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Cantor's Grundlagen and the Paradoxes of Set Theory
"... This paper was written in honor of Charles Parsons, from whom I have profited for many years in my study of the philosophy of mathematics and expect to continue profiting for many more years to come. In particular, listening to his lecture on "Sets and classes", published in [Parsons, 197 ..."
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This paper was written in honor of Charles Parsons, from whom I have profited for many years in my study of the philosophy of mathematics and expect to continue profiting for many more years to come. In particular, listening to his lecture on "Sets and classes", published in [Parsons, 1974], motivated my first attempts to understand proper classes and the realm of transfinite numbers. I read a version of the paper at the APA Central Division meeting in Chicago in May, 1998. I thank Howard Stein, who provided valuable criticisms of an earlier draft, ranging from the correction of spelling mistakes, through important historical remarks, to the correction of a mathematical mistake, and Patricia Blanchette, who commented on the paper at the APA meeting and raised two challenging points which have led to improvements in this final version
Beyond the axioms: The question of objectivity in mathematics
"... I will be discussing the axiomatic conception of mathematics, the modern version of which is clearly due to Hilbert... ..."
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I will be discussing the axiomatic conception of mathematics, the modern version of which is clearly due to Hilbert...
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 There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between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There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between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
History
"... Proof theory was created early in the 20th century by David Hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics— in arithmetic (number theory), analysis and set theory. Already in his famous “Mathematical problems ” of 1900 [Hilbert, 1900] he raised, as the seco ..."
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Proof theory was created early in the 20th century by David Hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics— in arithmetic (number theory), analysis and set theory. Already in his famous “Mathematical problems ” of 1900 [Hilbert, 1900] he raised, as the second problem, that of proving the consistency of the arithmetic of the real numbers. In 1904, in “On the foundations of logic and arithmetic ” [Hilbert, 1905], he for the first time initiated his own program for proving consistency. 1.1 Consistency Whence his concern for consistency? The history of the concept of consistency in mathematics has yet to be written; but there are some things we can mention. There is of course a long tradition of skepticism. For example Descartes considered the idea we are deceived by a malicious god and that the simplest arithmetical truths might be inconsistent—but that was simply an empty skepticism. On the other hand, as we now know, in view of Gödel’s incom
Primitive Recursive Arithmetic and its Role in the Foundations of Arithmetic: Historical and Philosophical Reflections In Honor of Per MartinLöf on the Occasion of His Retirement
"... We discuss both the historical roots of Skolem’s primitive recursive arithmetic, its essential role in the foundations of arithmetic, its relation to the finitism of Hilbert and Bernays, and its relation to Kant’s philosophy of mathematics. 1. Skolem tells us in the Concluding Remark of his seminal ..."
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We discuss both the historical roots of Skolem’s primitive recursive arithmetic, its essential role in the foundations of arithmetic, its relation to the finitism of Hilbert and Bernays, and its relation to Kant’s philosophy of mathematics. 1. Skolem tells us in the Concluding Remark of his seminal paper on primitive recursive arithmetic (PRA), “The foundations of arithmetic established by means of the recursive mode of thought, without use of apparent variables ranging over infinite domains ” [1923], that the paper was written in 1919 after he had studied Whitehead and Russell’s Principia Mathematica and in reaction to that work. His specific complaint about the foundations of arithmetic (i.e. number theory) in that work was, as implied by his title, the essential role in it of logic and in particular quantification over infinite domains, even for the understanding of the most elementary propositions of arithmetic such as polynomial equations; and he set about to eliminate these infinitary quantifications by means of the “recursive mode of thought. ” On this ground, not only polynomial equations, but all primitive recursive formulas stand on their own feet without logical underpinning. 2. Skolem’s 1923 paper did not include a formal system of arithmetic, but as he noted in his 1946 address, “The development of recursive arithmetic” [1947], formalization of the methods used in that paper results in one of the many equivalent systems we refer to as PRA. Let me stop here and briefly describe one such system. ∗Is paper is loosely based on the Skolem Lecture that I gave at the University of Oslo in June, 2010. The present paper has profited, both with respect to what it now contains and with respect to what it no longer contains, from the discussion following that lecture. 1 We admit the following finitist types1 of objects: