This paper surveys some problems in the model theory of uncountable first order languages. These problems were first raised in Mal'cev. Their solutions involve the construction of infinite models which are "small" relative to the cardinality of their language. The most important of these problems concern extending the Upward Löwenheim- Skolem Theorem for uncountable languages. The earliest results relevant to this problem used the ultraproduct construction to obtain "small" elementary extensions. Stronger results have been obtained more recently by other methods. These are used to construct "small" elementary subsystems.

A critical evaluation of Cantor's number conception is undertaken against which the interpretations by Wang and Hallett of Cantoran set theory are measured. Wang takes Cantor's theory to tend to be a theory of numbers rather than a theory of sets, while Hallett takes Cantor as proposing an ordinal theory of cardinal numbers which however permits Cantor to accept ordinal numbers as given without defining them. The evidence presented, however, shows that Cantor conceived numbers, both cardinals and ordinals, as extensional objects, and while either Wang's or Hallett's interpretations eliminate certain difficulties of Cantoran set theory, neither one of them is an accurate depiction of Cantor's theory.