Inductive set

Inductive set

: "This article relates to the notion of inductive sets from descriptive set theory. For the notion in the context of the axiom of infinity, see Inductive set (axiom of infinity)."

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.

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

*


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Inductive set (axiom of infinity) — In the context of the axiom of infinity, an inductive set (also known as a successor set) is a set X with the property that, for every x in X, the successor x of x is also an element of X. An example of an inductive set is the set of natural… …   Wikipedia

  • Inductive reasoning aptitude — Inductive reasoning is a measurable aptitude for how well a person can identify a pattern within a large amount of data. Measurement is generally done in a timed test by showing four pictures or words and asking the test taker to identify which… …   Wikipedia

  • Inductive logic programming — (ILP) is a subfield of machine learning which uses logic programming as a uniform representation for examples, background knowledge and hypotheses. Given an encoding of the known background knowledge and a set of examples represented as a logical …   Wikipedia

  • Inductive reasoning — Induction or inductive reasoning, sometimes called inductive logic, is the process of reasoning in which the premises of an argument are believed to support the conclusion but do not entail it; i.e. they do not ensure its truth. Induction is a… …   Wikipedia

  • Inductive charging — Magne Charge wall, handheld, and floor mount Inductive charging uses an electromagnetic field to transfer energy between two objects. This is usually done with a charging station. Energy is sent through inductive coupling to an electrical device …   Wikipedia

  • Inductive bias — The inductive bias of a learning algorithm is the set of assumptions that the learner uses to predict outputs given inputs that it has not encountered (Mitchell, 1980).In machine learning, one aims to construct algorithms that are able to learn… …   Wikipedia

  • Inductive dimension — In the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind(X) or the large inductive dimension Ind(X). These are based on the observation that, in n… …   Wikipedia

  • Morse–Kelley set theory — In the foundation of mathematics, Morse–Kelley set theory (MK) or Kelley–Morse set theory (KM) is a first order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory (NBG). While von Neumann–Bernays–Gödel set theory …   Wikipedia

  • Von Neumann–Bernays–Gödel set theory — In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC. A statement in the language of ZFC is provable in NBG if and only …   Wikipedia

  • Scott–Potter set theory — An approach to the foundations of mathematics that is of relatively recent origin, Scott–Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”