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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a real closed field is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
F that has the same first-order properties as the field of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s. Some examples are the field of real numbers, the field of real
algebraic number In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is ...
s, and the field of
hyperreal number In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, for some integer n
s.


Definition

A real closed field is a field ''F'' in which any of the following equivalent conditions is true: #''F'' is elementarily equivalent to the real numbers. In other words, it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in ''F''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is true in the reals. #There is a
total order In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( re ...
on ''F'' making it an
ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...
such that, in this ordering, every positive element of ''F'' has a
square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
in ''F'' and any
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
of odd degree with
coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s in ''F'' has at least one
root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
in ''F''. #''F'' is a
formally real field In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field. Alternative definitions The definition given above ...
such that every polynomial of odd degree with coefficients in ''F'' has at least one root in ''F'', and for every element ''a'' of ''F'' there is ''b'' in ''F'' such that ''a'' = ''b''2 or ''a'' = −''b''2. #''F'' is not
algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra h ...
, but its
algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ...
is a
finite extension In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory—in ...
. #''F'' is not algebraically closed but the
field extension In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...
F(\sqrt\,) is algebraically closed. #There is an ordering on ''F'' that does not extend to an ordering on any proper
algebraic extension In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, every element of is a root of a non-zero polynomial with coefficients in . A field extens ...
of ''F''. #''F'' is a formally real field such that no proper algebraic extension of ''F'' is formally real. (In other words, the field is maximal in an algebraic closure with respect to the property of being formally real.) #There is an ordering on ''F'' making it an ordered field such that, in this ordering, the
intermediate value theorem In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two imp ...
holds for all polynomials over ''F'' with degree ''≥'' 0. # ''F'' is a weakly o-minimal ordered field.


Examples of real closed fields

* the field of real
algebraic numbers In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is an algebraic number, because it is a ...
* the field of
computable number In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers, computable reals, or recursive reals ...
s * the field of
definable number Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, \sqrt, c ...
s * the field of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s * the field of
Puiseux series In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series : \begin x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\ &=x^+ 2x^ + x^ + 2x^ + x^ + ...
with real coefficients * the
Levi-Civita field In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. It is usually denoted \mathcal. Each member a can be constructed ...
* the
hyperreal number In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, for some integer n
fields * the
superreal number In abstract algebra, the superreal numbers are a class of extensions of the real numbers, introduced by H. Garth Dales and W. Hugh Woodin as a generalization of the hyperreal numbers and primarily of interest in non-standard analysis, model theor ...
fields * the field of
surreal number In mathematics, the surreal number system is a total order, totally ordered proper class containing not only the real numbers but also Infinity, infinite and infinitesimal, infinitesimal numbers, respectively larger or smaller in absolute value th ...
s (this is a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...
, not a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
)


Real closure

If ''F'' is an ordered field, the Artin–Schreier theorem states that ''F'' has an algebraic extension, called the real closure ''K'' of ''F'', such that ''K'' is a real closed field whose ordering is an extension of the given ordering on ''F'', and is unique
up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech ...
a unique
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
of fields identical on ''F''Rajwade (1993) pp. 222–223 (note that every
ring homomorphism In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
between real closed fields automatically is order preserving, because ''x'' ≤ ''y'' if and only if ∃''z'' : ''y'' = ''x'' + ''z''2). For example, the real closure of the ordered field of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s is the field \mathbb_\mathrm of real algebraic numbers. The
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
is named for
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number t ...
and
Otto Schreier Otto Schreier (3 March 1901 in Vienna, Austria – 2 June 1929 in Hamburg, Germany) was a Jewish-Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups. Life His parents were the arch ...
, who proved it in 1926. If (''F'', ''P'') is an ordered field, and ''E'' is a
Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...
of ''F'', then by
Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...
there is a maximal ordered field extension (''M'', ''Q'') with ''M'' a subfield of ''E'' containing ''F'' and the order on ''M'' extending ''P''. This ''M'', together with its ordering ''Q'', is called the relative real closure of (''F'', ''P'') in ''E''. We call (''F'', ''P'') real closed relative to ''E'' if ''M'' is just ''F''. When ''E'' is the algebraic closure of ''F'' the relative real closure of ''F'' in ''E'' is actually the real closure of ''F'' described earlier.Efrat (2006) p. 177 If ''F'' is a field (no ordering compatible with the field operations is assumed, nor is it assumed that ''F'' is orderable) then ''F'' still has a real closure, which may not be a field anymore, but just a
real closed ring In mathematics, a real closed ring (RCR) is a commutative ring ''A'' that is a subring of a product ring, product of real closed fields, which is closed under continuous function, continuous Semialgebraic set, semi-algebraic functions defined over ...
. For example, the real closure of the field \mathbb(\sqrt 2) is the
ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
\mathbb_\mathrm \!\times \mathbb_\mathrm (the two copies correspond to the two orderings of \mathbb(\sqrt 2)). On the other hand, if \mathbb(\sqrt 2) is considered as an ordered subfield of \mathbb, its real closure is again the field \mathbb_\mathrm.


Decidability and quantifier elimination

The
language Language is a structured system of communication that consists of grammar and vocabulary. It is the primary means by which humans convey meaning, both in spoken and signed language, signed forms, and may also be conveyed through writing syste ...
of real closed fields \mathcal_\text includes symbols for the operations of addition and multiplication, the constants 0 and 1, and the order relation (as well as equality, if this is not considered a logical symbol). In this language, the (first-order) theory of real closed fields, \mathcal_\text, consists of all sentences that follow from the following axioms: * the
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s of
ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...
s; * the axiom asserting that every positive number has a square root; * for every odd number d, the axiom asserting that all polynomials of degree d have at least one root. All of these axioms can be expressed in
first-order logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
(i.e. quantification ranges only over elements of the field). Note that \mathcal_\text is just the set of all first-order sentences that are true about the field of real numbers. Tarski showed that \mathcal_\text is
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
, meaning that any \mathcal_\text-sentence can be proven either true or false from the above axioms. Furthermore, \mathcal_\text is decidable, meaning that there is an
algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
to determine the truth or falsity of any such sentence. This was done by showing
quantifier elimination Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "\exists x such that ..." can be viewed as a question "When is there an x such ...
: there is an algorithm that, given any \mathcal_\text-
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
, which may contain
free variable In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for f ...
s, produces an equivalent quantifier-free formula in the same free variables, where ''equivalent'' means that the two formulas are true for exactly the same values of the variables. Tarski's proof uses a generalization of
Sturm's theorem In mathematics, the Sturm sequence of a univariate polynomial is a sequence of polynomials associated with and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real number, real R ...
. Since the truth of quantifier-free formulas without free variables can be easily checked, this yields the desired decision procedure. These results were obtained and published in 1948. The
Tarski–Seidenberg theorem In mathematics, the Tarski–Seidenberg theorem states that a set in (''n'' + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto ''n''-dimensional space, and the resulting set is still defin ...
extends this result to the following ''projection theorem''. If is a real closed field, a formula with free variables defines a subset of , the set of the points that satisfy the formula. Such a subset is called a
semialgebraic set In mathematics, a basic semialgebraic set is a set defined by polynomial equalities and polynomial inequalities, and a semialgebraic set is a finite union of basic semialgebraic sets. A semialgebraic function is a function with a semialgebraic gr ...
. Given a subset of variables, the ''projection'' from to is the
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
that maps every -tuple to the -tuple of the components corresponding to the subset of variables. The projection theorem asserts that a projection of a semialgebraic set is a semialgebraic set, and that there is an algorithm that, given a quantifier-free formula defining a semialgebraic set, produces a quantifier-free formula for its projection. In fact, the projection theorem is equivalent to quantifier elimination, as the projection of a semialgebraic set defined by the formula is defined by :(\exists x) P(x,y), where and represent respectively the set of eliminated variables, and the set of kept variables. The decidability of a first-order theory of the real numbers depends dramatically on the primitive operations and functions that are considered (here addition and multiplication). Adding other functions symbols, for example, the
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
or the exponential function, can provide undecidable theories; see
Richardson's theorem In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, , ln 2, and exponential and sine functions. It was proved in 1968 by the mathematician and computer s ...
and
Decidability of first-order theories of the real numbers In mathematical logic, a first-order language of the real numbers is the set of all well-formed sentences of first-order logic that involve universal and existential quantifiers and logical combinations of equalities and inequalities of expressi ...
. Furthermore, the completeness and decidability of the first-order theory of the real numbers (using addition and multiplication) contrasts sharply with Gödel's and
Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical compute ...
's results about the incompleteness and undecidability of the first-order theory of the natural numbers (using addition and multiplication). There is no contradiction, since the statement "''x'' is an integer" cannot be formulated as a first-order formula in the language \mathcal_\text.


Complexity of deciding 𝘛rcf

Tarski's original algorithm for
quantifier elimination Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "\exists x such that ..." can be viewed as a question "When is there an x such ...
has nonelementary
computational complexity In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations ...
, meaning that no tower :2^ can bound the execution time of the algorithm if is the size of the input formula. The
cylindrical algebraic decomposition In mathematics, cylindrical algebraic decomposition (CAD) is a notion, along with an algorithm to compute it, that is fundamental for computer algebra and real algebraic geometry. Given a set ''S'' of polynomials in R''n'', a cylindrical algebraic ...
, introduced by
George E. Collins George E. Collins (born on January 10, 1928 in Stuart, Iowa – and died on November 21, 2017 in Madison, Wisconsin) was an American mathematician and computer scientist. He is the inventor of Garbage collection (computer science), garbage collec ...
, provides a much more practicable algorithm of complexity :d^ where is the total number of variables (free and bound), is the product of the degrees of the polynomials occurring in the formula, and is
big O notation Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
. Davenport and Heintz (1988) proved that this worst-case complexity is nearly optimal for quantifier elimination by producing a family of formulas of length , with quantifiers, and involving polynomials of constant degree, such that any quantifier-free formula equivalent to must involve polynomials of degree 2^ and length 2^, where \Omega(n) is
big Omega notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by German mathematicians Paul ...
. This shows that both the time complexity and the space complexity of quantifier elimination are intrinsically
double exponential Double exponential may refer to: * A double exponential function ** Double exponential time, a task with time complexity roughly proportional to such a function ** 2-EXPTIME, the complexity class of decision problems solvable in double-exponentia ...
. For the decision problem, Ben-Or, Kozen, and Reif (1986) claimed to have proved that the theory of real closed fields is decidable in exponential space, and therefore in double exponential time, but their argument (in the case of more than one variable) is generally held as flawed; see Renegar (1992) for a discussion. For purely existential formulas, that is for formulas of the form : where stands for either or , the complexity is lower. Basu and Roy (1996) provided a well-behaved algorithm to decide the truth of such an existential formula with complexity of arithmetic operations and polynomial space.


Order properties

A crucially important property of the real numbers is that it is an
Archimedean field In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, as ...
, meaning it has the Archimedean property that for any real number, there is an
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
larger than it in
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
. Note that this statement is not expressible in the first-order language of ordered fields, since it is not possible to quantify over integers in that language. There are real-closed fields that are non-Archimedean; for example, any field of
hyperreal number In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, for some integer n
s is real closed and non-Archimedean. These fields contain infinitely large (larger than any integer) and infinitesimal (positive but smaller than any positive rational) elements. The Archimedean property is related to the concept of
cofinality In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''. Formally, :\operatorname(A) = \inf \ This definition of cofinality relies o ...
. A set ''X'' contained in an ordered set ''F'' is cofinal in ''F'' if for every ''y'' in ''F'' there is an ''x'' in ''X'' such that ''y'' < ''x''. In other words, ''X'' is an unbounded sequence in ''F''. The cofinality of ''F'' is the
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
of the smallest cofinal set, which is to say, the size of the smallest cardinality giving an unbounded sequence. For example,
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s are cofinal in the reals, and the cofinality of the reals is therefore \aleph_0. We have therefore the following invariants defining the nature of a real closed field ''F'': * The cardinality of ''F''. * The cofinality of ''F''. To this we may add * The weight of ''F'', which is the minimum size of a dense subset of ''F''. These three cardinal numbers tell us much about the order properties of any real closed field, though it may be difficult to discover what they are, especially if we are not willing to invoke the
generalized continuum hypothesis In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
. There are also particular properties that may or may not hold: * A field ''F'' is complete if there is no ordered field ''K'' properly containing ''F'' such that ''F'' is dense in ''K''. If the cofinality of ''F'' is ''κ'', this is equivalent to saying
Cauchy sequence In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are le ...
s indexed by ''κ'' are convergent in ''F''. * An ordered field ''F'' has the eta set property η''α'', for the
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
''α'', if for any two subsets ''L'' and ''U'' of ''F'' of cardinality less than \aleph_\alpha such that every element of ''L'' is less than every element of ''U'', there is an element ''x'' in ''F'' with ''x'' larger than every element of ''L'' and smaller than every element of ''U''. This is closely related to the model-theoretic property of being a
saturated model In mathematical logic, and particularly in its subfield model theory, a saturated model ''M'' is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is \al ...
; any two real closed fields are η''α'' if and only if they are \aleph_\alpha-saturated, and moreover two η''α'' real closed fields both of cardinality \aleph_\alpha are
order isomorphic In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be cons ...
.


The generalized continuum hypothesis

The characteristics of real closed fields become much simpler if we are willing to assume the
generalized continuum hypothesis In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
. If the continuum hypothesis holds, all real closed fields with
cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \bold\mathfrak c (lowercase Fraktur "c") or \ ...
and having the ''η''1 property are order isomorphic. This unique field ''Ϝ'' can be defined by means of an
ultrapower The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All fact ...
, as \mathbb^/\mathbf, where M is a maximal ideal not leading to a field order-isomorphic to \mathbb. This is the most commonly used hyperreal number field in
nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...
, and its uniqueness is equivalent to the continuum hypothesis. (Even without the continuum hypothesis we have that if the cardinality of the continuum is \aleph_\beta then we have a unique ''η''''β'' field of size \aleph_\beta.) Moreover, we do not need ultrapowers to construct ''Ϝ'', we can do so much more constructively as the subfield of series with a
countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
number of nonzero terms of the field \mathbb G of
formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...
on a
totally ordered In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( r ...
abelian
divisible group In mathematics, specifically in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an ''n''th multiple for each positiv ...
''G'' that is an ''η''1 group of cardinality \aleph_1 . ''Ϝ'' however is not a complete field; if we take its completion, we end up with a field ''Κ'' of larger cardinality. ''Ϝ'' has the cardinality of the continuum, which by hypothesis is \aleph_1, ''Κ'' has cardinality \aleph_2, and contains ''Ϝ'' as a dense subfield. It is not an ultrapower but it ''is'' a hyperreal field, and hence a suitable field for the usages of nonstandard analysis. It can be seen to be the higher-dimensional analogue of the real numbers; with cardinality \aleph_2 instead of \aleph_1, cofinality \aleph_1 instead of \aleph_0, and weight \aleph_1 instead of \aleph_0, and with the ''η''1 property in place of the ''η''0 property (which merely means between any two real numbers we can find another).


Elementary Euclidean geometry

Tarski's axioms Tarski's axioms are an axiom system for Euclidean geometry, specifically for that portion of Euclidean geometry that is formulable in first-order logic with identity (i.e. is formulable as an elementary theory). As such, it does not require an u ...
are an axiom system for the first-order ("elementary") portion of
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
. Using those axioms, one can show that the points on a line form a real closed field R, and one can introduce coordinates so that the Euclidean plane is identified with R2 . Employing the decidability of the theory of real closed fields, Tarski then proved that the elementary theory of Euclidean geometry is complete and decidable.


Notes


References

* * Basu, Saugata, Richard Pollack, and
Marie-Françoise Roy Marie-Françoise Roy (born 28 April 1950 in Paris) is a French mathematician noted for her work in real algebraic geometry. She has been Professor of Mathematics at the University of Rennes 1 since 1985 and in 2009 was made a ''Chevalier'' of th ...
(2003) "Algorithms in real algebraic geometry" in ''Algorithms and computation in mathematics''. Springer.
online version
* Michael Ben-Or, Dexter Kozen, and John Reif,
The complexity of elementary algebra and geometry
', Journal of Computer and Systems Sciences 32 (1986), no. 2, pp. 251–264. * Caviness, B F, and Jeremy R. Johnson, eds. (1998) ''Quantifier elimination and cylindrical algebraic decomposition''. Springer. *
Chen Chung Chang Chen Chung Chang () was a mathematician who worked in model theory. He obtained his PhD from Berkeley in 1955 on "Cardinal and Ordinal Factorization of Relation Types" under Alfred Tarski. He wrote the standard text on model theory. Chang's co ...
and
Howard Jerome Keisler Howard Jerome Keisler (born 3 December 1936) is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis. His Ph.D. advisor was Alfred Tarski a ...
(1989) ''Model Theory''. North-Holland. * Dales, H. G., and W. Hugh Woodin (1996) ''Super-Real Fields''. Oxford Univ. Press. * * * Macpherson, D., Marker, D. and Steinhorn, C., ''Weakly o-minimal structures and real closed fields'', Trans. of the American Math. Soc., Vol. 352, No. 12, 1998. * Mishra, Bhubaneswar (1997)
Computational Real Algebraic Geometry
" in ''Handbook of Discrete and Computational Geometry''. CRC Press. 2004 edition, p. 743. * * * *
Alfred Tarski Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...
(1951) ''A Decision Method for Elementary Algebra and Geometry''. Univ. of California Press. *


External links

{{Commons category
''Real Algebraic and Analytic Geometry Preprint Server''''Model Theory preprint server''