## Franklin's Notes

### Real closed field

A real closed field is a field which is elementarily equivalent to $\mathbb{R}$ as a model of the theory of fields in first-order logic. Some examples of real closed fields are

• The algebraic numbers $\mathbb{A}$

• The computable numbers

• The definable real numbers

• Any model of the reals constructed from a nonstandard model of arithmetic

• The surreal numbers

• The Puiseux series with real coefficients