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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.
Independence Structures In Set Theory
, 1996
"... This article, based on an invited lecture at the Logic Colloquium '93 in Keele, is a sequel to van Lambalgen [1992]. Apart from presenting new results, it differs from its predecessor in the following respects: (i) the presentation of the axioms is simplified, following some suggestions of Wo ..."
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This article, based on an invited lecture at the Logic Colloquium '93 in Keele, is a sequel to van Lambalgen [1992]. Apart from presenting new results, it differs from its predecessor in the following respects: (i) the presentation of the axioms is simplified, following some suggestions of Wojciech Buszkowski, (ii) the axioms have been strengthened, and (iii) the philosophical discussion has (hopefully) been improved. The article has appeared in W. Hodges et al (eds.), Logic: from Foundations to Applications (European Logic Colloquium), Oxford University Press 1996
Global Reflection Principles
, 2012
"... Reflection Principles are commonly thought to produce only strong axioms of infinity consistent with V = L. It would be desirable to have some notion of strong reflection to remedy this, and we have proposed Global Reflection Principles based on a somewhat Cantorian view of the universe. Such princi ..."
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Reflection Principles are commonly thought to produce only strong axioms of infinity consistent with V = L. It would be desirable to have some notion of strong reflection to remedy this, and we have proposed Global Reflection Principles based on a somewhat Cantorian view of the universe. Such principles justify the kind of cardinals needed for, inter alia, Woodin’s ΩLogic. 1 To say that the universe of all sets is an unfinished totality does not mean objective undeterminateness, but merely a subjective inability to finish it. Gödel, in Wang, [17] 1 Reflection Principles in Set Theory Historically reflection principles are associated with attempts to say that no one notion, idea, or statement can capture our whole view of the universe of sets V = ⋃ α∈On Vα where On is the class of all ordinals. That no one idea can pin down the universe of all sets has firm historical roots (see the quotation from Cantor later or the following): The Universe of sets cannot be uniquely characterized (i.e. distinguished from all its initial segments) by any internal structural property of the membership relation in it, which is expressible in any logic of finite or transfinite type, including infinitary logics of any cardinal number. Gödel: Wang ibid. Indeed once set theory was formalized by the (first order version of) the axioms and schemata of Zermelo with the additions of Skolem and Fraenkel, it was seen that reflection of first order formulae ϕ(v0, , vn) in the language of set theory L∈ ˙ could actually be proven:
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
A Puzzle for Structuralism
, 2003
"... Structuralism is the view that the subjectmatter of a theory of pure mathematics is a mathematical structure. Different versions of structuralism tell different stories about what mathematical structures are.1 But assuming that they agree about when a model2 exemplifies a mathematical structure, t ..."
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Structuralism is the view that the subjectmatter of a theory of pure mathematics is a mathematical structure. Different versions of structuralism tell different stories about what mathematical structures are.1 But assuming that they agree about when a model2 exemplifies a mathematical structure, they can agree about the conditions under which a sentence of pure mathematics is true: