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Is Complexity a Source of Incompleteness?
 IS COMPLEXITY A SOURCE OF INCOMPLETENESS
, 2004
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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.
Why philosophers should care about computational complexity
 In Computability: Gödel, Turing, Church, and beyond (eds
, 2012
"... One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed casethat onewouldbe wrong. In particular, I arguethat computational complexity theory—the field that ..."
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One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed casethat onewouldbe wrong. In particular, I arguethat computational complexity theory—the field that studies the resources (such as time, space, and randomness) needed to solve computational problems—leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume’s problem of induction, Goodman’s grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing
What does it mean to say that logic is formal
, 2000
"... Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and ..."
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Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say that logic is distinctively formal? Three things: logic is said to be formal (or “topicneutral”) (1) in the sense that it provides constitutive norms for thought as such, (2) in the sense that it is indifferent to the particular identities of objects, and (3) in the sense that it abstracts entirely from the semantic content of thought. Though these three notions of formality are by no means equivalent, they are frequently run together. The reason, I argue, is that modern talk of the formality of logic has its source in Kant, and these three notions come together in the context of Kant’s transcendental philosophy. Outside of this context (e.g., in Frege), they can come apart. Attending to this
GÖDEL AND SET THEORY
"... Kurt Gödel (1906–1978) with his work on the constructible universe L established the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of firstorder logic as the framework and a matter of method for set theory and secured the ..."
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Kurt Gödel (1906–1978) with his work on the constructible universe L established the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of firstorder logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of settheoretic constructions and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel’s work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges
My fourty years on his shoulders
, 2007
"... Gödel's legacy is still very much in evidence. We will not attempt to properly discuss the full impact of his work and all of the ongoing important research programs that it suggests. This would require a book length manuscript. Indeed, there are several books discussing the Gödel legacy from m ..."
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Gödel's legacy is still very much in evidence. We will not attempt to properly discuss the full impact of his work and all of the ongoing important research programs that it suggests. This would require a book length manuscript. Indeed, there are several books discussing the Gödel legacy from many points of view, including, for example, [Wa87], [Wa96], [Da05], and the historically comprehensive five volume set [Go,8603]. In sections 27 we briefly discuss some research projects that are suggested by some of his most famous contributions. In sections 811 we discuss some highlights of a main recurrent theme in our own research, which amounts to an expansion of the Gödel incompleteness phenomenon in a critical direction.
Dialectical Logic and Indiscrete Models
"... These laws [of logic] are the work of thought itself, and not a fact which [thought] finds and must submit to.2 G. W. F. Hegel 1830 Why are the objects that we want to take into account in dialectical logic not severally independent [isolated] things? What is there to relate entities internally as ..."
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These laws [of logic] are the work of thought itself, and not a fact which [thought] finds and must submit to.2 G. W. F. Hegel 1830 Why are the objects that we want to take into account in dialectical logic not severally independent [isolated] things? What is there to relate entities internally as distinct from externally?3 Kerruish and Petersen [With classical logic] the objective state of affairs is determined before we start reasoning.4 Kerruish and Petersen It might be argued that if α is a state, then all the information that is relevant to possible transitions is contained in α alone.5 J. M. Dunn Uwe Petersen6 offers a unique perspective on the philosophical significance of substructural logics.7 He argues that such logics encourage us to revisit Hegelian dialectical theses. The dialectical features that interest Petersen reside in proof theory. I note that these prooftheoretic characteristics have correlates in a certain class of models for nonclassical logics. Ternary operations on indices play a prominent role in the models for many deviant logics. Michael Dunn embraces an uncivilized interpretation8 of such operations — the resulting monsters provide our correlates.9 ∗University of Massachusetts at Dartmouth,
Absolute Infinity ∗
, 2012
"... This article is concerned with reflection principles in the context of Cantor’s conception of the set theoretic universe. We argue that within a Cantorian conception of the set theoretic universe reflection principles can be formulated that confer intrinsic plausibility to strong axioms of infinity. ..."
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This article is concerned with reflection principles in the context of Cantor’s conception of the set theoretic universe. We argue that within a Cantorian conception of the set theoretic universe reflection principles can be formulated that confer intrinsic plausibility to strong axioms of infinity. How can I talk to you, I have no words... Virgin Prunes, I am God 1