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  1. Frege's Theorem in Plural Logic.Simon Hewitt - manuscript
    A version of Frege's theorem can be proved in a plural logic with pair abstraction. We talk through this and discuss the philosophical implications of the result.
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  2. Frege's Theorem and Foundations for Arithmetic.Edward N. Zalta - Spring 2015 - In Stanford Encyclopedia of Philosophy.
    The principal goal of this entry is to present Frege's Theorem (i.e., the proof that the Dedekind-Peano axioms for number theory can be derived in second-order logic supplemented only by Hume's Principle) in the most logically perspicuous manner. We strive to present Frege's Theorem by representing the ideas and claims involved in the proof in clear and well-established modern logical notation. This prepares one to better prepared to understand Frege's own notation and derivations, and read Frege's original work (whether in (...)
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  3. Frege's Theorem. [REVIEW]P. Ebert - 2014 - Philosophical Quarterly 64 (254):166-169.
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  4. Review of Frege's Theorem[REVIEW]G. Aldo Antonelli - 2012 - International Studies in the Philosophy of Science 26 (2):219-222.
  5. Richard G. Heck, Jr. , Frege's Theorem . Reviewed By. [REVIEW]Manuel Bremer - 2012 - Philosophy in Review 32 (4):319-325.
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  6. Richard G. Heck, Jr.: Frege’s Theorem. [REVIEW]John P. Burgess - 2012 - Journal of Philosophy 109 (12):728-733.
  7. Frege’s Theorem by Richard G. Heck, Jr. [REVIEW]John P. Burgess - 2012 - Journal of Philosophy 109 (12):728-732.
  8. Reading Frege's Grundgesetze.Richard Heck - 2012 - Oxford University Press.
    Richard G. Heck presents a new account of Gottlob Frege's Grundgesetze der Arithmetik, or Basic Laws of Arithmetic, which establishes it as a neglected masterpiece at the center of Frege's philosophy. He explores Frege's philosophy of logic, and argues that Frege knew that his proofs could be reconstructed so as to avoid Russell's Paradox.
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  9. Frege's Result: Frege's Theorem and Related Matters.Hirotoshi Tabata - 2012 - Frontiers of Philosophy in China 7 (3):351-366.
    One of the remarkable results of Frege’s Logicism is Frege’s Theorem, which holds that one can derive the main truths of Peano arithmetic from Hume’s Principle (HP) without using Frege’s Basic Law V. This result was rediscovered by the Neo-Fregeans and their allies. However, when applied in developing a more advanced theory of mathematics, their fundamental principles—the abstraction principles—incur some problems, e.g., that of inflation. This paper finds alternative paths for such inquiry in extensionalism and object theory.
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  10. A Logic for Frege's Theorem.Richard Heck - 2011 - In Frege’s Theorem: An Introduction. Oxford University Press.
    It has been known for a few years that no more than Pi-1-1 comprehension is needed for the proof of "Frege's Theorem". One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no part of my view that, say, (...)
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  11. Frege's Theorem.Richard G. Heck - 2011 - Clarendon Press.
    The book begins with an overview that introduces the Theorem and the issues surrounding it, and explores how the essays that follow contribute to our understanding of those issues.
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  12. Frege's Logic, Theorem, and Foundations for Arithmetic.Edward N. Zalta - 2010 - Stanford Encyclopedia of Philosophy.
    This entry explains Frege's Theorem by using the modern notation of the predicate calculus. Frege's Theorem is that the Dedekind-Peano axioms for number theory are derivable from Hume's Principle, given the axioms and rules of second-order logic. Frege's methodology for defining the natural numbers and for the derivation of the Dedekind-Peano axioms are sketched in some detail.
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  13. Hume’s Principle and Axiom V Reconsidered: Critical Reflections on Frege and His Interpreters.Matthias Schirn - 2006 - Synthese 148 (1):171-227.
    In this paper, I shall discuss several topics related to Frege's paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege's notion of evidence and its interpretation by Jeshion, the introduction (...)
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  14. Predicative Fragments of Frege Arithmetic.Øystein Linnebo - 2004 - Bulletin of Symbolic Logic 10 (2):153-174.
    Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume’s Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be one-to-one correlated. According to Frege’s Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume’s Principle, the other, with the underlying second-order logic—and (...)
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  15. On the Philosophical Interest of Frege Arithmetic.William Demopoulos - 2003 - Philosophical Books 44 (3):220-228.
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  16. Frege’s Theorem: AN INTRODUCTION.Richard Heck Jr - 2003 - Manuscrito 26 (2):471-503.
    Frege's work was largely devoted to an attempt to argue that the'basic laws of arithmetic' are truths of logic. That attempt had both philosophical and formal aspects. The present note offers an introduction to both of these, so that readers will be able to appreciate contemporary discussions of the philosophical significance of 'Frege's Theorem'.
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  17. Why Frege Would Not Be a Neo‐Fregean.Marco Ruffino - 2003 - Mind 112 (445):51-78.
    In this paper, I seek to clarify an aspect of Frege's thought that has been only insufficiently explained in the literature, namely, his notion of logical objects. I adduce some elements of Frege's philosophy that elucidate why he saw extensions as natural candidates for paradigmatic cases of logical objects. Moreover, I argue (against the suggestion of some contemporary scholars, in particular, Wright and Boolos) that Frege could not have taken Hume's Principle instead of Axiom V as a fundamental law of (...)
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  18. Critical Notice of Michael Dummett's Frege: Philosophy of Mathematics.Crispin Wright - 2001 - Philosophical Books 1995:89-102.
    The aim of this critical notice is to elucidate Dummett's contributions to the issues surrounding Frege's contextual definition of number (the number of Fs equals the number of Gs if the Fs and the Gs are in one-one correspondence) and the interpretation of "Frege's theorem" -- the theorem that the second order theory consisting of the contextual definition implies the infinity of the natural numbers. To do so, we focus on Dummett's account of the context principle, his discussion of Frege's (...)
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  19. Frege's Theorem and His Logicism.Hirotoshi Tabata - 2000 - History and Philosophy of Logic 21 (4):265-295.
    As is well known, Frege gave an explicit definition of number (belonging to some concept) in ?68 of his Die Grundlagen der Arithmetik.
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  20. Frege's Theorem in a Constructive Setting.John L. Bell - 1999 - Journal of Symbolic Logic 64 (2):486-488.
    then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map ν be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., (...)
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  21. Frege’s Theorem: An Introduction.Richard Heck - 1999 - The Harvard Review of Philosophy 7 (1):56-73.
    A brief, non-technical introduction to technical and philosophical aspects of Frege's philosophy of arithmetic. The exposition focuses on Frege's Theorem, which states that the axioms of arithmetic are provable, in second-order logic, from a single non-logical axiom, "Hume's Principle", which itself is: The number of Fs is the same as the number of Gs if, and only if, the Fs and Gs are in one-one correspondence.
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  22. Frege’s Theorem.Richard Heck - 1999 - The Harvard Review of Philosophy 7 (1):56-73.
    A brief, non-technical introduction to technical and philosophical aspects of Frege's philosophy of arithmetic. The exposition focuses on Frege's Theorem, which states that the axioms of arithmetic are provable, in second-order logic, from a single non-logical axiom, "Hume's Principle", which itself is: The number of Fs is the same as the number of Gs if, and only if, the Fs and Gs are in one-one correspondence.
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  23. On the Philosophical Significance of Frege's Theorem.Crispin Wright - 1997 - In Richard G. Heck (ed.), Language, Thought, and Logic: Essays in Honour of Michael Dummett. Oxford University Press. pp. 201--44.
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  24. Frege's Theorem and the Peano Postulates.George Boolos - 1995 - Bulletin of Symbolic Logic 1 (3):317-326.
    Two thoughts about the concept of number are incompatible: that any zero or more things have a (cardinal) number, and that any zero or more things have a number (if and) only if they are the members of some one set. It is Russell's paradox that shows the thoughts incompatible: the sets that are not members of themselves cannot be the members of any one set. The thought that any (zero or more) things have a number is Frege's; the thought (...)
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  25. Frege's Theorem and the Peano Postulates.George Boolos - 1995 - Bulletin of Symbolic Logic 1 (3):317-326.
  26. The Development of Arithmetic in Frege's Grundgesetze der Arithmetik.Richard Heck - 1993 - Journal of Symbolic Logic 58 (2):579-601.
    Frege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system--Axiom V--which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does prove each of (...)
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  27. IX—Saving Frege From Contradiction.George Boolos - 1987 - Proceedings of the Aristotelian Society 87 (1):137-152.
  28. Saving Frege From Contradiction.George Boolos - 1987 - Proceedings of the Aristotelian Society 87:137--151.
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  29. Frege's Conception of Numbers as Objects.Crispin Wright - 1983 - Aberdeen University Press.
  30. Frege's Theory of Numbers.Charles Parsons - 1965 - In M. Black (ed.), Philosophy in America. Cornell University Press. pp. 180-203.