Jeff B. Paris

A Mathematical Incompleteness in Peano Arithmetic

Publisher Summary This chapter investigates a reasonably natural theorem of finitary combinatorics, a simple extension of the Finite Ramsey Theorem. The chapter demonstrates that this theorem, while true, is not provable in Peano arithmetic. The chapter works extensively with the partition calculus. In the proof of the main theorem, the chapter relies on various proof-theoretic results, in particular, on Godels Second Incompleteness Theorem. It is possible, however, to prove the main theorem using only model-theoretic methods. This is the approach where a general model-theoretic methodology (called indicator functions) for producing such results is developed.

The Type Theoretic Interpretation of Constructive Set Theory

By adding to Martin-LSfs intuitionistic theory of types a ‘type of sets’ we give a constructive interpretation of constructive set theory. constructive version of the classical conception of the cumulative hierarchy of sets. This interpretation is a

A note on the inevitability of maximum entropy

Abstract It is shown that, assuming natural principles of independence and consistency, the method of maximum entropy provides the only consistent model of inductive inference. This paper is related to earlier results of Shore and Johnson. The relation is briefly discussed in an appendix.

Provability of the pigeonhole principle and the existence of infinitely many primes

In this note we shall be interested in the following problems. Problem 1. Can I Δ 0 ⊢ ∀ x ∃ y > x ( y is prime)? Here I Δ 0 is Peano arithmetic with the induction axiom restricted to bounded (i.e. Δ 0 ) formulae. Problem 2. Can I Δ 0 ⊢ Δ 0 PHP? Here Δ 0 PHP (Δ 0 pigeonhole principle) is the schema for θ ∈ Δ 0 , or equivalently in I Δ 0 , for a Δ 0 formula F ( x,y ) written . By obtaining partial solutions to Problem 2 we shall show that Problem 1 has a positive solution if I Δ 0 is replaced by I Δ 0 + ∀ xx log( x ) exists. Our notation will be entirely standard (see for example [3] and [4]). In particular all logarithms will be to the base 2 and in expressions like log( x ), (1 + e ) x , etc. we shall always mean the integer part of these quantities. Concerning Problem 2 we remark that it is shown in [5] that for k ∈ N and F ∈ Δ 0 , As far as we know this is the best result of this form, in that we do not know how to replace log( z ) k by anything larger. However, as we shall show in Theorem 1, we can do much better if we increase the difference between the sizes of the domain and range of F . In what follows let M be a countable nonstandard model of I Δ 0 , and let be those subsets of M defined by Δ 0 formulae with parameters from M . Theorem 1. For k ∈ N and F ∈ Δ 0 , Here log 0 ( x ) = x , log k + 1 ( x ) = log(log k ( x )). Proof. To simplify matters, consider first the case k = 1. So assume M ⊨ a log(a) exists and with and a > 1. The idea of the proof is the following.

COMMON SENSE AND MAXIMUM ENTROPY

This paper concerns the question of how to draw inferences common sensically from uncertain knowledge. Since the early work of Shore and Johnson (1980), Paris and Vencovská (1990), and Csiszár (1989), it has been known that the Maximum Entropy Inference Process is the only inference process which obeys certain common sense principles of uncertain reasoning. In this paper we consider the present status of this result and argue that within the rather narrow context in which we work this complete and consistent mode of uncertain reasoning is actually characterised by the observance of just a single common sense principle (or slogan).

The liar paradox and fuzzy logic

Can one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr( x ) saying “ x is true” and satisfying the “dequotation schema” for all sentences φ? This problem is investigated in the frame of Łukasiewicz infinitely valued logic.

On the applicability of maximum entropy to inexact reasoning

Abstract It is shown that under a certain interpretation of belief maximum entropy arises naturally in inexact reasoning. Practical and theoretical consequences of this method are then discussed.

Some Independence Results for Peano Arithmetic

In this paper we shall outline a purely model theoretic method for obtaining independence results for Peanos first order axioms (P). The method is of interest in that it provides for the first time elementary combinatorial statements about the natural numbers which are not provable in P. We give several examples of such statements. Central to this exposition will be the notion of an indicator. Indicators were introduced by L. Kirby and the author in [3] although they had occurred implicitly in earlier papers, for example Friedman [1]. The main result on indicators which we shall need (Lemma 1) was proved by Laurie Kirby and the author in the summer of 1976 but it was not until early in the following year that the author realised that this lemma could be used to give independence results. The first combinatorial independence results obtained were essentially statements about certain finite games and consequently were not immediately meaningful (see Example 2). This shortcoming was remedied by Leo Harrington who, upon hearing an incorrect version of our results, noticed a beautifully simply independent combinatorial statement. We outline this result in Example 3. An alternative, more detailed, proof may be found in [5]. Clearly Laurie Kirby and Leo Harrington have made a very significant contribution to this paper and we wish to express our sincere thanks to them.

A Hierarchy of Cuts in Models of Arithmetic

In this paper we show that it is possible to classify most of the natural families of cuts considered to date in terms of a single hierarchy. This classification gives conservation and independence results for fragments of arithmetic.

In defense of the maximum entropy inference process

Abstract This paper is a sequel to an earlier result of the authors that in making inferences from certain probabilistic knowledge bases the maximum entropy inference process, ME , is the only inference process respecting “common sense.” This result was criticized on the grounds that the probabilistic knowledge bases considered are unnatural and that ignorance of dependence should not be identified with statistical independence. We argue against these criticisms and also against the more general criticism that ME is representation dependent. In a final section, however, we provide a criticism of our own of ME , and of inference processes in general, namely that they fail to satisfy compactness.