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On reflection principles
 Ann. Pure Appl. Logic
, 2009
"... Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justi ..."
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Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified. Taking as our starting point Tait’s work on general reflection principles, we prove a series of limitative results concerning this approach. These results collectively show that general reflection principles are either weak (in that they are consistent relative to the Erdös cardinal κ(ω)) or inconsistent. The philosophical significance of these results is discussed.
Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
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Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
Unfolding finitist arithmetic
, 2010
"... The concept of the (full) unfolding U(S) of a schematic system S is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted S? The program to determine U(S) for various systems S of foundational significan ..."
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The concept of the (full) unfolding U(S) of a schematic system S is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted S? The program to determine U(S) for various systems S of foundational significance was previously carried out for a system of nonfinitist arithmetic, NFA; it was shown that U(NFA) is prooftheoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system of finitist arithmetic, FA, and for an extension of that by a form BR of the socalled Bar Rule. It is shown that U(FA) and U(FA + BR) are prooftheoretically equivalent, respectively, to Primitive Recursive Arithmetic, PRA, and to Peano Arithmetic, PA.
Carnap on the foundations of logic and mathematics
, 2009
"... Throughout most of his philosophical career Carnap upheld and defended three distinctive philosophical positions: (1) The thesis that the truths of logic and mathematics are analytic and hence without content and purely formal. (2) The thesis that radical pluralism holds in pure mathematics in that ..."
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Throughout most of his philosophical career Carnap upheld and defended three distinctive philosophical positions: (1) The thesis that the truths of logic and mathematics are analytic and hence without content and purely formal. (2) The thesis that radical pluralism holds in pure mathematics in that any consistent system of postulates is equally legitimate and that there is no question of justification in mathematics but only the question of which system is most expedient for the purposes of empirical science. (3) A minimalist conception of philosophy in which most traditional questions are rejected as pseudoquestions and the task of philosophy is identified with the metatheoretic study of the sciences. In this paper, I will undertake a detailed analysis of Carnap’s defense of the first and second thesis. This will involve an examination of his most technical work The Logical Syntax of Language (1934), along with the monograph “Foundations of Logic and Mathematics ” (1939). These are the main works in which Carnap defends his views concerning the nature of truth and radical
Gödel on Intuition and on Hilbert’s finitism
"... There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the con ..."
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There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, Gödel’s writings represent a smooth evolution, with just one rather small doublereversal, of his view of finitism. He used the term “finit ” (in German) or “finitary ” or “finitistic ” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [Gödel, 1938a] and the lecture notes for a lecture at Yale [Gödel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of firstorder number theory, P A; but starting in the Dialectica paper
In Defense of the Ideal 2nd DRAFT
"... This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size ..."
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This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size of α. The question of the grounds for doing so is perhaps the central question of the workshop. But I want to discuss another question which, as I said, arises at the periphery: How do we know that there are structures 〈Rα, ∈ ∩R2 α〉? How do we know that there exist such things as sets and how do we know that, given such things, the axioms we write down are true of them? These seem very primitive questions, but the skepticism implicit in them has deep (and ancient) roots. In particular, they are questions about ideal objects in general, and not just about the actual infinite. I want to explain why I think the questions (as intended) are empty and the skepticism unfounded. 1 I will be expanding the argument of the first part of my paper “Proof and truth: the Platonism of mathematics”[1986a]. 2 The argument in question
Kant and the natural numbers
"... The ontological status of mathematical objects is perhaps the most important unsolved problem in the philosophy of mathematics. It is thoughtprovoking that within the philosophy of mathematics there is no agreement on what the mathematical objects generally are. It is, moreover, surprising that n ..."
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The ontological status of mathematical objects is perhaps the most important unsolved problem in the philosophy of mathematics. It is thoughtprovoking that within the philosophy of mathematics there is no agreement on what the mathematical objects generally are. It is, moreover, surprising that not even the question
Construction and Schemata in Mathematics
"... scandal in philosophy is the problem of free will ” [17, p. 205]. I very much agree with Suppes that the problem of the free will is a major puzzle, which we should try to get a better understanding of by examining the deeper issues connected with the free will. This essay, however, does not treat t ..."
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scandal in philosophy is the problem of free will ” [17, p. 205]. I very much agree with Suppes that the problem of the free will is a major puzzle, which we should try to get a better understanding of by examining the deeper issues connected with the free will. This essay, however, does not treat the problem of the free will. It concerns the problems of the ontology and epistemology of mathematics. In genereal, the problems of the philosophy of mathematics are just as old and—if it makes sense to talk about solvability of such problems— perhaps just as unsolved as the problem of the free will. Mathematics is a very important ingredient of knowledge. In its most simple form mathematics plays a necessary role in our understanding of the surrounding world and is necessary for solving simple problems of ordinary life. At the other end of the simplicityscale we find the mathematics as used in science. Also here mathematics has a necessary role in our descriptions of nature and the way in which we are involved with it and each other. It truly amazes me that there seems to be very little consensus with respect to the ontology and epistemology
—Carnap, The Logical Syntax of Language
"... “... before us lies the boundless ocean of unlimited possibilities.” ..."