Remarks on applicative theories (Q2566067)

\(\text{TON}\) is a first-order predicate calculus with, as basic well-formed formulas, the equations of strong combinatory logic. In addition, basic arithmetical functions over \(N\) and definition by cases over \(\text{IN}\), using the same equality, are assumed. The author shows that certain models of the extended untyped \(\lambda\)-calculus are also models of \(\text{ TON}\) together with a number of interesting and surprising extra uniformity, reflection, enumeration and choice principles. Similar results are obtained for some variants and extensions of \(\text{ TON}\).