every totally ordered subset has an upper bound, i.e., it is a set fulfilling the assumption of Zorn's lemma.
References
*
Descriptive set theory
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