15 found
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  1. On the Question of Whether the Mind Can Be Mechanized, I: From Gödel to Penrose.Peter Koellner - 2018 - Journal of Philosophy 115 (7):337-360.
    In this paper I address the question of whether the incompleteness theorems imply that “the mind cannot be mechanized,” where this is understood in the specific sense that “the mathematical outputs of the idealized human mind do not coincide with the mathematical outputs of any idealized finite machine.” Gödel argued that his incompleteness theorems implied a weaker, disjunctive conclusion to the effect that either “the mind cannot be mechanized” or “mathematical truth outstrips the idealized human mind.” Others, most notably, Lucas (...)
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  2. On the question of absolute undecidability.Peter Koellner - 2010 - In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: essays for his centennial. Ithaca, NY: Association for Symbolic Logic. pp. 153-188.
    The paper begins with an examination of Gödel's views on absolute undecidability and related topics in set theory. These views are sharpened and assessed in light of recent developments. It is argued that a convincing case can be made for axioms that settle many of the questions undecided by the standard axioms and that in a precise sense the program for large cardinals is a complete success “below” CH. It is also argued that there are reasonable scenarios for settling CH (...)
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  3. On the Question of Whether the Mind Can Be Mechanized, II: Penrose’s New Argument.Peter Koellner - 2018 - Journal of Philosophy 115 (9):453-484.
    Gödel argued that his incompleteness theorems imply that either “the mind cannot be mechanized” or “there are absolutely undecidable sentences.” In the precursor to this paper I examined the early arguments for the first disjunct. In the present paper I examine the most sophisticated argument for the first disjunct, namely, Penrose’s new argument. It turns out that Penrose’s argument requires a type-free notion of truth and a type-free notion of absolute provability. I show that there is a natural such system, (...)
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  4. On reflection principles.Peter Koellner - 2009 - Annals of Pure and Applied Logic 157 (2-3):206-219.
    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 ) or inconsistent. The philosophical significance of these (...)
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  5. Large cardinals beyond choice.Joan Bagaria, Peter Koellner & W. Hugh Woodin - 2019 - Bulletin of Symbolic Logic 25 (3):283-318.
    The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V or HOD is “far” from V. The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the (...)
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  6. Strong logics of first and second order.Peter Koellner - 2010 - Bulletin of Symbolic Logic 16 (1):1-36.
    In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant (...)
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  7. Independence and large cardinals.Peter Koellner - 2010 - Stanford Encyclopedia of Philosophy.
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  8. The Search for New Axioms.Peter Koellner - 2003 - Dissertation, Massachusetts Institute of Technology
    The independence results in set theory invite the search for new and justified axioms. In Chapter 1 I set the stage by examining three approaches to justifying the axioms of standard set theory and argue that the approach via reflection principles is the most successful. In Chapter 2 I analyse the limitations of ZF and use this analysis to set up a mathematically precise minimal hurdle which any set of new axioms must overcome if it is to effect a significant (...)
     
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  9. Incompatible Ω-Complete Theories.Peter Koellner & W. Hugh Woodin - 2009 - Journal of Symbolic Logic 74 (4):1155 - 1170.
    In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and $V^{B1} $ and $V^{B2} $ are generic extensions of V satisfying CH then $V^{B1} $ and $V^{B2} $ agree on all $\Sigma _1^2 $ -statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for $\Sigma _1^2 $ Moreover. CH is the unique $\Sigma _1^2 $ -statement (...)
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  10. Infinity up on Trial: Reply to Feferman.Peter Koellner - 2016 - Journal of Philosophy 113 (5/6):247-260.
    In this paper I examine Feferman’s reasons for maintaining that while the statements of first-order number theory are “completely clear'” and “completely definite,”' many of the statements of analysis and set theory are “inherently vague'” and “indefinite.”' I critique his four central arguments and argue that in the end the entire case rests on the brute intuition that the concept of subsets of natural numbers—along with the richer concepts of set theory—is not “clear enough to secure definiteness.” My response to (...)
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  11. On a Purported Proof that the Mind Is Not a Machine.Peter Koellner - 2018 - Thought: A Journal of Philosophy 7 (2):91-96.
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  12.  31
    (2 other versions)Of the association for symbolic logic.Sergei Artemov, Peter Koellner, Michael Rabin, Jeremy Avigad, Wilfried Sieg, William Tait & Haim Gaifman - 2006 - Bulletin of Symbolic Logic 12 (3-4):503.
  13.  21
    Foundations of Mathematics.Andrés Eduardo Caicedo, James Cummings, Peter Koellner & Paul B. Larson (eds.) - 2016 - American Mathematical Society.
    This volume contains the proceedings of the Logic at Harvard conference in honor of W. Hugh Woodin's 60th birthday, held March 27–29, 2015, at Harvard University. It presents a collection of papers related to the work of Woodin, who has been one of the leading figures in set theory since the early 1980s. The topics cover many of the areas central to Woodin's work, including large cardinals, determinacy, descriptive set theory and the continuum problem, as well as connections between set (...)
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  14. Feferman on Set Theory: Infinity up on Trial.Peter Koellner - 2017 - In Gerhard Jäger & Wilfried Sieg (eds.), Feferman on Foundations: Logic, Mathematics, Philosophy. Cham: Springer.
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    (1 other version)The Cambridge companion to Bertrand Russell. [REVIEW]Peter Koellner - 2005 - Bulletin of Symbolic Logic 11 (1):72-77.