(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, meaning it has the Archimedean property that for any real number, there is an integer larger than it in absolute value. An equivalent statement is that for any real number, there are integers both larger and smaller. Such real closed fields that are not Archimedean, are non-Archimedean ordered fields. For example, any field of hyperreal numbers is real closed and non-Archimedean. 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 . 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. 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 sequences indexed by κ are convergent in F. An ordered