__ Theoretical Mathematics__ is the study of the

In order to clarify the __foundations of mathematics__, the fields of __mathematical logic__ and __set theory__ were developed.

**Mathematical logic**includes the mathematical study of__logic__and the applications of formal logic to other areas of mathematics.**Set theory**is the branch of mathematics that studies__sets__or collections of objects.**Category theory**, which deals in an abstract way with__mathematical structures__and relationships between them, is still in development.

__ Mathematical Logic__ is a subfield of

*The unifying themes in mathematical logic include the study of the expressive power of **formal systems** and the **deductive** power of formal **proof** systems. *

Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of __foundations of mathematics__. This study began in the late 19th century with the development of __axiomatic__ frameworks for __geometry__, __arithmetic__, and __analysis__. In the early 20th century it was shaped by __David Hilbert__'s __program__ to prove the consistency of foundational theories. Results of __Kurt Gödel__, __Gerhard Gentzen__, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in __reverse mathematics__) rather than trying to find theories in which all of mathematics can be developed.

**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__. The objects of study are models of__theories__in a__formal language__. A set of__sentences__in a__formal language__is called a**theory**; a**model**of a theory is a structure (e.g. an__interpretation__) that satisfies the sentences of that theory.**Recursion theory**

Computability theory**recursion theory**, is a branch of__mathematical logic__, of__computer science__, and of the__theory of computation__that originated in the 1930s with the study of__computable functions__and__Turing degrees__. The field has since expanded to include the study of generalized computability and definability. In these areas, recursion theory overlaps with__proof theory__and__effective descriptive set theory__.**Proof theory**

Proof theory__mathematical logic__that represents__proofs__as formal__mathematical objects__, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined__data structures__such as plain lists, boxed lists, or__trees__, which are constructed according to the__axioms__and__rules of inference__of the logical system. As such, proof theory is__syntactic__in nature, in contrast to__model theory__, which is__semantic__in nature.

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

Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all __mathematical objects__. From set theory's inception, some mathematicians __have objected to it__ as a __foundation for mathematics__. The most common objection to set theory, one __Kronecker__ voiced in set theory's earliest years, starts from the __constructivist__ view that mathematics is loosely related to computation. A different objection put forth by __Henri Poincaré__ is that defining sets using the axiom schemas of __specification__ and __replacement__, as well as the axiom of __power set__, introduces __impredicativity__, a type of __circularity__, into the definitions of mathematical objects. __Category theorists__ have proposed __topos theory__ as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as __constructivism__, finite set theory, and __computable__ set theory. Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for __pointless topology__ and __Stone spaces__. An active area of research is the __univalent foundations__ and related to it __homotopy type theory__. Here, sets may be defined as certain kinds of types, with __universal properties__ of sets arising from __higher inductive types__. Principles such as the __axiom of choice__ and the __law of the excluded middle__ appear in a spectrum of different forms, some of which can be proven, others which correspond to the classical notions; this allows for a detailed discussion of the effect of these axioms on mathematics.

**Combinatorial set theory**

**Combinatorial set theory**concerns extensions of finite__combinatorics__to infinite sets. This includes the study of__cardinal arithmetic__and the study of extensions of__Ramsey's theorem__such as the__Erdős–Rado theorem__.**Descriptive set theory**

Descriptive set theory__real line__and, more generally, subsets of__Polish spaces__. It begins with the study of__pointclasses__in the__Borel hierarchy__and extends to the study of more complex hierarchies such as the__projective hierarchy__and the__Wadge hierarchy__. Many properties of__Borel sets__can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.**Fuzzy set theory**

In set theory as__Cantor__defined and__Zermelo__and__Fraenkel__axiomatized, an object is either a member of a set or not. In__fuzzy set theory__this condition was relaxed by__Lotfi A. Zadeh__so an object has a*degree of membership*in a set, a number between 0 and 1. For example, the degree of membership of a person in the set of "tall people" is more flexible than a simple yes or no answer and can be a real number such as 0.75.**Inner model theory**

An**inner model**of Zermelo–Fraenkel set theory (ZF) is a transitive__class__that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the__constructible universe__*L*developed by Gödel. One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether a model*V*of ZF satisfies the__continuum hypothesis__or the__axiom of choice__, the inner model*L*constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together with these two principles is consistent.**Large cardinals**

A**large cardinal**is a cardinal number with an extra property. Many such properties are studied, including__inaccessible cardinals__,__measurable cardinals__, and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in Zermelo-Fraenkel set theory.**Determinacy**

Determinacy__axiom of determinacy__(AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the__Wadge degrees__have an elegant structure.**Forcing**

__Paul Cohen__invented the method of__forcing__while searching for a__model__of__ZFC__in which the__continuum hypothesis__fails, or a model of ZF in which the__axiom of choice__fails. Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the__natural numbers__without changing any of the__cardinal numbers__of the original model. Forcing is also one of two methods for proving__relative consistency__by finitistic methods, the other method being__Boolean-valued models__.**Cardinal invariants**

A**cardinal invariant**is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of__meagre sets__of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory.**Set-theoretic topology**

**Set-theoretic topology**studies questions of__general topology__that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem is the__normal Moore space question__, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.

__ Category Theory__ formalizes

**Magmas (Algebra)**

In__abstract algebra__, a**magma**(or**groupoid**; not to be confused with__groupoids__in__category theory__) is a basic kind of__algebraic structure__. Specifically, a magma consists of a__set__equipped with a single__binary operation__. The binary operation must be__closed__by definition but no other properties are imposed.**Topological Manifold**

In__topology__, a branch of__mathematics__, a**topological manifold**is a__topological space__(which may also be a__separated space__) which locally resembles__real__*n*-__dimensional__space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. There are a number of different types of differentiable manifolds,**differentiable manifold, smooth manifold,**, and**analytic manifold****complex manifold****.****Metric spaces**

In__mathematics__, a**metric space**is a__set__for which__distances__between all__members__of the set are defined. Those distances, taken together, are called a__metric__on the set. A metric on a space induces__topological properties__like__open__and__closed sets__, which lead to the study of more abstract__topological spaces__. The most familiar metric space is__3-dimensional Euclidean space__. In fact, a "metric" is the generalization of the__Euclidean metric__arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the__straight____line segment__connecting them. Other metric spaces occur for example in__elliptic geometry__and__hyperbolic geometry__, where distance on a__sphere__measured by angle is a metric, and the__hyperboloid model__of hyperbolic geometry is used by__special relativity__as a metric space of__velocities__.**R-modules**

In__mathematics__, a**module**is one of the fundamental__algebraic structures__used in__abstract algebra__. A**module over a**is a generalization of the notion of**ring**__vector space__over a__field__, wherein the corresponding__scalars__are the elements of an arbitrary given ring (with identity) and a multiplication (on the left and/or on the right) is defined between elements of the ring and elements of the module. Thus, a module, like a vector space, is an additive__abelian group__; a product is defined between elements of the ring and elements of the module that is distributive over the addition operation of each parameter and is__compatible__with the ring multiplication.**Rings**

In__mathematics__, a**ring**is one of the fundamental__algebraic structures__used in__abstract algebra__. It consists of a__set__equipped with two__binary operations__that generalize the__arithmetic operations__of__addition__and__multiplication__. Through this generalization, theorems from__arithmetic__are extended to non-numerical objects such as__polynomials__,__series__,__matrices__and__functions__. A ring is an__abelian group__with a second__binary operation__that is associative, is__distributive__over the abelian group operation, and has an__identity element__(this last property is not required by some authors. By extension from the__integers__, the abelian group operation is called*addition*and the second binary operation is called*multiplication***Sets**

I__mathematics__, a**set**is a collection of distinct objects, considered as an__object__in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century,__set theory__is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In__mathematics education__, elementary topics from set theory such as__Venn diagrams__are taught at a young age, while more advanced concepts are taught as part of a university degree.**Groups**

I__mathematics__, a**group**is an__algebraic structure__consisting of a__set__of__elements__equipped with an__operation__that combines any two elements to form a third element and that satisfies four conditions called the group__axioms__, namely__closure__,__associativity__,__identity__and__invertibility__. One of the most familiar examples of a group is the set of__integers__together with the__addition__operation, but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, applies much more widely. It allows entities with highly diverse mathematical origins in__abstract algebra__and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.**Topological spaces**

In__topology__and related branches of__mathematics__, a**topological space**may be defined as a__set__of__points__, along with a set of__neighbourhoods__for each point, satisfying a set of__axioms__relating points and neighbourhoods. The definition of a topological space relies only upon__set theory__and is the most general notion of a__mathematical space__that allows for the definition of concepts such as__continuity__,__connectedness__, and__convergence__. Other spaces, such as__manifolds__and__metric spaces__, are specializations of topological spaces with extra__structures__or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called__point-set topology__or__general topology__.**Uniform spaces**

In the__mathematical__field of__topology__, a**uniform space**is a__set__with a**uniform**. Uniform spaces are**structure**__topological spaces__with additional structure that is used to define__uniform properties__such as__completeness__,__uniform continuity__and__uniform convergence__. The conceptual difference between uniform and__topological structures__is that, in a uniform space, one can formalize certain notions of relative closeness and closeness of points. In other words, ideas like "*x*is closer to*a*than*y*is to*b*" make sense in uniform spaces. By comparison, in a general topological space, given sets*A,B*it is meaningful to say that a point*x*is*arbitrarily close*to*A*(i.e., in the closure of A), or perhaps that*A*is a*smaller neighborhood*of*x*than*B*, but notions of closeness of points and relative closeness are not described well by topological structure alone.**Vector spaces**

In algebra, given a__ring__*R*, the**category of left modules**over*R*is the category whose objects are all left__modules__over*R*and whose morphisms are all__module homomorphisms__between left*R*-modules. For example, when*R*is the ring of integers**Z**, it is the same thing as the__category of abelian groups__. The**category of right modules**is defined in a similar way. The__category__**K-Vect**(some authors use**Vect***K*) has all__vector spaces__over a fixed__field__*K*as__objects__and*K*__-linear transformations__as__morphisms__. Since vector spaces over*K*(as a field) are the same thing as__modules__over the__ring__*K*,**K-Vect**is a special case of**R-Mod**, the category of left*R*-modules. Much of__linear algebra__concerns the description of**K-Vect**. For example, the__dimension theorem for vector spaces__says that the__isomorphism classes__in**K-Vect**correspond exactly to the__cardinal numbers__, and that**K-Vect**is__equivalent__to the__subcategory__of**K-Vect**which has as its objects the free vector spaces*Kn*, where*n*is any cardinal number.

**In **__ Theory of Computation__ (

**Automata theory**

Automata theory is the study of abstract machines (or more appropriately, abstract 'mathematical' machines or systems) and the computational problems that can be solved using these machines. These abstract machines are called automata. Automata comes from the Greek word (Αυτόματα) which means that something is doing something by itself. Automata theory is also closely related to__formal language__theory, as the automata are often classified by the class of formal languages they are able to recognize. An automaton can be a finite representation of a formal language that may be an infinite set. Automata are used as theoretical models for computing machines, and are used for proofs about computability.**Computability theory**

Computability theory deals primarily with the question of the extent to which a problem is solvable on a computer. The statement that the__halting problem__cannot be solved by a Turing machine is one of the most important results in computability theory, as it is an example of a concrete problem that is both easy to formulate and impossible to solve using a Turing machine. Much of computability theory builds on the halting problem result. Another important step in computability theory was__Rice's theorem__, which states that for all non-trivial properties of partial functions, it is__undecidable__whether a Turing machine computes a partial function with that property.**Computational complexity theory**

__Complexity theory__considers not only whether a problem can be solved at all on a computer, but also how efficiently the problem can be solved. Two major aspects are considered: time complexity and space complexity, which are respectively how many steps does it take to perform a computation, and how much memory is required to perform that computation. In order to analyze how much time and space a given__algorithm__requires, computer scientists express the time or space required to solve the problem as a function of the size of the input problem.

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