Subsets of this set that do not have a canonical measure are named for Vitali.
The first element of the signature of a number field is the number of embeddings of the field in this set. This set of numbers comprises the standard parts of numbers used in non-standard analysis. Subsets of this set that do not have a canonical measure are named for Vitali. One construction builds these numbers as equivalence classes of Cauchy sequences, while another uses upper-bounded intervals of rational numbers called Dedekind cuts. The "extended" version of this number system includes positive and negative infinity, and this set can be written as the interval between those two infinities. For 10 points, identify this set that comprises the number line, which is represented by a double-struck capital R.
The first element of the signature of a number field is the number of embeddings of the field in this set.
real numbers [or the real line; or the reals] &lt;Thompson&gt;