The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.

this paper. The exposition here diverges from the presentation given at the conference in two regards. Firstly, the talk began with a broad introduction, explaining the current rationale and goals of ordinal-theoretic proof theory, which take the place of the original Hilbert Program. Since this part of the talk is now incorporated in the first two sections of the BSL-paper [48] there is no point in reproducing it here. Secondly, we shall omit those parts of the talk concerned with infinitary proof systems of ramified set theory as they can also be found in [48] and even more detailed in [45]. Thirdly, thanks to the aforementioned omissions, the advantage of present paper over the talk is to allow for a much more detailed account of the actual information furnished by ordinal analyses and the role of large cardinal hypotheses in devising ordinal representation systems. 2 Observations on ordinal analyses

Abstract. We introduce an operational set theory in the style of [5] and [17]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical of explicit mathematics [14]. In particular, one has non-extensional operations (or rules) alongside extensional constructive sets. Operations are in general partial and a limited form of self–application is permitted. The system we introduce here is a fully explicit, finitely axiomatised system of constructive sets and operations, which is shown to be as strong as HA. 1.

Abstract not found