In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.
A real closed field is a field F in which any of the following equivalent conditions is true:
F(\sqrt{-1})
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 a unique isomorphism of fields identical on F[2] (note that every ring homomorphism between real closed fields automatically is order preserving, because x ≤ y if and only if ∃z : y = x + z2). For example, the real closure of the ordered field of rational numbers is the field
Ralg
If (F, P) is an ordered field, and E is a Galois extension of F, then by Zorn's lemma 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.[3]
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 areal closed ring. For example, the real closure of the field
Q(\sqrt2)
Ralg x Ralg
Q(\sqrt2)
Q(\sqrt2)
R
Ralg
The language of real closed fields
l{L}rcf
l{T}rcf
d
d
All of these axioms can be expressed in first-order logic (i.e. quantification ranges only over elements of the field). Note that
l{T}rcf
Tarski showed that
l{T}rcf
l{L}rcf
l{T}rcf
l{L}rcf
The Tarski–Seidenberg theorem 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. Given a subset of variables, the projection from to is the function 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
(\existsx)P(x,y),
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 or the exponential function, can provide undecidable theories; see Richardson's theorem and Decidability of first-order theories of the real numbers.
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'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
l{L}rcf
Tarski's original algorithm for quantifier elimination has nonelementary computational complexity, 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, introduced by George E. Collins, provides a much more practicable algorithm of complexity
2O(n) | |
d |
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\Omega(n) | |
2 |
2\Omega(n) | |
2 |
,
\Omega(n)
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.
A crucially important property of the real numbers is that it is an Archimedean field, meaning it has the Archimedean property that for any real number, there is an integer larger than it in absolute value. 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 numbers 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. 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 of the smallest cofinal set, which is to say, the size of the smallest cardinality giving an unbounded sequence. For example, natural numbers are cofinal in the reals, and the cofinality of the reals is therefore
\aleph0
We have therefore the following invariants defining the nature of a real closed field F:
To this we may add
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. There are also particular properties that may or may not hold:
\aleph\alpha
\aleph\alpha
\aleph\alpha
The characteristics of real closed fields become much simpler if we are willing to assume the generalized continuum hypothesis. If the continuum hypothesis holds, all real closed fields with cardinality of the continuum and having the η1 property are order isomorphic. This unique field Ϝ can be defined by means of an ultrapower, as
RN/M
R
\aleph\beta
\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 number of nonzero terms of the field
R[[G]]
\aleph1
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
\aleph1
\aleph2
\aleph2
\aleph1
\aleph1
\aleph0
\aleph1
\aleph0
Tarski's axioms are an axiom system for the first-order ("elementary") portion of Euclidean geometry. 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.