## Set theory

In computer science, a set is a collection of certain values without any particular order. It corresponds with the mathematical concept of set, but with the restriction that it has to be finite. Disregarding sequence, it is the same as a list. A set can be seen as an associative array where the value of each key-value pair is ignored. ...more on Wikipedia about "Set (computer science)"

In set theory as usually formulated, referring to the set of all sets typically leads to a paradox. The reason for this is the form of Zermelo's axiom of separation: for any ...more on Wikipedia about "Set of all sets"

Set theory is the mathematical theory of sets, which represent collections of abstract objects. It encompasses the everyday notions, introduced in primary school, of collections of objects, and the elements of, and membership in, such collections. In most modern mathematical formalisms, set theory provides the language in which mathematical objects are described. It is (along with logic and the predicate calculus) one of the axiomatic foundations for mathematics, allowing mathematical objects to be constructed formally from the undefined terms of "set", and "set membership". It is in its own right a branch of mathematics and an active field of ongoing mathematical research. ...more on Wikipedia about "Set theory"

In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by indicating the properties that its members must satisfy. This is also known as set comprehension. ...more on Wikipedia about "Set-builder notation"

A set-theoretic definition of the natural numbers which does work in ZFC and related theories was proposed by John von Neumann. It allows the discussion of natural numbers in a system based (as modern mathematics is) on axiomatic set theory. ...more on Wikipedia about "Set-theoretic definition of natural numbers"

In mathematics, the limit of a sequence of sets A1, A2, ... is a set whose elements are determined by the sequence in either of two equivalent ways: ...more on Wikipedia about "Set-theoretic limit"

PROPOSITION 1: For any sets A, B, and C: ...more on Wikipedia about "Simple theorems in the algebra of sets"

In mathematics, a singleton is a set with exactly one element. For example, the set {0} is a singleton. Note that a set such as is also a singleton: the only element is a set (which itself is however not a singleton). ...more on Wikipedia about "Singleton (mathematics)"

A Skolem hull is an construction from mathematical logic. ...more on Wikipedia about "Skolem hull"

In mathematical logic, specifically set theory, Skolem's paradox is a direct result of the (downward) Löwenheim-Skolem theorem, which states that every model of a sentence of a first-order language has an elementarily equivalent countable submodel. ...more on Wikipedia about "Skolem's paradox"

In mathematics, particularly in set theory and model theory, there are at least three notions of stationary set: ...more on Wikipedia about "Stationary set"

In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing that a unique formal interpretation ...more on Wikipedia about "Stratification (mathematics)"

In set theory and its applications throughout mathematics, a subclass is a class contained in some other class in the same way that a subset is a set contained in some other set. ...more on Wikipedia about "Subclass (set theory)"

In mathematics, especially in set theory, a set A is a subset of a set B, if A is "contained" inside B. The relationship of one set being a subset of another is called inclusion. Every set is a subset of itself. ...more on Wikipedia about "Subset" Don't hesitate to contact stuff on www.shortopedia.com

That the set above is nonempty follows from Hartogs' theorem, which says for a well-orderable cardinal, we can construct a larger one. The minimum actually exists because the ordinals are well-ordered. It is therefore immediate that there is no cardinal number in between κ and κ+. A successor cardinal is a cardinal which is κ+ for some cardinal κ. In the infinite case, the successor operation skips over many ordinal numbers; in fact, every infinite cardinal is a limit ordinal. Therefore, the successor operation on cardinals gains a lot of power in the infinite case (relative the ordinal successorship operation), and consequently the cardinal numbers are a very "sparse" subclass of the ordinals. We define the sequence of alephs (via the axiom of replacement) via this operation, through all the ordinal numbers as follows: ...more on Wikipedia about "Successor cardinal"

When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation S to get the next higher one. Using von Neumann's ordinal numbers (the standard ordinals used in set theory), we have, for any ordinal number, ...more on Wikipedia about "Successor ordinal"

In mathematics, a cardinal number κ is called superstrong iff there exists an elementary embedding j : VM from V into a transitive inner model M with critical point κ and Vj(κ)M. ...more on Wikipedia about "Superstrong cardinal"

(Surjection) In mathematics, a function f is said to be surjective if and only if its values span its whole codomain; that is, for every y in the codomain, there is at least one x in the domain such that f(x) = y. ...more on Wikipedia about "Surjection"

In mathematics, Suslin's problem is a question about orders posed by M. Suslin in the early 1920s. In the 1960s, it was proved that the question is independent of the standard axiomatic system of set theory known as ZFC: the statement can neither be proven nor disproven from those axioms. ...more on Wikipedia about "Suslin's problem"

In mathematics, the symmetric difference of two sets is the set of elements which are in one of either set, but not in both. ...more on Wikipedia about "Symmetric difference"

In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a. ...more on Wikipedia about "Symmetric relation"

In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. This means that if we denote one such relation by ≤ then the following statements hold for all a, b and c in X: ...more on Wikipedia about "Total order"

In mathematics, a binary relation R over a set X is total if it holds for all a and b in X that a is related to b or b is related to a (or both). ...more on Wikipedia about "Total relation"

Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. It may be regarded as one of three forms of mathematical induction. ...more on Wikipedia about "Transfinite induction"

Transfinite numbers, also known as infinite numbers, are numbers that are not finite. These numbers were first considered by Indian Jaina mathematicians around the 4th century BC. ** ** ...more on Wikipedia about "Transfinite number" Evergreen shortopedia!!!