 This is not the original type theory of Russell. Leaving aside Russell’s use of “propositional functions” instead of classes and relations, the system of Principia Mathematica (Whitehead &amp; Russell 1910–1913), hereinafter PM fails to be a set theory because it has separate types for relations (propositional functions of arity \(\gt 1)\). It was not until Norbert Wiener observed in 1914 that it was possible to define the ordered pair as a set (his definition of \(\lt x, y \gt\) was not the current \(\{\{x\},\{x, y\}\}\), due to Kuratowski (1921), but \(\{\{\{x\}, \varnothing \},\{\{y\}\}\})\) that it became clear that it is possible to code relation types into set types. Russell frequently said in English that relations could be understood as sets of pairs (or longer tuples) but he had no implementation of this idea (in fact, he defined ordered pairs as relations in PM rather than the now usual reverse!) For a discussion of the history of this simplified type theory, see Wang 1970.