Philosophy of Mathematics

by Øystein Linnebo

STT distinguishes sharply between individuals, classes of individuals, classes of classes of individuals, etc. As Russell suggests, we may think of these as respectively individuals, families, clans of families, etc. In fact, the distinction between individuals and classes of the various levels is so sharp that each level has its own set of variables and constants, each with a superscript that indicates its level or type (as it is also called). For example, variables for individuals and for “clans” are of type 0 and 2, respectively. Furthermore, a membership claim s ∈ t is considered well-formed only if the type of t is exactly one higher than that of s. Thus, while it is permissible to say that an individual is a member of a family, or a family of a clan, it is impermissible to say that an individual is a member of a clan or of another individual. It follows that the condition meant to define the Russell set, namely ‘x ∉ x,’ is ill-formed. Russell’s paradox is thus blocked by a grammatical prohibition.