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The semantic tableaux version of the second incompleteness theorem extends almost to Robinson's Arithmetic Q
 in Automated Reasoning with Semantic Tableaux and Related Methods, SpringerVerlag LNCS 1847
"... Abstract. We will generalize the Second Incompleteness Theorem almost to the level of Robinson’s System Q. We will prove there exists a Π1 sentence V, such that if α is any finite consistent extension of Q+V then α will be unable to prove its Semantic Tableaux consistency. ..."
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Abstract. We will generalize the Second Incompleteness Theorem almost to the level of Robinson’s System Q. We will prove there exists a Π1 sentence V, such that if α is any finite consistent extension of Q+V then α will be unable to prove its Semantic Tableaux consistency.
An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency
 Journal of Symbolic Logic
"... Abstract. This article will study a class of deduction systems that allow for a limited use of the modus ponens method of deduction. We will show that it is possible to devise axiom systems α that can recognize their consistency under a deduction system D provided that: 1) α treats multiplication as ..."
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Abstract. This article will study a class of deduction systems that allow for a limited use of the modus ponens method of deduction. We will show that it is possible to devise axiom systems α that can recognize their consistency under a deduction system D provided that: 1) α treats multiplication as a 3way relation (rather than as a total function), and that 2)D does not allow for the use of a modus ponens methodology above essentially the levels of Π1 and Σ1 formulae. Part of what will make this boundarycase exception to the Second Incompleteness Theorem interesting is that we will also characterize generalizations of the Second Incompleteness Theorem that take force when we only slightly weaken the assumptions of our boundarycase exceptions in any of several further directions.
Growing commas  a study of sequentiality and concatenation
 DEPARTMENT OF PHILOSOPHY, UTRECHT UNIVERSITY
, 2007
"... In his paper [Grz05], Andrzej Grzegorczyk introduces a theory of concatenation TC. We show that TC does not define pairing. We determine a reasonable extension of TC that is sequential, i.e., has a good sequence coding. ..."
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In his paper [Grz05], Andrzej Grzegorczyk introduces a theory of concatenation TC. We show that TC does not define pairing. We determine a reasonable extension of TC that is sequential, i.e., has a good sequence coding.
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.
WHAT IS THE RIGHT NOTION OF SEQUENTIALITY?
"... Abstract. In this paper we give an informally semirigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion msequentiality as the notion that precisely captures the intuitions behind sequentiality. ..."
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Abstract. In this paper we give an informally semirigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion msequentiality as the notion that precisely captures the intuitions behind sequentiality. The paper provides all relevant separating examples minus one. 1.
Isle of Consciousness. According to Penrose
"... $UK 16.99 There was a time when cultured Englishmen would embark on a Grand Tour of Europe, visiting important cities and inspecting monuments and vistas. The book of Penrose is a Grand ..."
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$UK 16.99 There was a time when cultured Englishmen would embark on a Grand Tour of Europe, visiting important cities and inspecting monuments and vistas. The book of Penrose is a Grand
Book Review Philosophy of Mathematics: An Introduction to the World of Proofs and Pictures
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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
Abstract Bounded Arithmetic, Proof Complexity and Two Papers of Parikh
"... 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. 1 ..."
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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. 1