Summary. From 1931 until late in his life (at least 1970) Gödel called for the pursuit of new axioms for mathematics to settle both undecided number-theoretical propositions (of the form obtained in his incompleteness results) and undecided set-theoretical propositions (in particular CH). As to the nature of these, Gödel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of infinity. In particular, he speculated (in his 1946 Princeton remarks) that there might be a uniform (though non-decidable) rationale for the choice of the latter. Despite the intense exploration of the “higher infinite” in the last 30-odd years, no single rationale of that character has emerged. Moreover, CH still remains undecided by such axioms, though they have been demonstrated to have many other interesting set-theoretical consequences.

In this paper, I present a new very general notion of the “unfolding” closure of schematically axiomatized formal systems S which provides a uniform systematic means of expanding in an essential way both the language and axioms (and hence theorems) of such systems S. Reporting joint work with T. Strahm, a characterization is given in more familiar terms in the case that S is a basic system of non-finitist arithmetic. When reflective closure is applied to suitable systems of set theory, one is able to derive cardinal axioms as theorems. It is an open question how these may be characterized in terms of current notions in that subject.

Why new axioms?

Gödel's published statements over the years (from 1931 to 1972) pointing to the need for new axioms to settle both undecided number-theoretic and set-theoretic propositions are rather well known. They are most easily cited by reference to the first two volumes of the edition of his Collected Works. A number of less familiar statements of a similar character from his unpublished essays and lectures are now available in the third volume of that edition.

Given the ready accessibility of these sources, there is no need for extensive quotation, though several representative passages are singled out below for special attention.

A central but still unsettled question in formal theories about discourse interpretation is: What are the key theoretical structures on which discourse interpretation should depend? If we take our cue from theories that analyze the meanings of individual sentences, the meaning of the discourse's parts should determine the meaning of the whole; some sort of principle of compositionality of meaning must hold at the level of discourse interpretation. So a theory of discourse interpretation must develop from an account of discourse structure.

Unlike the syntactic structure of a sentence, the discourse structure of a text is not a structure studied by syntacticians or governed principally by syntactic concerns. It has to be inferred from a variety of knowledge sources. Recent work on discourse structure in AI, philosophy and linguistics has shown that discourse structure depends on numerous information sources— compositional semantic principles, lexical semantics, pragmatic principles, and information about the speaker's and interpreter's mental states. So a theory of discourse interpretation must in fact also be a theory of semantics and pragmatics and their interaction—a theory of the pragmatics-semantics interface.

Such a theory linking together pragmatics and semantics brings up a foundational question about frameworks. Pragmatics, though not often formalized, has often made appeal to different types of logical principles than semantics. While semantic theories have typically used a classical, monotonic, logical framework, pragmatic theories appear to best couched within nonmonotonic logic. How should we model the interaction of these multiple knowledge sources needed to construct discourse structure, or the interaction between defeasible pragmatic principles and nondefeasible semantic principles?

Another fundamental difference between pragmatics and semantics also requires resolution. The model-theoretic approach to semantics is thoroughly entrenched, whereas in pragmatics appeals are often made to representations of information, beliefs and other mental states of the participants. How should a discourse context be thought of—as a structured representation, or model-theoretically? How should we model in a logically precise fashion the updating of discourse contexts with new information?

One of the good things about recent developments in theories of discourse interpretation is that we know at least some of the answers to these questions. I will try to provide a guide to some strategies for building such theories.

We begin with logic.

This book does not require any prior knowledge of formal logic. First we expand upon some points made in the introduction to the book. All of our notions of computability (for real numbers, continuous functions, and beyond) are based on the notion of a recursive function. (N denotes the set of non-negative integers.) On the other hand, this book does not require a detailed knowledge of recursion theory. For reasons to be explained below, an intuitive understanding of that theory will suffice.

We continue for now to consider only functions from N to N Intuitively, a recursive function is simply a “computable” function. More precisely, a recursive function is a function which is computable by a Turing machine. The weight of fifty years experience leads to the conclusion that “recursive function” is the correct definition of the intuitive notion of a computable function. The definition is as solid as the definition of a group or a vector space. By now, the theory of recursive functions is highly developed.

However, as we have said, this book does not require a detailed knowledge of that theory. We avoid the need for heavy technical prerequisites in two ways.

1. 1. Whenever we prove that some process is computable, we actually give the algorithm which produces the computation. As we shall see, some of these algorithms are quite intricate. But each of them is built up from scratch, so that the book is self-contained.

2. 2. To prove that certain processes are not computable, we shall find that it suffices to know one basic result from recusive function theory—the existence of a recursively enumerable nonrecursive set. This we now discuss.

Imagine a computer which has been programmed to produce the values of a function a from nonnegative integers to nonnegative integers. We set the program in motion, and have the computer list the values a(0), a(l), a(2),… in order. This set A of values, a subset of the natural numbers, is an example of a “recursively enumerable set”. If we take a general all purpose computer—e.g. a Turing machine —and consider the class of all such programs, we obtain the class of all recursively enumerable sets.

Summary. lnspired by Buchholz’ technique of operator controlled derivations (which were introduced for simplifying Pohlers’ method of local predicativity) a straightforward, perspicuous and conceptually simple method for characterizing the provably recursive functions of Peano arithmetic in terms of Kreisel's ordinal recursive functions is given. Since only amazingly little proof and hierarchy theory is used, the paper is intended to make the field of ordinally informative proof theory accessible even to non-prooftheorists whose knowledge in mathematical logic does not exceed a first graduate level course.

Introduction and Motivation

A fascinating result of ordinally informative proof theory due to Kreisel (1952) is as follows:

Theorem: (*)

The provably recursive functions of Peano arithmetic are exactly the ordinal recursive functions.

Folklore (proof-theoretic) proofs for (*) [cf., for example, Schwichtenberg (1977), Takeuti (1987), Buchholz (1991) or Friedman and Sheard (1995) for such proofs] rely on non trivial metamathematical evaluations of the Gentzenor Schütte-style proof-theoretic analyses of Peano arithmetic. Alternatively a proof- and recursion-theoretic analysis of Gödel's 1958 functional interpretation of Heyting arithmetic can be employed for proving (*), cf. for example [Tait (1965), Buchholz (1980), Weiermann (1995)]. A proof of (*) which does not rely on metamathematical considerations - like primitive recursive stipulations of codes of infinite proof-trees - has been given in [Buchholz (1987), Buchholz and Wainer (1987)]. A proof of (*) using the slow growing hierarchy is given in [Arai (1991)]. A local predicativity style proof - which generalizes uniformly to theories of proof-theoretic strength less than or equal to K PM, cf. [Rathjen (1991)] - of (*) has been given in [Weiermann (1993), Blankertz and Weiermann (1995)]. Other proofs for (*) which are based on model theory can be found, for example, in [Hajek and Pudlak (1993)]. Buchholz (1992) introduced the technique of operator controlled derivations which allows a simplified and conceptually improved exposition of Pohlers’ local predicativity. One aim of the present paper is to give a contribution to the following question (Buchholz, private communication, 1993): Is it possible to use operator controlled derivations to give a proof for (*) - and generalizations of (*) - which is technically smooth?