Real Closed Field - The Generalized Continuum Hypothesis

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 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 non-standard 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 the Sierpiński group.

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).

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