A consistent combinatory logic with an inverse to equality

In a previous paper [9] a demonstrably consistent type-free system of combinatory logic CΔ was presented. This system contained a moderately strong exten-sionality principle, the usual equations of combinatory logic, Boolean propositional connectives, unrestricted quantifiers, an unrestricted abstraction principle (“comprehension axiom”), and a limited principle of excluded middle. It also contained elementary arithmetic in its entirety, avoiding arithmetical Gödel incompleteness [14] by being ω-complete. CΔ probably can be shown to contain certain important parts of mathematical analysis, such as the theory of continuous functions, which are contained by the somewhat similar (but nonextensional) system K′ [6], [7]. A stronger system CΓ, having these same properties and more, will be formulated in the present paper. Elsewhere [13] it will be shown that the system Q of my book, Elements of combinatory logic (Yale University Press, New Haven and London, 1974), referred to below as ECL, is essentially a subsystem of CΓ, so that the consistency of CΓ guarantees that of Q.CΓ will be stronger than both CΔ and Q in having an operator ‘ɿ’ (inverted Greek iota) which denotes the inverse of equality in the sense that ‘ɿ(= a) = a’ is a theorem of the system. Thus ‘ɿ’ denotes a function or operator which operates on a unit class to give the only member of that class. (Here ‘=a’ denotes the unit class whose only member is denoted by ‘a’.) If uninverted Greek iota, ‘ι’, is used as an abbreviation for ‘=’, the above equation could be written as ‘ɿ(ιa) = a’. Here ‘ιa’ is analogous to Bertrand Russell's well-known notation ‘ιa’, denoting a unit class. This same operator ‘ɿ’ makes it possible to transform one-many or many-one relations into the corresponding functions or operators, a kind of transformation not usually available in systems of combinatory logic [1], [2]. It also provides a method for restricting functions by restricting the corresponding relations.