Inaccessible cardinalIn set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal κ is strongly inaccessible if it is uncountable, it is not a sum of fewer than κ cardinals smaller than κ, and implies . The term "inaccessible cardinal" is ambiguous. Until about 1950, it meant "weakly inaccessible cardinal", but since then it usually means "strongly inaccessible cardinal".
LogicLogic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language.
Independence (mathematical logic)In mathematical logic, independence is the unprovability of a sentence from other sentences. A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that σ is false. Sometimes, σ is said (synonymously) to be undecidable from T; this is not the same meaning of "decidability" as in a decision problem. A theory T is independent if each axiom in T is not provable from the remaining axioms in T.
Constructible universeIn mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis".
Universe (mathematics)In mathematics, and particularly in set theory, , type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation. In set theory, universes are often classes that contain (as elements) all sets for which one hopes to prove a particular theorem. These classes can serve as inner models for various axiomatic systems such as ZFC or Morse–Kelley set theory. Universes are of critical importance to formalizing concepts in inside set-theoretical foundations.
Von Neumann universeIn set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC. The concept is named after John von Neumann, although it was first published by Ernst Zermelo in 1930. The rank of a well-founded set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set.
Elementary equivalenceIn model theory, a branch of mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences. If N is a substructure of M, one often needs a stronger condition. In this case N is called an elementary substructure of M if every first-order σ-formula φ(a1, ..., an) with parameters a1, ..., an from N is true in N if and only if it is true in M. If N is an elementary substructure of M, then M is called an elementary extension of N.
Measurable cardinalIn mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal κ, or more generally on any set. For a cardinal κ, it can be described as a subdivision of all of its subsets into large and small sets such that κ itself is large, ∅ and all singletons , α ∈ κ are small, complements of small sets are large and vice versa. The intersection of fewer than κ large sets is again large.
Forcing (mathematics)In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe to a larger universe by introducing a new "generic" object . Forcing was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory.
Zermelo–Fraenkel set theoryIn set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.
New FoundationsIn mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name. Much of this entry discusses NF with urelements (NFU), an important variant of NF due to Jensen and clarified by Holmes. In 1940 and in a revision in 1951, Quine introduced an extension of NF sometimes called "Mathematical Logic" or "ML", that included proper classes as well as sets.
Model theoryIn mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself.
Axiom of determinacyIn mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy. Steinhaus and Mycielski's motivation for AD was its interesting consequences, and suggested that AD could be true in the smallest natural model L(R) of a set theory, which accepts only a weak form of the axiom of choice (AC) but contains all real and all ordinal numbers.
Transfinite numberIn mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term transfinite was coined in 1895 by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were, nevertheless, not finite.
Gödel's incompleteness theoremsGödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.
Continuum hypothesisIn mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that there is no set whose cardinality is strictly between that of the integers and the real numbers, or equivalently, that any subset of the real numbers is finite, is countably infinite, or has the same cardinality as the real numbers. In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: , or even shorter with beth numbers: .
Set theorySet theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory.
