field F has the eta set property ηα, for the ordinal number α, if for any two subsets L and U of F of cardinality less than 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; any two real closed fields are ηα if and only if they are -saturated, and moreover two ηα real closed fields both of cardinality are order isomorphic. The generalized continuum hypothesis 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 , where M is a maximal ideal not leading to a field order-isomorphic to . This is the most commonly used hyperreal number field in nonstandard analysis, and its uniqueness is equivalent to the continuum hypothesis. (Even without the continuum hypothesis we have that if the cardinality of the continuum is then we have a unique ηβ field of size ηβ.) 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 of formal power series on a totally ordered abelian divisible group G that is an η1 group of cardinality . 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 , Κ has cardinality , 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 instead of , cofinality instead of , and weight instead of , and with the η1 property in place of the η0 property (which merely means between any two real numbers we can find another). Examples of real closed fields the real algebraic numbers the computable