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Set theory

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176-475: Set 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

352-415: A cumulative hierarchy of sets. New Foundations takes a different approach; it allows objects such as the set of all sets at the cost of restrictions on its set-existence axioms. The system of Kripke–Platek set theory is closely related to generalized recursion theory. Two famous statements in set theory are the axiom of choice and the continuum hypothesis . The axiom of choice, first stated by Zermelo,

528-710: A mathematical function as a relation from one set (the domain ) to another set (the range ). Mathematical logic Mathematical logic is the study of formal logic within mathematics . Major subareas include model theory , proof theory , set theory , and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics . Since its inception, mathematical logic has both contributed to and been motivated by

704-1117: A common denominator. This can be achieved by scaling the first number with the denominator of the second number while scaling the second number with the denominator of the first number. For instance, 1 3 + 1 2 = 1 ⋅ 2 3 ⋅ 2 + 1 ⋅ 3 2 ⋅ 3 = 2 6 + 3 6 = 5 6 {\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{2}}={\tfrac {1\cdot 2}{3\cdot 2}}+{\tfrac {1\cdot 3}{2\cdot 3}}={\tfrac {2}{6}}+{\tfrac {3}{6}}={\tfrac {5}{6}}} . Two rational numbers are multiplied by multiplying their numerators and their denominators respectively, as in 2 3 ⋅ 2 5 = 2 ⋅ 2 3 ⋅ 5 = 4 15 {\displaystyle {\tfrac {2}{3}}\cdot {\tfrac {2}{5}}={\tfrac {2\cdot 2}{3\cdot 5}}={\tfrac {4}{15}}} . Dividing one rational number by another can be achieved by multiplying

880-434: A correspondence between syntax and semantics in first-order logic. Gödel used the completeness theorem to prove the compactness theorem , demonstrating the finitary nature of first-order logical consequence . These results helped establish first-order logic as the dominant logic used by mathematicians. In 1931, Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems , which proved

1056-615: A cumulative hierarchy. NF and NFU include a "set of everything", relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold. Despite NF's ontology not reflecting the traditional cumulative hierarchy and violating well-foundedness, Thomas Forster has argued that it does reflect an iterative conception of set . Systems of constructive set theory , such as CST, CZF, and IZF, embed their set axioms in intuitionistic instead of classical logic . Yet other systems accept classical logic but feature

1232-430: A definition of the real numbers in terms of Dedekind cuts of rational numbers, a definition still employed in contemporary texts. Georg Cantor developed the fundamental concepts of infinite set theory. His early results developed the theory of cardinality and proved that the reals and the natural numbers have different cardinalities. Over the next twenty years, Cantor developed a theory of transfinite numbers in

1408-400: A finitistic system together with a principle of transfinite induction . Gentzen's result introduced the ideas of cut elimination and proof-theoretic ordinals , which became key tools in proof theory. Gödel gave a different consistency proof, which reduces the consistency of classical arithmetic to that of intuitionistic arithmetic in higher types. The first textbook on symbolic logic for

1584-415: A formalized mathematical statement, whether the statement is true or false. Ernst Zermelo gave a proof that every set could be well-ordered , a result Georg Cantor had been unable to obtain. To achieve the proof, Zermelo introduced the axiom of choice , which drew heated debate and research among mathematicians and the pioneers of set theory. The immediate criticism of the method led Zermelo to publish

1760-561: A foundation for modern mathematics, there has been support for the idea of introducing the basics of naive set theory early in mathematics education . In the US in the 1960s, the New Math experiment aimed to teach basic set theory, among other abstract concepts, to primary school students, but was met with much criticism. The math syllabus in European schools followed this trend, and currently includes

1936-521: A foundational theory for mathematics. Fraenkel proved that the axiom of choice cannot be proved from the axioms of Zermelo's set theory with urelements . Later work by Paul Cohen showed that the addition of urelements is not needed, and the axiom of choice is unprovable in ZF. Cohen's proof developed the method of forcing , which is now an important tool for establishing independence results in set theory. Leopold Löwenheim and Thoralf Skolem obtained

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2112-494: A function as a rule for computation, or a smooth graph, were no longer adequate. Weierstrass began to advocate the arithmetization of analysis , which sought to axiomatize analysis using properties of the natural numbers. The modern (ε, δ)-definition of limit and continuous functions was already developed by Bolzano in 1817, but remained relatively unknown. Cauchy in 1821 defined continuity in terms of infinitesimals (see Cours d'Analyse, page 34). In 1858, Dedekind proposed

2288-442: A general identity element since 1 is not the neutral element for the base. Exponentiation and logarithm are neither commutative nor associative. Different types of arithmetic systems are discussed in the academic literature. They differ from each other based on what type of number they operate on, what numeral system they use to represent them, and whether they operate on mathematical objects other than numbers. Integer arithmetic

2464-422: 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 . A cardinal invariant

2640-410: A limited amount of basic numerals, which directly refer to certain numbers. The system governs how these basic numerals may be combined to express any number. Numeral systems are either positional or non-positional. All early numeral systems were non-positional. For non-positional numeral systems, the value of a digit does not depend on its position in the numeral. The simplest non-positional system

2816-421: A milestone in recursion theory and proof theory, but has also led to Löb's theorem in modal logic. The method of forcing is employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics. The mathematical field of category theory uses many formal axiomatic methods, and includes the study of categorical logic , but category theory is not ordinarily considered

2992-485: 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. The study of inner models is common in the study of determinacy and large cardinals , especially when considering axioms such as

3168-561: A model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory , and they are a key reason for the prominence of first-order logic in mathematics. Gödel's incompleteness theorems establish additional limits on first-order axiomatizations. The first incompleteness theorem states that for any consistent, effectively given (defined below) logical system that

3344-460: A more complex non-positional numeral system . They have additional symbols for numbers like 10, 100, 1000, and 10,000. These symbols can be combined into a sum to more conveniently express larger numbers. For instance, the numeral for 10,405 uses one time the symbol for 10,000, four times the symbol for 100, and five times the symbol for 1. A similar well-known framework is the Roman numeral system . It has

3520-475: 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 . Within homotopy type theory, a set may be regarded as a homotopy 0-type, with universal properties of sets arising from the inductive and recursive properties of higher inductive types . Principles such as

3696-403: A new concept – the computable function – had been discovered, and that this definition was robust enough to admit numerous independent characterizations. In his work on the incompleteness theorems in 1931, Gödel lacked a rigorous concept of an effective formal system; he immediately realized that the new definitions of computability could be used for this purpose, allowing him to state

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3872-711: A nonstandard membership relation. These include rough set theory and fuzzy set theory , in which the value of an atomic formula embodying the membership relation is not simply True or False . The Boolean-valued models of ZFC are a related subject. An enrichment of ZFC called internal set theory was proposed by Edward Nelson in 1977. Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs , manifolds , rings , vector spaces , and relational algebras can all be defined as sets satisfying various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and

4048-639: A number, it is also possible to multiply by its reciprocal . The reciprocal of a number is 1 divided by that number. For instance, 48 ÷ 8 = 48 × 1 8 {\displaystyle 48\div 8=48\times {\tfrac {1}{8}}} . The multiplicative identity element is 1 and the multiplicative inverse of a number is the reciprocal of that number. For example, 13 × 1 = 13 {\displaystyle 13\times 1=13} and 13 × 1 13 = 1 {\displaystyle 13\times {\tfrac {1}{13}}=1} . Multiplication

4224-485: A particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has

4400-490: A plane. Further branches of number theory are probabilistic number theory , which employs methods from probability theory , combinatorial number theory , which relies on the field of combinatorics , computational number theory , which approaches number-theoretic problems with computational methods, and applied number theory, which examines the application of number theory to fields like physics , biology , and cryptography . Influential theorems in number theory include

4576-422: A portion of set theory directly in their semantics. The most well studied infinitary logic is L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} . In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it

4752-553: A positive number as its base. The same is true for the logarithm of positive real numbers as long as the logarithm base is positive and not 1. Irrational numbers involve an infinite non-repeating series of decimal digits. Because of this, there is often no simple and accurate way to express the results of arithmetic operations like 2 + π {\displaystyle {\sqrt {2}}+\pi } or e ⋅ 3 {\displaystyle e\cdot {\sqrt {3}}} . In cases where absolute precision

4928-507: A range of values if one does not know the precise magnitude, for example, because of measurement errors . Interval arithmetic includes operations like addition and multiplication on intervals, as in [ 1 , 2 ] + [ 3 , 4 ] = [ 4 , 6 ] {\displaystyle [1,2]+[3,4]=[4,6]} and [ 1 , 2 ] × [ 3 , 4 ] = [ 3 , 8 ] {\displaystyle [1,2]\times [3,4]=[3,8]} . It

5104-478: A rough division of contemporary mathematical logic into four areas: Additionally, sometimes the field of computational complexity theory is also included as part of mathematical logic. Each area has a distinct focus, although many techniques and results are shared among multiple areas. The borderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are not always sharp. Gödel's incompleteness theorem marks not only

5280-485: A second exposition of his result, directly addressing criticisms of his proof. This paper led to the general acceptance of the axiom of choice in the mathematics community. Skepticism about the axiom of choice was reinforced by recently discovered paradoxes in naive set theory . Cesare Burali-Forti was the first to state a paradox: the Burali-Forti paradox shows that the collection of all ordinal numbers cannot form

5456-423: A separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of

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5632-404: A series of publications. In 1891, he published a new proof of the uncountability of the real numbers that introduced the diagonal argument , and used this method to prove Cantor's theorem that no set can have the same cardinality as its powerset . Cantor believed that every set could be well-ordered , but was unable to produce a proof for this result, leaving it as an open problem in 1895. In

5808-444: A series of two operations, it does not matter which operation is carried out first. This is the case for multiplication, for example, since ( 5 × 4 ) × 2 {\displaystyle (5\times 4)\times 2} is the same as 5 × ( 4 × 2 ) {\displaystyle 5\times (4\times 2)} . Addition is an arithmetic operation in which two numbers, called

5984-454: A set A . If o is a member (or element ) of A , the notation o ∈ A is used. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }. Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets. A derived binary relation between two sets is the subset relation, also called set inclusion . If all

6160-457: A set. Very soon thereafter, Bertrand Russell discovered Russell's paradox in 1901, and Jules Richard discovered Richard's paradox . Zermelo provided the first set of axioms for set theory. These axioms, together with the additional axiom of replacement proposed by Abraham Fraenkel , are now called Zermelo–Fraenkel set theory (ZF). Zermelo's axioms incorporated the principle of limitation of size to avoid Russell's paradox. In 1910,

6336-402: A similar role in the sciences , like physics and economics . Arithmetic is present in many aspects of daily life , for example, to calculate change while shopping or to manage personal finances . It is one of the earliest forms of mathematics education that students encounter. Its cognitive and conceptual foundations are studied by psychology and philosophy . The practice of arithmetic

6512-439: A special type of rational numbers since their denominator is a power of 10. For instance, 0.3 is equal to 3 10 {\displaystyle {\tfrac {3}{10}}} , and 25.12 is equal to 2512 100 {\displaystyle {\tfrac {2512}{100}}} . Every rational number corresponds to a finite or a repeating decimal . Irrational numbers are numbers that cannot be expressed through

6688-831: A spectacular blunder in Remarks on the Foundations of Mathematics : Wittgenstein attempted to refute Gödel's incompleteness theorems after having only read the abstract. As reviewers Kreisel , Bernays , Dummett , and Goodstein all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such as Crispin Wright begun to rehabilitate Wittgenstein's arguments. 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

6864-571: A stronger limitation than the one established by the Löwenheim–Skolem theorem. The second incompleteness theorem states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached. Many logics besides first-order logic are studied. These include infinitary logics , which allow for formulas to provide an infinite amount of information, and higher-order logics , which include

7040-402: A subfield of mathematical logic. Because of its applicability in diverse fields of mathematics, mathematicians including Saunders Mac Lane have proposed category theory as a foundational system for mathematics, independent of set theory. These foundations use toposes , which resemble generalized models of set theory that may employ classical or nonclassical logic. Mathematical logic emerged in

7216-500: A variation of the logical system of Boole and Schröder but adding quantifiers. Peano was unaware of Frege's work at the time. Around the same time Richard Dedekind showed that the natural numbers are uniquely characterized by their induction properties. Dedekind proposed a different characterization, which lacked the formal logical character of Peano's axioms. Dedekind's work, however, proved theorems inaccessible in Peano's system, including

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7392-581: Is computable ; this is not true in classical theories of arithmetic such as Peano arithmetic . Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras . Set theory

7568-556: Is exponentiation by squaring . It breaks down the calculation into a number of squaring operations. For example, the exponentiation 3 65 {\displaystyle 3^{65}} can be written as ( ( ( ( ( 3 2 ) 2 ) 2 ) 2 ) 2 ) 2 × 3 {\displaystyle (((((3^{2})^{2})^{2})^{2})^{2})^{2}\times 3} . By taking advantage of repeated squaring operations, only 7 individual operations are needed rather than

7744-406: Is 0 and the additive inverse of a number is the negative of that number. For instance, 13 + 0 = 13 {\displaystyle 13+0=13} and 13 + ( − 13 ) = 0 {\displaystyle 13+(-13)=0} . Addition is both commutative and associative. Multiplication is an arithmetic operation in which two numbers, called the multiplier and

7920-426: Is 0. 3 . Every repeating decimal expresses a rational number. Real number arithmetic is the branch of arithmetic that deals with the manipulation of both rational and irrational numbers. Irrational numbers are numbers that cannot be expressed through fractions or repeated decimals, like the root of 2 and π . Unlike rational number arithmetic, real number arithmetic is closed under exponentiation as long as it uses

8096-530: 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 refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from

8272-442: Is a partial list of them: Some basic sets of central importance are the set of natural numbers , the set of real numbers and the empty set —the unique set containing no elements. The empty set is also occasionally called the null set , though this name is ambiguous and can lead to several interpretations. The Power set of a set A , denoted P ( A ) {\displaystyle {\mathcal {P}}(A)} ,

8448-517: Is a particular formal system of logic . Its syntax involves only finite expressions as well-formed formulas , while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse . Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it

8624-545: Is a prime number that has no other prime factorization. Euclid's theorem states that there are infinitely many prime numbers. Fermat's last theorem is the statement that no positive integer values can be found for a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} , to solve the equation a n + b n = c n {\displaystyle a^{n}+b^{n}=c^{n}} if n {\displaystyle n}

8800-741: 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 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

8976-465: Is a relatively crude method, with some unintuitive subtleties; explicitly keeping track of an estimate or upper bound of the approximation error is a more sophisticated approach. In the example, the person's height might be represented as 1.62 ± 0.005 meters or 63.8 ± 0.2 inches . In performing calculations with uncertain quantities, the uncertainty should be propagated to calculated quantities. When adding or subtracting two or more quantities, add

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9152-902: Is a similar process in which the last preserved digit is increased by one if the next digit is 5 or greater but remains the same if the next digit is less than 5, so that the rounded number is the best approximation of a given precision for the original number. For instance, if the number π is rounded to 4 decimal places, the result is 3.142 because the following digit is a 5, so 3.142 is closer to π than 3.141. These methods allow computers to efficiently perform approximate calculations on real numbers. In science and engineering, numbers represent estimates of physical quantities derived from measurement or modeling. Unlike mathematically exact numbers such as π or ⁠ 2 {\displaystyle {\sqrt {2}}} ⁠ , scientifically relevant numerical data are inherently inexact, involving some measurement uncertainty . One basic way to express

9328-479: Is a subset of B , but A is not equal to B . Also, 1, 2, and 3 are members (elements) of the set {1, 2, 3} , but are not subsets of it; and in turn, the subsets, such as {1} , are not members of the set {1, 2, 3} . More complicated relations can exist; for example, the set {1} is both a member and a proper subset of the set {1, {1}} . Just as arithmetic features binary operations on numbers , set theory features binary operations on sets. The following

9504-505: Is about calculations with positive and negative integers . Rational number arithmetic involves operations on fractions of integers. Real number arithmetic is about calculations with real numbers , which include both rational and irrational numbers . Another distinction is based on the numeral system employed to perform calculations. Decimal arithmetic is the most common. It uses the basic numerals from 0 to 9 and their combinations to express numbers . Binary arithmetic, by contrast,

9680-450: Is an inverse of the operation " ∘ {\displaystyle \circ } " if it fulfills the following condition: t ⋆ s = r {\displaystyle t\star s=r} if and only if r ∘ s = t {\displaystyle r\circ s=t} . Commutativity and associativity are laws governing the order in which some arithmetic operations can be carried out. An operation

9856-436: Is applied to another element. For example, the identity element of addition is 0 since any sum of a number and 0 results in the same number. The inverse element is the element that results in the identity element when combined with another element. For instance, the additive inverse of the number 6 is -6 since their sum is 0. There are not only inverse elements but also inverse operations . In an informal sense, one operation

10032-516: Is at least thousands and possibly tens of thousands of years old. Ancient civilizations like the Egyptians and the Sumerians invented numeral systems to solve practical arithmetic problems in about 3000 BCE. Starting in the 7th and 6th centuries BCE, the ancient Greeks initiated a more abstract study of numbers and introduced the method of rigorous mathematical proofs . The ancient Indians developed

10208-546: Is between set theory and recursion theory . It includes the study of lightface pointclasses , and is closely related to hyperarithmetical theory . In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable. A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations . This has important applications to

10384-583: Is both commutative and associative. Exponentiation is an arithmetic operation in which a number, known as the base, is raised to the power of another number, known as the exponent. The result of this operation is called the power. Exponentiation is sometimes expressed using the symbol ^ but the more common way is to write the exponent in superscript right after the base. Examples are 2 4 = 16 {\displaystyle 2^{4}=16} and 3 {\displaystyle 3} ^ 3 = 27 {\displaystyle 3=27} . If

10560-449: Is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms , the Gödel sentence holds for

10736-403: Is closed under division as long as the divisor is not 0. Both integer arithmetic and rational number arithmetic are not closed under exponentiation and logarithm. One way to calculate exponentiation with a fractional exponent is to perform two separate calculations: one exponentiation using the numerator of the exponent followed by drawing the nth root of the result based on the denominator of

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10912-401: Is commutative if the order of the arguments can be changed without affecting the results. This is the case for addition, for instance, 7 + 9 {\displaystyle 7+9} is the same as 9 + 7 {\displaystyle 9+7} . Associativity is a rule that affects the order in which a series of operations can be carried out. An operation is associative if, in

11088-439: Is denoted  V {\displaystyle V} . Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams . The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and

11264-431: Is greater than 2 {\displaystyle 2} . Rational number arithmetic is the branch of arithmetic that deals with the manipulation of numbers that can be expressed as a ratio of two integers. Most arithmetic operations on rational numbers can be calculated by performing a series of integer arithmetic operations on the numerators and the denominators of the involved numbers. If two rational numbers have

11440-468: Is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism . As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark. Gödel's completeness theorem established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if

11616-439: Is infinite without repeating decimals. The set of rational numbers together with the set of irrational numbers makes up the set of real numbers. The symbol of the real numbers is R {\displaystyle \mathbb {R} } . Even wider classes of numbers include complex numbers and quaternions . A numeral is a symbol to represent a number and numeral systems are representational frameworks. They usually have

11792-476: Is more flexible than a simple yes or no answer and can be a real number such as 0.75. 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

11968-427: Is not closed under division. This means that when dividing one integer by another integer, the result is not always an integer. For instance, 7 divided by 2 is not a whole number but 3.5. One way to ensure that the result is an integer is to round the result to a whole number. However, this method leads to inaccuracies as the original value is altered. Another method is to perform the division only partially and retain

12144-417: Is not required, the problem of calculating arithmetic operations on real numbers is usually addressed by truncation or rounding . For truncation, a certain number of leftmost digits are kept and remaining digits are discarded or replaced by zeros. For example, the number π has an infinite number of digits starting with 3.14159.... If this number is truncated to 4 decimal places, the result is 3.141. Rounding

12320-437: Is often treated as a special case of addition: instead of subtracting a positive number, it is also possible to add a negative number. For instance 14 − 8 = 14 + ( − 8 ) {\displaystyle 14-8=14+(-8)} . This helps to simplify mathematical computations by reducing the number of basic arithmetic operations needed to perform calculations. The additive identity element

12496-447: Is one of many counterintuitive results of the axiom of choice. Arithmetic Arithmetic is an elementary branch of mathematics that studies numerical operations like addition , subtraction , multiplication , and division . In a wider sense, it also includes exponentiation , extraction of roots , and taking logarithms . Arithmetic systems can be distinguished based on the type of numbers they operate on. Integer arithmetic

12672-446: Is possible to say that an object is a whole number using a formula of L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} such as Higher-order logics allow for quantification not only of elements of the domain of discourse , but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having

12848-930: Is restricted to the study of integers and focuses on their properties and relationships such as divisibility , factorization , and primality . Traditionally, it is known as higher arithmetic. Numbers are mathematical objects used to count quantities and measure magnitudes. They are fundamental elements in arithmetic since all arithmetic operations are performed on numbers. There are different kinds of numbers and different numeral systems to represent them. The main kinds of numbers employed in arithmetic are natural numbers , whole numbers, integers , rational numbers , and real numbers . The natural numbers are whole numbers that start from 1 and go to infinity. They exclude 0 and negative numbers. They are also known as counting numbers and can be expressed as { 1 , 2 , 3 , 4 , . . . } {\displaystyle \{1,2,3,4,...\}} . The symbol of

13024-491: Is strictly greater than the rank of any of its elements. For example, the empty set is assigned rank 0, while the set {{}} containing only the empty set is assigned rank 1. For each ordinal α {\displaystyle \alpha } , the set V α {\displaystyle V_{\alpha }} is defined to consist of all pure sets with rank less than α {\displaystyle \alpha } . The entire von Neumann universe

13200-674: 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. The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo–Fraenkel theory, is much greater than that of constructive mathematics, to the point that Solomon Feferman has said that "all of scientifically applicable analysis can be developed [using predicative methods]". Ludwig Wittgenstein condemned set theory philosophically for its connotations of mathematical platonism . He wrote that "set theory

13376-418: 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. 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

13552-517: Is the unary numeral system . It relies on one symbol for the number 1. All higher numbers are written by repeating this symbol. For example, the number 7 can be represented by repeating the symbol for 1 seven times. This system makes it cumbersome to write large numbers, which is why many non-positional systems include additional symbols to directly represent larger numbers. Variations of the unary numeral systems are employed in tally sticks using dents and in tally marks . Egyptian hieroglyphics had

13728-462: Is the branch of arithmetic that deals with the manipulation of positive and negative whole numbers. Simple one-digit operations can be performed by following or memorizing a table that presents the results of all possible combinations, like an addition table or a multiplication table . Other common methods are verbal counting and finger-counting . For operations on numbers with more than one digit, different techniques can be employed to calculate

13904-428: Is the inverse of another operation if it undoes the first operation. For example, subtraction is the inverse of addition since a number returns to its original value if a second number is first added and subsequently subtracted, as in 13 + 4 − 4 = 13 {\displaystyle 13+4-4=13} . Defined more formally, the operation " ⋆ {\displaystyle \star } "

14080-413: Is the inverse of exponentiation. The logarithm of a number x {\displaystyle x} to the base b {\displaystyle b} is the exponent to which b {\displaystyle b} must be raised to produce x {\displaystyle x} . For instance, since 1000 = 10 3 {\displaystyle 1000=10^{3}} ,

14256-400: Is the set whose members are all of the possible subsets of A . For example, the power set of {1, 2} is { {}, {1}, {2}, {1, 2} } . Notably, P ( A ) {\displaystyle {\mathcal {P}}(A)} contains both A and the empty set. A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set containing only

14432-642: Is the study of sets , which are abstract collections of objects. Many of the basic notions, such as ordinal and cardinal numbers, were developed informally by Cantor before formal axiomatizations of set theory were developed. The first such axiomatization , due to Zermelo, was extended slightly to become Zermelo–Fraenkel set theory (ZF), which is now the most widely used foundational theory for mathematics. Other formalizations of set theory have been proposed, including von Neumann–Bernays–Gödel set theory (NBG), Morse–Kelley set theory (MK), and New Foundations (NF). Of these, ZF, NBG, and MK are similar in describing

14608-401: Is the theorem that a line contains at least two points, or that circles of the same radius whose centers are separated by that radius must intersect. Hilbert developed a complete set of axioms for geometry , building on previous work by Pasch. The success in axiomatizing geometry motivated Hilbert to seek complete axiomatizations of other areas of mathematics, such as the natural numbers and

14784-469: Is used by most computers and represents numbers as combinations of the basic numerals 0 and 1. Computer arithmetic deals with the specificities of the implementation of binary arithmetic on computers . Some arithmetic systems operate on mathematical objects other than numbers, such as interval arithmetic and matrix arithmetic. Arithmetic operations form the basis of many branches of mathematics, such as algebra , calculus , and statistics . They play

14960-698: Is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers". Wittgenstein identified mathematics with algorithmic human deduction; the need for a secure foundation for mathematics seemed, to him, nonsensical. Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radical constructivism and finitism . Meta-mathematical statements — which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory — are not mathematics. Few modern philosophers have adopted Wittgenstein's views after

15136-529: The Organon , found wide application and acceptance in Western science and mathematics for millennia. The Stoics , especially Chrysippus , began the development of predicate logic . In 18th-century Europe, attempts to treat the operations of formal logic in a symbolic or algebraic way had been made by philosophical mathematicians including Leibniz and Lambert , but their labors remained isolated and little known. In

15312-449: The Axiom of Choice have recently seen applications in evolutionary dynamics , enhancing the understanding of well-established models of evolution and interaction. Set theory is a major area of research in mathematics with many interrelated subfields: Combinatorial set theory concerns extensions of finite combinatorics to infinite sets. This includes the study of cardinal arithmetic and

15488-599: The Burali-Forti paradox . Axiomatic set theory was originally devised to rid set theory of such paradoxes. The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy . Such systems come in two flavors, those whose ontology consists of: The above systems can be modified to allow urelements , objects that can be members of sets but that are not themselves sets and do not have any members. The New Foundations systems of NFU (allowing urelements ) and NF (lacking them), associate with Willard Van Orman Quine , are not based on

15664-591: The Hindu–Arabic numeral system , the radix is 10. This means that the first digit is multiplied by 10 0 {\displaystyle 10^{0}} , the next digit is multiplied by 10 1 {\displaystyle 10^{1}} , and so on. For example, the decimal numeral 532 stands for 5 ⋅ 10 2 + 3 ⋅ 10 1 + 2 ⋅ 10 0 {\displaystyle 5\cdot 10^{2}+3\cdot 10^{1}+2\cdot 10^{0}} . Because of

15840-500: The Löwenheim–Skolem theorem , which says that first-order logic cannot control the cardinalities of infinite structures. Skolem realized that this theorem would apply to first-order formalizations of set theory, and that it implies any such formalization has a countable model . This counterintuitive fact became known as Skolem's paradox . In his doctoral thesis, Kurt Gödel proved the completeness theorem , which establishes

16016-428: The absolute uncertainties of each summand together to obtain the absolute uncertainty of the sum. When multiplying or dividing two or more quantities, add the relative uncertainties of each factor together to obtain the relative uncertainty of the product. When representing uncertainty by significant digits, uncertainty can be coarsely propagated by rounding the result of adding or subtracting two or more quantities to

16192-494: The axiom of choice and the law of the excluded middle can be formulated in a manner corresponding to the classical formulation in set theory or perhaps in a spectrum of distinct ways unique to type theory. Some of these principles may be proven to be a consequence of other principles. The variety of formulations of these axiomatic principles allows for a detailed analysis of the formulations required in order to derive various mathematical results. As set theory gained popularity as

16368-416: The axiom of choice ) is still the best-known and most studied. Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity , and has various applications in computer science (such as in

16544-505: The constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by Errett Bishop 's influential book Foundations of Constructive Analysis . A different objection put forth by Henri Poincaré

16720-494: The fundamental theorem of arithmetic , Euclid's theorem , and Fermat's last theorem . According to the fundamental theorem of arithmetic, every integer greater than 1 is either a prime number or can be represented as a unique product of prime numbers. For example, the number 18 is not a prime number and can be represented as 2 × 3 × 3 {\displaystyle 2\times 3\times 3} , all of which are prime numbers. The number 19 , by contrast,

16896-584: The lattice method . Computer science is interested in multiplication algorithms with a low computational complexity to be able to efficiently multiply very large integers, such as the Karatsuba algorithm , the Schönhage–Strassen algorithm , and the Toom–Cook algorithm . A common technique used for division is called long division . Other methods include short division and chunking . Integer arithmetic

17072-403: The natural and real numbers can be derived within set theory, as each of these number systems can be defined by representing their elements as sets of specific forms. Set theory as a foundation for mathematical analysis , topology , abstract algebra , and discrete mathematics is likewise uncontroversial; mathematicians accept (in principle) that theorems in these areas can be derived from

17248-431: The quotient . The symbols of division are ÷ {\displaystyle \div } and / {\displaystyle /} . Examples are 48 ÷ 8 = 6 {\displaystyle 48\div 8=6} and 29.4 / 1.4 = 21 {\displaystyle 29.4/1.4=21} . Division is often treated as a special case of multiplication: instead of dividing by

17424-421: The real line . This would prove to be a major area of research in the first half of the 20th century. The 19th century saw great advances in the theory of real analysis , including theories of convergence of functions and Fourier series . Mathematicians such as Karl Weierstrass began to construct functions that stretched intuition, such as nowhere-differentiable continuous functions . Previous conceptions of

17600-568: The remainder . For example, 7 divided by 2 is 3 with a remainder of 1. These difficulties are avoided by rational number arithmetic, which allows for the exact representation of fractions. A simple method to calculate exponentiation is by repeated multiplication. For instance, the exponentiation of 3 4 {\displaystyle 3^{4}} can be calculated as 3 × 3 × 3 × 3 {\displaystyle 3\times 3\times 3\times 3} . A more efficient technique used for large exponents

17776-466: The 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of naive set theory . After the discovery of paradoxes within naive set theory (such as Russell's paradox , Cantor's paradox and the Burali-Forti paradox ), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without

17952-403: The 19th century. Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry. In logic, the term arithmetic refers to the theory of the natural numbers . Giuseppe Peano published a set of axioms for arithmetic that came to bear his name ( Peano axioms ), using

18128-448: The 64 operations required for regular repeated multiplication. Methods to calculate logarithms include the Taylor series and continued fractions . Integer arithmetic is not closed under logarithm and under exponentiation with negative exponents, meaning that the result of these operations is not always an integer. Number theory studies the structure and properties of integers as well as

18304-647: The Latin term " arithmetica " which derives from the Ancient Greek words ἀριθμός (arithmos), meaning "number", and ἀριθμητική τέχνη (arithmetike tekhne), meaning "the art of counting". There are disagreements about its precise definition. According to a narrow characterization, arithmetic deals only with natural numbers . However, the more common view is to include operations on integers , rational numbers , real numbers , and sometimes also complex numbers in its scope. Some definitions restrict arithmetic to

18480-494: The West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity . Especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870–1874, and was motivated by Cantor's work in real analysis . Set theory begins with a fundamental binary relation between an object o and

18656-457: The accuracy and speed with which arithmetic calculations could be performed. Arithmetic is the fundamental branch of mathematics that studies numbers and their operations. In particular, it deals with numerical calculations using the arithmetic operations of addition , subtraction , multiplication , and division . In a wider sense, it also includes exponentiation , extraction of roots , and logarithm . The term "arithmetic" has its root in

18832-435: The addends, are combined into a single number, called the sum. The symbol of addition is + {\displaystyle +} . Examples are 2 + 2 = 4 {\displaystyle 2+2=4} and 6.3 + 1.26 = 7.56 {\displaystyle 6.3+1.26=7.56} . The term summation is used if several additions are performed in a row. Counting is a type of repeated addition in which

19008-401: The axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice). A large cardinal

19184-510: The base can be understood from context. So, the previous example can be written log 10 ⁡ 1000 = 3 {\displaystyle \log _{10}1000=3} . Exponentiation and logarithm do not have general identity elements and inverse elements like addition and multiplication. The neutral element of exponentiation in relation to the exponent is 1, as in 14 1 = 14 {\displaystyle 14^{1}=14} . However, exponentiation does not have

19360-440: The claim that every even number is a sum of two prime numbers . Algebraic number theory employs algebraic structures to analyze the properties of and relations between numbers. Examples are the use of fields and rings , as in algebraic number fields like the ring of integers . Geometric number theory uses concepts from geometry to study numbers. For instance, it investigates how lattice points with integer coordinates behave in

19536-469: The collection is nonempty, the lack of a general, concrete rule by which the choice can be made renders the axiom nonconstructive. Stefan Banach and Alfred Tarski showed that the axiom of choice can be used to decompose a solid ball into a finite number of pieces which can then be rearranged, with no scaling, to make two solid balls of the original size. This theorem, known as the Banach–Tarski paradox ,

19712-608: The completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis. Another type of logics are fixed-point logic s that allow inductive definitions , like one writes for primitive recursive functions . One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic . Lindström's theorem implies that

19888-447: The concept of zero and the decimal system , which Arab mathematicians further refined and spread to the Western world during the medieval period. The first mechanical calculators were invented in the 17th century. The 18th and 19th centuries saw the development of modern number theory and the formulation of axiomatic foundations of arithmetic. In the 20th century, the emergence of electronic calculators and computers revolutionized

20064-412: The consistency of elementary arithmetic, respectively; the tenth was to produce a method that could decide whether a multivariate polynomial equation over the integers has a solution. Subsequent work to resolve these problems shaped the direction of mathematical logic, as did the effort to resolve Hilbert's Entscheidungsproblem , posed in 1928. This problem asked for a procedure that would decide, given

20240-679: The context of proof theory. At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems . These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language . The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic . First-order logic

20416-488: The decimal fraction notation. Modified versions of integer calculation methods like addition with carry and long multiplication can be applied to calculations with decimal fractions. Not all rational numbers have a finite representation in the decimal notation. For example, the rational number 1 3 {\displaystyle {\tfrac {1}{3}}} corresponds to 0.333... with an infinite number of 3s. The shortened notation for this type of repeating decimal

20592-522: The degree of certainty about each number's value and avoid false precision is to round each measurement to a certain number of digits, called significant digits , which are implied to be accurate. For example, a person's height measured with a tape measure might only be precisely known to the nearest centimeter, so should be presented as 1.62 meters rather than 1.6217 meters. If converted to imperial units, this quantity should be rounded to 64 inches or 63.8 inches rather than 63.7795 inches, to clearly convey

20768-576: The desired level of accuracy. The Taylor series or the continued fraction method can be utilized to calculate logarithms. The decimal fraction notation is a special way of representing rational numbers whose denominator is a power of 10. For instance, the rational numbers 1 10 {\displaystyle {\tfrac {1}{10}}} , 371 100 {\displaystyle {\tfrac {371}{100}}} , and 44 10000 {\displaystyle {\tfrac {44}{10000}}} are written as 0.1, 3.71, and 0.0044 in

20944-652: The distinction between the natural and the whole numbers by including 0 in the set of natural numbers. The set of integers encompasses both positive and negative whole numbers. It has the symbol Z {\displaystyle \mathbb {Z} } and can be expressed as { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . } {\displaystyle \{...,-2,-1,0,1,2,...\}} . Based on how natural and whole numbers are used, they can be distinguished into cardinal and ordinal numbers . Cardinal numbers, like one, two, and three, are numbers that express

21120-407: The early decades of the 20th century, the main areas of study were set theory and formal logic. The discovery of paradoxes in informal set theory caused some to wonder whether mathematics itself is inconsistent, and to look for proofs of consistency. In 1900, Hilbert posed a famous list of 23 problems for the next century. The first two of these were to resolve the continuum hypothesis and prove

21296-475: The effect of the digits' positions, the numeral 532 differs from the numerals 325 and 253 even though they have the same digits. Another positional numeral system used extensively in computer arithmetic is the binary system , which has a radix of 2. This means that the first digit is multiplied by 2 0 {\displaystyle 2^{0}} , the next digit by 2 1 {\displaystyle 2^{1}} , and so on. For example,

21472-399: The empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets. Sets in

21648-408: The exponent is a natural number then exponentiation is the same as repeated multiplication, as in 2 4 = 2 × 2 × 2 × 2 {\displaystyle 2^{4}=2\times 2\times 2\times 2} . Roots are a special type of exponentiation using a fractional exponent. For example, the square root of a number is the same as raising the number to

21824-458: The exponent. For example, 5 2 3 = 5 2 3 {\displaystyle 5^{\frac {2}{3}}={\sqrt[{3}]{5^{2}}}} . The first operation can be completed using methods like repeated multiplication or exponentiation by squaring. One way to get an approximate result for the second operation is to employ Newton's method , which uses a series of steps to gradually refine an initial guess until it reaches

22000-421: The field of numerical calculations. When understood in a wider sense, it also includes the study of how the concept of numbers developed, the analysis of properties of and relations between numbers, and the examination of the axiomatic structure of arithmetic operations. Arithmetic is closely related to number theory and some authors use the terms as synonyms. However, in a more specific sense, number theory

22176-483: The first number with the reciprocal of the second number. This means that the numerator and the denominator of the second number change position. For example, 3 5 : 2 7 = 3 5 ⋅ 7 2 = 21 10 {\displaystyle {\tfrac {3}{5}}:{\tfrac {2}{7}}={\tfrac {3}{5}}\cdot {\tfrac {7}{2}}={\tfrac {21}{10}}} . Unlike integer arithmetic, rational number arithmetic

22352-450: The first volume of Principia Mathematica by Russell and Alfred North Whitehead was published. This seminal work developed the theory of functions and cardinality in a completely formal framework of type theory , which Russell and Whitehead developed in an effort to avoid the paradoxes. Principia Mathematica is considered one of the most influential works of the 20th century, although the framework of type theory did not prove popular as

22528-423: The importance of the incompleteness theorem for some time. Gödel's theorem shows that a consistency proof of any sufficiently strong, effective axiom system cannot be obtained in the system itself, if the system is consistent, nor in any weaker system. This leaves open the possibility of consistency proofs that cannot be formalized within the system they consider. Gentzen proved the consistency of arithmetic using

22704-443: The incompleteness (in a different meaning of the word) of all sufficiently strong, effective first-order theories. This result, known as Gödel's incompleteness theorem , establishes severe limitations on axiomatic foundations for mathematics, striking a strong blow to Hilbert's program. It showed the impossibility of providing a consistency proof of arithmetic within any formal theory of arithmetic. Hilbert, however, did not acknowledge

22880-491: The incompleteness theorems in generality that could only be implied in the original paper. Numerous results in recursion theory were obtained in the 1940s by Stephen Cole Kleene and Emil Leon Post . Kleene introduced the concepts of relative computability, foreshadowed by Turing, and the arithmetical hierarchy . Kleene later generalized recursion theory to higher-order functionals. Kleene and Georg Kreisel studied formal versions of intuitionistic mathematics, particularly in

23056-502: The integer 1, called the numerator, by the integer 2, called the denominator. Other examples are 3 4 {\displaystyle {\tfrac {3}{4}}} and 281 3 {\displaystyle {\tfrac {281}{3}}} . The set of rational numbers includes all integers, which are fractions with a denominator of 1. The symbol of the rational numbers is Q {\displaystyle \mathbb {Q} } . Decimal fractions like 0.3 and 25.12 are

23232-559: 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. The Handbook of Mathematical Logic in 1977 makes

23408-569: The layman was written by Lewis Carroll , author of Alice's Adventures in Wonderland , in 1896. Alfred Tarski developed the basics of model theory . Beginning in 1935, a group of prominent mathematicians collaborated under the pseudonym Nicolas Bourbaki to publish Éléments de mathématique , a series of encyclopedic mathematics texts. These texts, written in an austere and axiomatic style, emphasized rigorous presentation and set-theoretic foundations. Terminology coined by these texts, such as

23584-493: The left. This process is repeated until all digits have been added. Other methods used for integer additions are the number line method, the partial sum method, and the compensation method. A similar technique is utilized for subtraction: it also starts with the rightmost digit and uses a "borrow" or a negative carry for the column on the left if the result of the one-digit subtraction is negative. A basic technique of integer multiplication employs repeated addition. For example,

23760-458: The leftmost last significant decimal place among the summands, and by rounding the result of multiplying or dividing two or more quantities to the least number of significant digits among the factors. (See Significant figures § Arithmetic .) More sophisticated methods of dealing with uncertain values include interval arithmetic and affine arithmetic . Interval arithmetic describes operations on intervals . Intervals can be used to represent

23936-690: The letter A "), which may be useful when learning computer programming , since Boolean logic is used in various programming languages . Likewise, sets and other collection-like objects, such as multisets and lists , are common datatypes in computer science and programming . In addition to that, sets are commonly referred to in mathematical teaching when talking about different types of numbers (the sets N {\displaystyle \mathbb {N} } of natural numbers , Z {\displaystyle \mathbb {Z} } of integers , R {\displaystyle \mathbb {R} } of real numbers , etc.), and when defining

24112-492: The logarithm base 10 of 1000 is 3. The logarithm of x {\displaystyle x} to base b {\displaystyle b} is denoted as log b ⁡ ( x ) {\displaystyle \log _{b}(x)} , or without parentheses, log b ⁡ x {\displaystyle \log _{b}x} , or even without the explicit base, log ⁡ x {\displaystyle \log x} , when

24288-444: The members of set A are also members of set B , then A is a subset of B , denoted A ⊆ B . For example, {1, 2} is a subset of {1, 2, 3} , and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A

24464-457: The mid-19th century as a subfield of mathematics, reflecting the confluence of two traditions: formal philosophical logic and mathematics. Mathematical logic, also called 'logistic', 'symbolic logic', the ' algebra of logic ', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the nineteenth century with the aid of an artificial notation and a rigorously deductive method. Before this emergence, logic

24640-487: The middle of the nineteenth century, George Boole and then Augustus De Morgan presented systematic mathematical treatments of logic. Their work, building on work by algebraists such as George Peacock , extended the traditional Aristotelian doctrine of logic into a sufficient framework for the study of foundations of mathematics . In 1847, Vatroslav Bertić made substantial work on algebraization of logic, independently from Boole. Charles Sanders Peirce later built upon

24816-407: The multiplicand is a natural number then multiplication is the same as repeated addition, as in 2 × 3 = 2 + 2 + 2 {\displaystyle 2\times 3=2+2+2} . Division is the inverse of multiplication. In it, one number, known as the dividend, is split into several equal parts by another number, known as the divisor. The result of this operation is called

24992-433: The multiplicand, are combined into a single number called the product . The symbols of multiplication are × {\displaystyle \times } , ⋅ {\displaystyle \cdot } , and *. Examples are 2 × 3 = 6 {\displaystyle 2\times 3=6} and 0.3 ⋅ 5 = 1.5 {\displaystyle 0.3\cdot 5=1.5} . If

25168-482: The natural numbers but cannot be proved. Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent ,

25344-484: The natural numbers is N {\displaystyle \mathbb {N} } . The whole numbers are identical to the natural numbers with the only difference being that they include 0. They can be represented as { 0 , 1 , 2 , 3 , 4 , . . . } {\displaystyle \{0,1,2,3,4,...\}} and have the symbol N 0 {\displaystyle \mathbb {N} _{0}} . Some mathematicians do not draw

25520-512: The number 1 is continuously added. Subtraction is the inverse of addition. In it, one number, known as the subtrahend, is taken away from another, known as the minuend. The result of this operation is called the difference. The symbol of subtraction is − {\displaystyle -} . Examples are 14 − 8 = 6 {\displaystyle 14-8=6} and 45 − 1.7 = 43.3 {\displaystyle 45-1.7=43.3} . Subtraction

25696-430: The number 13 is written as 1101 in the binary notation, which stands for 1 ⋅ 2 3 + 1 ⋅ 2 2 + 0 ⋅ 2 1 + 1 ⋅ 2 0 {\displaystyle 1\cdot 2^{3}+1\cdot 2^{2}+0\cdot 2^{1}+1\cdot 2^{0}} . In computing, each digit in the binary notation corresponds to one bit . The earliest positional system

25872-486: The only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic. Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing. Intuitionistic logic

26048-547: The power of 1 2 {\displaystyle {\tfrac {1}{2}}} and the cube root of a number is the same as raising the number to the power of 1 3 {\displaystyle {\tfrac {1}{3}}} . Examples are 4 = 4 1 2 = 2 {\displaystyle {\sqrt {4}}=4^{\frac {1}{2}}=2} and 27 3 = 27 1 3 = 3 {\displaystyle {\sqrt[{3}]{27}}=27^{\frac {1}{3}}=3} . Logarithm

26224-418: The precision of the measurement. When a number is written using ordinary decimal notation, leading zeros are not significant, and trailing zeros of numbers not written with a decimal point are implicitly considered to be non-significant. For example, the numbers 0.056 and 1200 each have only 2 significant digits, but the number 40.00 has 4 significant digits. Representing uncertainty using only significant digits

26400-414: The product of 3 × 4 {\displaystyle 3\times 4} can be calculated as 3 + 3 + 3 + 3 {\displaystyle 3+3+3+3} . A common technique for multiplication with larger numbers is called long multiplication . This method starts by writing the multiplier above the multiplicand. The calculation begins by multiplying the multiplier only with

26576-411: The quantity of objects. They answer the question "how many?". Ordinal numbers, such as first, second, and third, indicate order or placement in a series. They answer the question "what position?". A number is rational if it can be represented as the ratio of two integers. For instance, the rational number 1 2 {\displaystyle {\tfrac {1}{2}}} is formed by dividing

26752-441: The ratio of two integers. They are often required to describe geometric magnitudes. For example, if a right triangle has legs of the length 1 then the length of its hypotenuse is given by the irrational number 2 {\displaystyle {\sqrt {2}}} . π is another irrational number and describes the ratio of a circle 's circumference to its diameter . The decimal representation of an irrational number

26928-488: 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. 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

27104-534: The relations and laws between them. Some of the main branches of modern number theory include elementary number theory , analytic number theory , algebraic number theory , and geometric number theory . Elementary number theory studies aspects of integers that can be investigated using elementary methods. Its topics include divisibility , factorization , and primality . Analytic number theory, by contrast, relies on techniques from analysis and calculus. It examines problems like how prime numbers are distributed and

27280-506: The relevant definitions and the axioms of set theory. However, it remains that few full derivations of complex mathematical theorems from set theory have been formally verified, since such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath , includes human-written, computer-verified derivations of more than 12,000 theorems starting from ZFC set theory, first-order logic and propositional logic . ZFC and

27456-404: The result by using several one-digit operations in a row. For example, in the method addition with carries , the two numbers are written one above the other. Starting from the rightmost digit, each pair of digits is added together. The rightmost digit of the sum is written below them. If the sum is a two-digit number then the leftmost digit, called the "carry", is added to the next pair of digits to

27632-406: The rightmost digit of the multiplicand and writing the result below, starting in the rightmost column. The same is done for each digit of the multiplicand and the result in each case is shifted one position to the left. As a final step, all the individual products are added to arrive at the total product of the two multi-digit numbers. Other techniques used for multiplication are the grid method and

27808-418: The same denominator then they can be added by adding their numerators and keeping the common denominator. For example, 2 7 + 3 7 = 5 7 {\displaystyle {\tfrac {2}{7}}+{\tfrac {3}{7}}={\tfrac {5}{7}}} . A similar procedure is used for subtraction. If the two numbers do not have the same denominator then they must be transformed to find

27984-439: The start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The axiom of determinacy (AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of

28160-469: The structure of the real number line to the study of the consistency of large cardinals . Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor : " On a Property of the Collection of All Real Algebraic Numbers ". Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in

28336-401: The study of invariants in many fields of mathematics. 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"

28512-765: The study of extensions of Ramsey's theorem such as the Erdős–Rado theorem . Descriptive set theory is the study of subsets of the 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. The field of effective descriptive set theory

28688-410: 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

28864-477: The subject at different levels in all grades. Venn diagrams are widely employed to explain basic set-theoretic relationships to primary school students (even though John Venn originally devised them as part of a procedure to assess the validity of inferences in term logic ). Set theory is used to introduce students to logical operators (NOT, AND, OR), and semantic or rule description (technically intensional definition ) of sets (e.g. "months starting with

29040-437: The symbols I, V, X, L, C, D, M as its basic numerals to represent the numbers 1, 5, 10, 50, 100, 500, and 1000. A numeral system is positional if the position of a basic numeral in a compound expression determines its value. Positional numeral systems have a radix that acts as a multiplicand of the different positions. For each subsequent position, the radix is raised to a higher power. In the common decimal system, also called

29216-414: The theory of relational algebra ), philosophy , formal semantics , and evolutionary dynamics . Its foundational appeal, together with its paradoxes , and its implications for the concept of infinity and its multiple applications have made set theory an area of major interest for logicians and philosophers of mathematics . Contemporary research into set theory covers a vast array of topics, ranging from

29392-454: The theory of mathematical relations can be described in set theory. Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica , it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second-order logic . For example, properties of

29568-402: The turn of the century. The two-dimensional notation Frege developed was never widely adopted and is unused in contemporary texts. From 1890 to 1905, Ernst Schröder published Vorlesungen über die Algebra der Logik in three volumes. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of

29744-553: The uniqueness of the set of natural numbers (up to isomorphism) and the recursive definitions of addition and multiplication from the successor function and mathematical induction. In the mid-19th century, flaws in Euclid's axioms for geometry became known. In addition to the independence of the parallel postulate , established by Nikolai Lobachevsky in 1826, mathematicians discovered that certain theorems taken for granted by Euclid were not in fact provable from his axioms. Among these

29920-420: The von Neumann universe are organized into a cumulative hierarchy , based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by transfinite recursion ) an ordinal number α {\displaystyle \alpha } , known as its rank. The rank of a pure set X {\displaystyle X} is defined to be the least ordinal that

30096-463: The words bijection , injection , and surjection , and the set-theoretic foundations the texts employed, were widely adopted throughout mathematics. The study of computability came to be known as recursion theory or computability theory , because early formalizations by Gödel and Kleene relied on recursive definitions of functions. When these definitions were shown equivalent to Turing's formalization involving Turing machines , it became clear that

30272-430: The work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885. Gottlob Frege presented an independent development of logic with quantifiers in his Begriffsschrift , published in 1879, a work generally considered as marking a turning point in the history of logic. Frege's work remained obscure, however, until Bertrand Russell began to promote it near

30448-735: Was developed by ancient Babylonians and had a radix of 60. Arithmetic operations are ways of combining, transforming, or manipulating numbers. They are functions that have numbers both as input and output. The most important operations in arithmetic are addition , subtraction , multiplication , and division . Further operations include exponentiation , extraction of roots , and logarithm . If these operations are performed on variables rather than numbers, they are sometimes referred to as algebraic operations . Two important concepts in relation to arithmetic operations are identity elements and inverse elements . The identity element or neutral element of an operation does not cause any change if it

30624-486: Was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle , which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic

30800-407: Was proved independent of ZF by Fraenkel, but has come to be widely accepted by mathematicians. It states that given a collection of nonempty sets there is a single set C that contains exactly one element from each set in the collection. The set C is said to "choose" one element from each set in the collection. While the ability to make such a choice is considered obvious by some, since each set in

30976-496: Was studied with rhetoric , with calculationes , through the syllogism , and with philosophy . The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics. Theories of logic were developed in many cultures in history, including China , India , Greece and the Islamic world . Greek methods, particularly Aristotelian logic (or term logic) as found in

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