Mathematical logic

thumb|Flowchart of using successive subtractions to find the greatest common divisor of number r and s|alt=In a loop, subtract the larger number against the smaller number. Halt the loop when the subtraction will make a number negative. Assess two numbers, whether one of them is equal to zero or not. If yes, take the other number as the greatest common divisor. If no, put the two numbers in the subtraction loop again. In mathematics and computer science, an algorithm () is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perf

branch of mathematics that studies sets, which are collections of objects

rigorous demonstration that a mathematical statement follows from its premises

subfield of mathematics

value that can change, usually with a context of an equation or operation

form of mathematical proof

meanings of symbols used in mathematics

theorem that a wide class of logical systems cannot be both consistent and complete

property that assigns truth values to k-tuples of individuals

alt=A set of number blocks. The blocks 1, 2, and 3 are in the foreground; six other blocks can be seen in the background|thumb|Number blocks, which can be used for counting Counting is the process of determining the number of elements of a finite set of objects; that is, determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counte

class=skin-invert-image|450px|thumb|Infinitesimals (ε) and infinities (ω) on the hyperreal number line (ε = 1/ω)

mathematical relationship asserting that two quantities have the same value

logical formula which is true in every possible interpretation

study of computable functions and Turing degrees

formal logic able to express concepts such as necessity, possibility, provability, obligation, knowledge etc.

study of classes of mathematical structures from the perspective of mathematical logic

axiomatic system for the natural numbers

study of the basic mathematical concepts

structure of a formal language

function that returns 1 if an element is present in a specified subset and 0 if absent; naturally isomorphic with a set's subsets

a totally ordered proper class containing the real numbers as well as hyperreal numbers such as infinity and infinitesimals.

branch of mathematical logic

concept in logic

concept of mathematical logic

In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as . The contrapositive of a statement has its antecedent and consequent negated and swapped.

mathematical expression that may form a separable part of an equation, a series, or another expression; used in in mathematical logic, universal algebra, and rewriting systems

thumb|The title page of the Principia Mathematica (shortened version), an important work of metamathematics

paradox in naïve set theory that there is no set of all cardinal numbers

in set theory, a set large enough such that most ordinary mathematical constructions can take place within it

In mathematics and computer science, the ; ) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers an inputted statement and answers "yes" or "no" according to whether it is universally valid, i.e., valid in every structure. Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936.

a paradox in which an arbitrary claim F is proved from the mere existence of a sentence C that says of itself “If C, then F”

theory of logic based on the orthocomplemented lattice of closed subspaces of a separable Hilbert space, regarded as the lattice of quantum propositions

attempt to formalize all of mathematics, based on a finite set of axioms

theorem that, for any signature 𝜎, any infinite 𝜎-structure 𝑀 and any infinite cardinal 𝜅≥|𝜎|, there is a 𝜎‐structure 𝑁 of cardinality 𝜅 that is either an elementary substructure or an elementary extension of 𝑀

assignment of each symbol and well-formed formula of a formal language a unique natural number

Mereology (; from Greek μέρος 'part' (root: μερε-, mere-) and the suffix -logy, 'study, discussion, science') is the philosophical study of part-whole relationships, also called parthood relationships. As a branch of metaphysics, mereology examines the connections between parts and their wholes, exploring how components interact within a system. This theory has roots in ancient philosophy, with significant contributions from Plato, Aristotle, and later, medieval and Renaissance thinkers like Thomas Aquinas and John Duns Scotus. Mereology was formally axiomatized in the 20th century by Polish l

theorem

In mathematics, linguistics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduction systems). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects.

relation that occurs when the order of the elements in a given relation is switched

finite sequence of symbols from a given alphabet that is part of a formal language

defining the elements in a set in terms of other elements in the set

mathematical function

term in mathematical logic

study of the semantics, or interpretations, of formal and natural languages

An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (for example, whether the set must be finite, or whether the list is allowed to contain repetitions) depend on the discipline of study and the context of a given problem.

fundamental concept in metalogic, and the term may be used without qualification with differing meanings depending on the context within mathematical logic

type of logic whose elements are concepts

form of mathematical proof

in logic, atomic formula (atom) or its negation

notion in mathematical logic

Concept in model theory

in mathematical logic, a well-formed formula with no free variables

theorem that truth in the standard model of a formal system cannot be defined within the system

The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is the quotient set of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal.

elementary operation on a natural number

set together with an interpretation of a given first-order language

Axiom used in logic and philosophy

semantics for modal logics

hypothetical universal science modeled on mathematics

well-quasi-ordering of finite trees