Inductive set
Encyclopedia
In descriptive set theory
, an inductive set of real number
s (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 Σ1n formula, for some natural number n, together with a real parameter.
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.
Descriptive set theory
In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces...
, an inductive set of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s (or more generally, an inductive subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of a Polish space
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish...
) is one that can be defined as the least fixed point of a monotone operation definable by a positive Σ1n formula, for some natural number n, together with a real parameter.
The inductive sets form a boldface pointclass; that is, they are closed under continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
preimages. In the Wadge hierarchy
Wadge hierarchy
In descriptive set theory, 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.- Wadge degrees :...
, they lie above the projective sets and below the sets in L(R)
L(R)
In set theory, L is the smallest transitive inner model of ZF containing all the ordinals and all the reals. It can be constructed in a manner analogous to the construction of L , by adding in all the reals at the start, and then iterating the definable powerset operation through all the...
. Assuming sufficient determinacy
Determinacy
In set theory, a branch of mathematics, determinacy is the study of under what circumstances one or the other player of a game must have a winning strategy, and the consequences of the existence of such strategies.-Games:...
, the class of inductive sets has the scale property
Scale property
In the mathematical discipline of descriptive set theory, a scale is a certain kind of object defined on a set of points in some Polish space...
and thus the prewellordering property.