### WHAT IS THE RIGHT NOTION OF SEQUENTIALITY?

"... Abstract. In this paper we give an informally semi-rigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion m-sequentiality as the notion that precisely captures the intuitions behind sequentiality. ..."

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Abstract. In this paper we give an informally semi-rigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion m-sequentiality as the notion that precisely captures the intuitions behind sequentiality. The paper provides all relevant separating examples minus one. 1.

### HUME’S PRINCIPLE, BEGINNINGS

, 2010

"... Abstract. In this note we derive Robinson’s Arithmetic from Hume’s Principle in the context of very weak theories of classes and relations. ..."

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Abstract. In this note we derive Robinson’s Arithmetic from Hume’s Principle in the context of very weak theories of classes and relations.

### STRICT PREDICATIVITY 1

"... The most basic notion of impredicativity applies to specifications or definitions of sets or classes. If a set b is specified as {x: A(x)} for some predicate A, then the specification is impredicative if A contains quantifiers such that the set b falls in the range of these quantifiers. Already in t ..."

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The most basic notion of impredicativity applies to specifications or definitions of sets or classes. If a set b is specified as {x: A(x)} for some predicate A, then the specification is impredicative if A contains quantifiers such that the set b falls in the range of these quantifiers. Already in the early history of the notion, it was extended to other cases, such as propositions in Russell’s discussions of the liar paradox. Mathematics will be predicative if it avoids impredicative definitions. The logicist reduction of the concept of natural number met a difficulty on this point, since the definition of ‘natural number ’ already given in the work of Frege and Dedekind is impredicative. More recently, it has been argued by Michael Dummett, the author, and Edward Nelson that more informal explanations of the concept of natural number are impredicative as well. That has the consequence that impredicativity is more pervasive in mathematics, and appears at lower levels, than the earlier debates about the issue generally presupposed. The appearance to the contrary resulted historically from the fact that many opponents of impredicative methods, in particular Poincaré and Weyl, were prepared to assume the natural numbers in their work. In this they were followed by later analysts of predicativity, in particular Kreisel, Schütte, and Feferman. The result was that the working conception of predicativity was of predicativity given the natural numbers. Thus Feferman characterized “the predicative conception ” as holding that “only the natural numbers can be

### POLYNOMIAL AND FUNCTIONAL INTERPRETATIONS: CONSTRUCTIVE FOUNDATIONS FOR NON-STANDARD ARITHMETIC

"... standard Methods. Gödel’s Dialectica interpretation (see [3]) has inspired many workers in the field of proof mining of theorems in classical analysis. Kohlenbach, Fer-reira and Oliva in particular have made use of the functional interpretation to obtain constructive or feasible versions of importa ..."

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standard Methods. Gödel’s Dialectica interpretation (see [3]) has inspired many workers in the field of proof mining of theorems in classical analysis. Kohlenbach, Fer-reira and Oliva in particular have made use of the functional interpretation to obtain constructive or feasible versions of important theorems in set the-ory or functional analysis. Gödel’s motivation was intuitionistic in extending the finitist point of view by a functional interpretation over all finite types. It is also the case for the Curry-Howard isomorphism between formulas and types (or sets). The polynomial translation (see [2]) of the functional interpretation is a more radical enterprise and attempts to define an isomor-phism between formulas and polynomials for a non-standard arithmetic, the Fermat-Kronecker theory of forms (or homogeneous polynomials) with the constructive principle of infinite descent substituting for Peano’s induction postulate for standard arithmetic. Neither bounded arithmetic (see [1]) for subsystems of Peano’s arithmetic, nor predicative arithmetic (see [4]) for

### The many faces of interpretability

"... Abstract. In this paper we discus work in progress on interpretability logics. We show how semantical considerations have allowed us to formulate non-trivial principles about formalized interpretability. In particular we falsify the conjecture about the nature of the interpretability logic of all re ..."

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Abstract. In this paper we discus work in progress on interpretability logics. We show how semantical considerations have allowed us to formulate non-trivial principles about formalized interpretability. In particular we falsify the conjecture about the nature of the interpretability logic of all reasonable arithmetical theories. We consider this an interesting example of how purely semantical considerations give new non-trivial facts about syntactical and arithmetical notions. In addition we give some apparatus that allows us to push ‘global ’ semantical properties into more ‘local ’ syntactical ones. With this apparatus, the rather wild behavior of the different interpretability logics are nicely formulated in a single notion that expresses their differences in a uniform way. This paper consists of three parts. We start of by giving a short introduction to interpretability logics. In the second part we discuss how a careful analysis of the modal semantical behaviour of interpretability logics lets us formulate non-trivial interpretability principles. In the third part we present a semantical bookkeeping tool which pushes ‘global ’ semantical considerations into ‘local’ syntactical ones. The hope is that this machinery will provide a general and uniform treatment of the bewildering field of modal interpretability logics. 1

### Division by three

, 1994

"... We prove without appeal to the Axiom of Choice that for any sets A and B, if there is a one-to-one correspondence between 3 × A and 3 × B then there is a one-to-one correspondence between A and B. The first such proof, due to Lindenbaum, was announced by Lindenbaum and Tarski in 1926, and subsequent ..."

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We prove without appeal to the Axiom of Choice that for any sets A and B, if there is a one-to-one correspondence between 3 × A and 3 × B then there is a one-to-one correspondence between A and B. The first such proof, due to Lindenbaum, was announced by Lindenbaum and Tarski in 1926, and subsequently ‘lost’; Tarski published an alternative proof in 1949. We argue that the proof presented here follows Lindenbaum’s original. AMS Classification numbers 03E10 (Primary); 03E25 (Secondary). 1

### Arithmetic Without the Successor Axiom

, 2006

"... The Successor Axiom asserts that every number has a successor, or in other words, that the number series goes on and on ad infinitum. The present work investigates a particular subsystem of Frege Arithmetic, called F, which turns out to be equivalent to second-order Peano Arithmetic minus the Succes ..."

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The Successor Axiom asserts that every number has a successor, or in other words, that the number series goes on and on ad infinitum. The present work investigates a particular subsystem of Frege Arithmetic, called F, which turns out to be equivalent to second-order Peano Arithmetic minus the Successor Axiom, and shows how this system can develop arithmetic up

### 1. Introduction.

"... From PSA 1992, vol. 2 (1993), pp. 442–455 (with corrections) Why a little bit goes a long way: ..."

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From PSA 1992, vol. 2 (1993), pp. 442–455 (with corrections) Why a little bit goes a long way:

### Bounded Arithmetic, Proof Complexity and Two Papers of Parikh

, 2002

"... This article surveys R. Parikh's work on feasibility, bounded arithmetic and the complexity of proofs. We discuss in depth two of Parikh's papers on these subjects and some of the subsequent progress in the areas of feasible arithmetic and lengths of proofs. ..."

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This article surveys R. Parikh's work on feasibility, bounded arithmetic and the complexity of proofs. We discuss in depth two of Parikh's papers on these subjects and some of the subsequent progress in the areas of feasible arithmetic and lengths of proofs.