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 in nity. 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

This paper presents a formalization of finite and infinite sequences in domain theory carried out in the theorem prover Isabelle. The results

The need to use partial functions arises frequently in formal descriptions of computer systems. However, most proof assistants are based on logics of total functions. One way to address this mismatch is to invent and mechanize a new logic. Another is to develop practical workarounds in existing settings. In this paper we take the latter course: we survey and compare methods used to support partiality in a mechanization of a higher order logic featuring only total functions. The techniques we discuss are generally applicable and are illustrated by relatively large examples. 1.

The unfolding of schematic formal systems is a novel concept which was initiated in Feferman [6]. This paper is mainly concerned with the proof-theoretic analysis of various unfolding systems for non-nitist arithmetic NFA. In particular, we examine two restricted unfoldings U 0 (NFA) and U 1 (NFA), as well as a full unfolding, U(NFA). The principal results then state: (i) U 0 (NFA) is equivalent to PA; (ii) U 1 (NFA) is equivalent to RA<! ; (iii) U(NFA) is equivalent to RA< 0 . Thus U(NFA) is proof-theoretically equivalent to predicative analysis.

A stream is an in nite sequence of data from a set A. A wide variety of algorithms and architectures operate continuously in time, processing streams of data, for example: hardware systems, embedded systems and emergent systems. Also many models of real number computation use streams. In this paper we study the construction of an algebra A of streams over a many-sorted algebra A of data. In particular, we show how an initial algebra specification for A can be constructed from one for A. One problem is that A is uncountable even when A is finite. To handle this, we work with infinitary terms called stream terms, and infinitary formulae that generalise conditional equations, called !-conditional stream equations. We state a Birkhoff-Mal'cev type theorem that shows conservativity for the many-sorted stream version of full infinitary first-order logic L! 1 ! over an !-conditional equational logic, and also that certain !-conditional stream equational theories have initial model...

We compare four different formalizations of possibly infinite sequences in theorem provers based on higher-order logic. The formalizations have been carried out in different proof tools, namely in Gordon's HOL, in Isabelle and in PVS. The comparison considers different logics and proof infrastructures, but emphasizes on the proof principles that are available for each approach. The different formalizations discussed have been used not only to mechanize proofs of different properties of possibly infinite sequences, but also for the verification of some non-trivial theorems of concurrency theory.