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Θ (set theory)

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In set theory, Θ (pronounced like the letter theta) is the least nonzero ordinal α such that there is no surjection from the reals onto α. If the axiom of choice (AC) holds (or even if the reals can be wellordered), then Θ is simply (2^)^+, the cardinal successor of the cardinality of the continuum. [1]

17 relations: Axiom of choice, Axiom of determinacy, Axiom of power set, Axiom schema of replacement, Burali-Forti paradox, Cardinality of the continuum, Class (set theory), Hartogs number, Infimum and supremum, Injective function, Model theory, Ordinal number, Prewellordering, Set theory, Surjective function, Theta, Well-order.

Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

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Axiom of determinacy

In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962.

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Axiom of power set

In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory.

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Axiom schema of replacement

In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set.

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Burali-Forti paradox

In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction.

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Cardinality of the continuum

In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum.

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Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.

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Hartogs number

In mathematics, specifically in axiomatic set theory, a Hartogs number is a particular kind of cardinal number.

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Infimum and supremum

In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.

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Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

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Model theory

In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.

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Ordinal number

In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.

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In set theory, a prewellordering is a binary relation \le that is transitive, total, and wellfounded (more precisely, the relation x\le y\land y\nleq x is wellfounded).

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Set theory

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.

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Surjective function

In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).

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Theta (uppercase Θ or ϴ, lowercase θ (which resembles digit 0 with horizontal line) or ϑ; θῆτα thē̂ta; Modern: θήτα| thī́ta) is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth.

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In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.

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Th (Set theory), Theta (set theory), Θ (Set theory).


[1] https://en.wikipedia.org/wiki/Θ_(set_theory)

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