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One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an approach will never be efficient enough for large, complex proofs. One alternative, commonly called reflection, is to analyze proofs using a second layer of logic, a metalogic, and so justify abbreviating or simplifying proofs, making the kinds of shortcuts humans often do or appealing to specialized decision algorithms. In this paper we contrast the fully-expansive LCF approach with the use of reflection. We put forward arguments to suggest that the inadequacy of the LCF approach has not been adequately demonstrated, and neither has the practical utility of reflection (notwithstanding its undoubted intellectual interest). The LCF system with which we are most concerned is the HOL proof ...

Reflection is a wide-ranging concept that has been studied independently in many different areas of science in general, and computer science in particular. Even in the sub-area of programming languages, it has been applied to different paradigms, especially the logic, functional and objectoriented ones. Partly because of different past influences, but also because researchers in these communities scarcely talk to each others, concepts have evolved separately, sometimes to the point where it is hard for people in one community to recognize similarities in the work of others, not to speak about cross-fertilization among them. In this paper, we propose a synthesis covering mainly the application of computation reflection to programming languages. We compare the different approaches and try to identify similar concepts hidden behind different names or constructs. We also point out the different emphasis that has been given to different concepts in each of them. We do not claim neither comp...

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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re

We present a critical discussion of the claim (most forcefully propounded by Chaitin) that algorithmic information theory sheds new light on G6del's first incompleteness theorem.

In [15],[16] Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "-substitution method (due to W. Ackermann), that for every theorem A (A prenex) of first-order Peano arithmetic PA one can find ordinal recursive functionals \Phi A of order type ! " 0 which realize the Herbrand normal form A of A. Subsequently more

The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin’s famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure of the strength of the theory. I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental.

A well-known result of D. Leivant states that, over basic Kalmar elementary arithmetic EA, the induction schema for \Sigma n formulas is equivalent to the uniform reflection principle for \Sigma n+1 formulas. We show that fragments of arithmetic axiomatized by various forms of induction rules admit a precise axiomatization in terms of reflection principles as well. Thus, the closure of EA under the induction rule for \Sigma n (or \Pi n+1 ) formulas is equivalent to ! times iterated \Sigma n reflection principle. Moreover, for k ! !, k times iterated \Sigma n reflection principle over EA precisely corresponds to the extension of EA by k nested applications of \Sigma n induction rule. The above relationship holds in greater generality than just stated. In fact, we give general formulas characterizing in terms of iterated reflection principles the extension of any given theory (containing EA) by k nested applications of \Sigma n or \Pi n induction rules. In particular, the closure of a ...

We study a propositional polymodal provability logic GLP introduced by G. Japaridze. The previous treatments of this logic, due to Japaridze and Ignatiev (see [11, 7]), heavily relied on some non-finitary principles such as transfinite induction up to #0 or reflection principles. In fact, the closed fragment of GLP gives rise to a natural system of ordinal notation for #0 that was used in [1] for a proof-theoretic analysis of Peano arithmetic and for constructing simple combinatorial independent statements.