this paper, it will turn out logicians have universally missed the true, exceedingly simple feature of ordinary first-order logic that makes it incapable of accommodating its own truth predicate. (See Section 4 below.) This defect will also be shown to be easy to overcome without transcending the first-order level. This eliminates once and for all the need of set theory for the purposes of a metatheory of logic.

Introduction 1.1 Historical background Early investigators of realizability were interested in metamathematical questions. In keeping with the traditions of the time they concentrated on interpretations of one formal system in another. They considered an ad hoc collection of increasingly ingenious interpretations mainly to establish consistency, independence and conservativity results. van Oosten's contribution to the Workshop (see van Oosten [56] and the extended account van Oosten [57]) gave inter alia an account of these concerns from a modern perspective. (One should also draw attention to realizability used to provide interpretations of Brouwer's theory of Choice Sequences. An early approach is in Kleene Vesley [28]; for modern work in the area consult Moschovakis [35], [36], [37].) In the early days of categorical logic one considered realizability as providing models for constructive mathematics; while the metamathematics could be retrieved by `coding' the mod

Abstract. Taking a `set ' to be a type together with an equivalence relation and adding an extensional choice axiom to the logical framework (a restricted version of constructive type theory) it is shown that any `set' can be well-ordered. Zermelo's rst proof from 1904 is followed, with a simpli cation to avoid using comparability of well-orderings. The proof has been formalised in the system AgdaLight. 1