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Harvey Friedman [100]Harvey M. Friedman [55]
  1. Jeremy Avigad & Harvey Friedman, Combining Decision Procedures for the Reals.
    We address the general problem of determining the validity of boolean combinations of equalities and inequalities between real-valued expressions. In particular, we consider methods of establishing such assertions using only restricted forms of distributivity. At the same time, we explore ways in which “local'’ decision or heuristic procedures for fragments of the theory of the reals can be amalgamated into global ones. Let $Tadd[QQ]$ be the first-order theory of the real numbers in the language with symbols $0, 1, +, -.
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  2. Jeremy Avigad, Steven Kieffer & Harvey Friedman, A Language for Mathematical Knowledge Management.
    We argue that the language of Zermelo Fraenkel set theory with definitions and partial functions provides the most promising bedrock semantics for communicating and sharing mathematical knowledge. We then describe a syntactic sugaring of that language that provides a way of writing remarkably readable assertions without straying far from the set-theoretic semantics. We illustrate with some examples of formalized textbook definitions from elementary set theory and point-set topology. We also present statistics concerning the complexity of these definitions, under various complexity (...)
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  3. Harvey Friedman, Boolean Relation Theory and the Incompleteness Phenomena.
    ENTIRE BOOK, SINGLE FILE. BOOLEAN RELATION THEORY AND THE INCOMPLETENESS PHENOMENA. 10/30/07 version. Same as 10/01/07 version with Preface added. 568 pages without Appendix B. See above for Appendix B by Francoise Point.
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  4. Harvey Friedman & J. Avigad, Combining Decision Procedures for the Reals.
    We address the general problem of determining the validity of boolean combinations of equalities and inequalities between real-valued expressions. In particular, we consider methods of establishing such assertions using only restricted forms of distributivity. At the same time, we explore ways in which “local” decision or heuristic procedures for fragments of the theory of the reals can be amalgamated into global ones.
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  5. Harvey M. Friedman, Agenda.
    In the Foundational Life, philosophy is commonly used as a method for choosing and analyzing fundamental concepts, and mathematics is commonly used for rigorous development. The mathematics informs the philosophy and the philosophy informs the mathematics.
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  6. Harvey M. Friedman, A Complete Theory of Everything: Satisfiability in the Universal Domain.
    Here we take the view that LPC(=) is applicable to structures whose domain is too large to be a set. This is not just a matter of class theory versus set theory, although it can be interpreted as such, and this interpretation is discussed briefly at the end.
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  7. Harvey M. Friedman, Applications of Large Cardinals to Graph Theory.
    Since then we have been engaged in the development of such results of greater relevance to mathematical practice. In January, 1997 we presented some new results of this kind involving what we call “jump free” classes of finite functions. This Jump Free Theorem is treated in section 2.
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  8. Harvey M. Friedman, Adjacent Ramsey Theory.
    Let k ≥ 2 and f:Nk Æ [1,k] and n ≥ 1 be such that there is no x1 < ... < xk+1 £ n such that f(x1,...,xk) = f(x1,...,xk+1). Then we want to find g:Nk+1 Æ [1,3] such that there is no x1 < ... < xk+2 £ n such that g(x1,...,xk+1) = g(x2,...,xk+2). This reducees adjacent Ramsey in k dimensions with k colors to adjacent Ramsey in k+1 dimensions with 3 colors.
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  9. Harvey M. Friedman, Concrete Mathematical Incompleteness.
    there are mathematical statements that cannot be proved or refuted using the usual axioms and rules of inference of mathematics.
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  10. Harvey M. Friedman, Clay Millenium Problem: P = Np.
    The equation P = NP concerns algorithms for deciding membership in sets. The consensus is that P ≠ NP, although some prominent experts guess otherwise.
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  11. Harvey M. Friedman, Decision Problems in Euclidean Geometry.
    We show the algorithmic unsolvability of a number of decision procedures in ordinary two dimensional Euclidean geometry, involving lines and integer points. We also consider formulations involving integral domains of characteristic 0, and ordered rings. The main tool is the solution to Hilbert's Tenth Problem. The limited number of facts used from recursion theory are isolated at the beginning.
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  12. Harvey M. Friedman, Equational Representations.
    We begin by presenting the language L(N,℘N,℘℘N). This is the standard language for presenting third order sentences, using its intended interpretation.
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  13. Harvey M. Friedman, Friedman@Math.Ohio-State.Edu.
    It has been accepted since the early part of the Century that there is no problem formalizing mathematics in standard formal systems of axiomatic set theory. Most people feel that they know as much as they ever want to know about how one can reduce natural numbers, integers, rationals, reals, and complex numbers to sets, and prove all of their basic properties. Furthermore, that this can continue through more and more complicated material, and that there is never a real problem.
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  14. Harvey M. Friedman, Foundations of Mathematics: Past, Present, and Future.
    It turns out, time and time again, in order to make serious progress in f.o.m., we need to take actual reasoning and actual development into account at precisely the proper level. If we take these into account too much, then we are faced with information that is just too difficult to create an exact science around - at least at a given state of development of f.o.m. And if we take these into account too little, our findings will not have (...)
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  15. Harvey M. Friedman, Finite Phase Transitions.
    This topic has been discussed earlier on the FOM email list in various guises. The common theme is: big numbers and long sequences associated with mathematical objects. See..
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  16. Harvey M. Friedman, From Russell's Paradox To.
    Russell’s way out of his paradox via the impredicative theory of types has roughly the same logical power as Zermelo set theory - which supplanted it as a far more flexible and workable axiomatic foundation for mathematics. We discuss some new formalisms that are conceptually close to Russell, yet simpler, and have the same logical power as higher set theory - as represented by the far more powerful Zermelo-Frankel set theory and beyond. END.
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  17. Harvey M. Friedman, Geometry Axioms.
    To prove this, we fix P(x) to be any polynomial of degree ≥ 1 with a positive and negative value. We define a critical interval to be any nonempty open interval on which P is strictly monotone and where P is not strictly monotone on any larger open interval. Here an open interval may not have endpoints in F, and may be infinite on the left or right or both sides. Obviously, the critical intervals are pairwise disjoint.
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  18. Harvey M. Friedman, Introduction.
    The use of x[y,z,w] rather than the more usual y Πx has many advantages for this work. One of them is that we have found a convenient way to eliminate any need for axiom schemes. All axioms considered are single sentences with clear meaning. (In one case only, the axiom is a conjunction of a manageable finite number of sentences).
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  19. Harvey M. Friedman, Issues in the Foundations of Mathematics.
    C. To what extent, and in what sense, is the natural hierarchy of logical strengths rep resented by familiar systems ranging from exponential function arithmetic to ZF + j:V Æ V robust?
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  20. Harvey M. Friedman, Kernel Structure Theory.
    We have been recently engaged in this search, and have announced a long series of successively simpler and more convincing examples. See [Fr09-10].
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  21. Harvey M. Friedman, Phenomena.
    We have been particularly interested in the demonstrable unremovability of machinery, which is a theme that can be pursued systematically starting at the most elementary level - the use of binary notation to represent integers; the use of rational numbers to solve linear equations; the use of real and complex numbers to solve polynomial equations; and the use of transcendental functions to solve differential equations.
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  22. Harvey M. Friedman, P01 INCOMPLETENESS: Finite Set Equations.
    Let R Õ [1,n]3k ¥ [1,n]k. We define R = {y Œ [1,n]k:($xŒA3)(R(x,y))}. We say that R is strictly dominating if and only if for all x,yŒ[1,n]k, if R(x,y) then max(x) < max(y).
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  23. Harvey M. Friedman, Philosophy 532 Philosophical Problems in Logic Lecture 1 9/25/02.
    This is widely accepted, inside and outside philosophy, but one can spend an entire career clarifying, justifying, and amplifying on this statement. Certainly a graduate student career.
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  24. Harvey M. Friedman, Philosophy 536 Philosophy of Mathematics Lecture 1 9/25/02.
    This distinction between logic and mathematics is subject to various criticisms and can be given various defenses. Nevertheless, the division seems natural enough and is commonly adopted in presentations of the standard foundations for mathematics.
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  25. Harvey M. Friedman, P 1 INCOMPLETENESS: Finite Set Equations.
    We say that R is strictly dominating if and only if for all x,yŒ[1,n], if R(x,y) then max(x) 3k ¥ [1,n], there exists A Õ [1,n] such that R = A. Furthermore, A Õ [1,n] is unique.
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  26. Harvey M. Friedman, Proofless Text.
    i. Proofless text is based on a variant of ZFC with free logic. Here variables always denote, but not all terms denote. If a term denotes, then all subterms must denote. The sets are all in the usual extensional cumulative hierarchy of sets. There are no urelements.
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  27. Harvey M. Friedman, Quadratic Axioms.
    We axiomatize EFA in strictly mathematical terms, involving only the ring operations, without extending the language by either exponentiation, finite sets of integers, or polynomials.
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  28. Harvey M. Friedman, Remarks On GÖDel Phenomena and the Field of Reals.
    A lot of the well known impact of the Gödel phenomena is in the form of painful messages telling us that certain major mathematical programs cannot be completed as intended. This aspect of Gödel – the delivery of bad news –is not welcomed, and defensive measures are now in place.
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  29. Harvey M. Friedman, Remarks On the Unknowable.
    The kind of unknowability I will discuss concerns the count of certain natural finite sets of objects. Even the situation with regard to our present strong formal systems is rather unclear. One can just profitably focus on that, putting aside issues of general unknowability.
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  30. Harvey M. Friedman, Selection for Borel Relations.
    We present several selection theorems for Borel relations, involving only Borel sets and functions, all of which can be obtained as consequences of closely related theorems proved in [DSR 96,99,01,01X] involving coanalytic sets. The relevant proofs given there use substantial set theoretic methods, which were also shown to be necessary. We show that none of our Borel consequences can be proved without substantial set theoretic methods. The results are established for Baire space. We give equivalents of some of the main (...)
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  31. Harvey M. Friedman, Sentential Reflection.
    We present two forms of “sentential reflection”, which are shown to be mutually interpretable with Z2 and ZFC, respectively.
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  32. Harvey M. Friedman, Strict Reverse Mathematics Draft.
    NOTE: This is an expanded version of my lecture at the special session on reverse mathematics, delivered at the Special Session on Reverse Mathematics held at the Atlanta AMS meeting, on January 6, 2005.
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  33. Harvey M. Friedman, Similar Subclasses.
    Reflection, in the sense of [Fr03a] and [Fr03b], is based on the idea that a category of classes has a subclass that is “similar” to the category. Here we present axiomatizations based on the idea that a category of classes that does not form a class has extensionally different subclasses that are “similar”. We present two such similarity principles, which are shown to interpret and be interpretable in certain set theories with large cardinal axioms.
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  34. Harvey M. Friedman, The Interpretation of Set Theory in Mathematical Predication Theory.
    This paper was referred to in the Introduction in our paper [Fr97a], “The Axiomatization of Set Theory by Separation, Reducibility, and Comprehension.” In [Fr97a], all systems considered used the axiom of Extensionality. This is appropriate in a set theoretic context.
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  35. Harvey M. Friedman, Table of Contents.
    In fact, Godel gave an important model of pure predication, where he showed that restricted comprehension without parameters is valid, but where restricted comprehension with parameters is not (although this invalidity was not established until Cohen). This is the model based on ordinal definability in set theory.
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  36. Harvey M. Friedman, Transfer Principles in Set Theory.
    1. Transfer principles from N to On. A. Mahlo cardinals. B. Weakly compact cardinals. C. Ineffable cardinals. D. Ramsey cardinals. E. Ineffably Ramsey cardinals. F. Subtle cardinals. G. From N to (...))
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  37. Harvey M. Friedman, The Upper Shift Kernel Theorems.
    We now fix A ⊆ Q. We study a fundamental class of digraphs associated with A, which we call the A-digraphs. An A,kdigraph is a digraph (Ak,E), where E is an order invariant subset of A2k in the following sense. For all x,y ∈ A2k, if x,y have the same order type then x ∈ E ↔ y ∈ E.
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  38. Harvey M. Friedman, Vigre Lectures.
    In mathematics, we back up our discoveries with rigorous deductive proofs. Mathematicians develop a keen instinctive sense of what makes a proof rigorous. In logic, we strive for a *theory* of rigorous proofs.
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  39. Harvey Friedman & Chris Miller, A Big Difference Between Interpretability and Definability in an Expansion of the Real Field.
    We say that E is R-sparse if f(Ek) has no interior, for each k 2 N and f : Rk ! R de nable in R. (Throughout, \de nable" means \de nable without parameters".) In this note, we consider the extent to which basic metric and topological properties of subsets of R de nable in (R;E)# are determined by the corresponding properties of subsets of R de nable in (R;E), when R is an o-minimal expansion of (R;<;+;0;1) and E is (...)
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  40. Harvey Friedman, A Consistency Proof for Elementary Algebra and Geometry.
    We give a consistency proof within a weak fragment of arithmetic of elementary algebra and geometry. For this purpose, we use EFA (exponential function arithmetic), and various first order theories of algebraically closed fields and real closed fields.
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  41. Harvey Friedman, A Complete Theory of Everything: Satisfiability in the Universal Domain Harvey M. Friedman October 10, 1999 Friedman@Math.Ohio-State.Edu. [REVIEW]
    Here we take the view that LPC(=) is applicable to structures whose domain is too large to be a set. This is not just a matter of class theory versus set theory, although it can be interpreted as such, and this interpretation is discussed briefly at the end.
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  42. Harvey Friedman, A Complete Theory of Everything: Satisfiability in the Universal Domain Harvey M. Friedman October 10, 1999 Friedman@Math.Ohio-State.Edu Www.Math.Ohio-State.Edu/~Friedman/. [REVIEW]
    Here we take the view that LPC(=) is applicable to structures whose domain is too large to be a set. This is not just a matter of class theory versus set theory, although it can be interpreted as such, and this interpretation is discussed briefly at the end.
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  43. Harvey Friedman, Adventures in the Verification of Mathematics.
    Mathematical statements arising from program verification are believed to be much easier to deal with than statements coming from serious mathematics. At least this is true for “normal programming”.
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  44. Harvey Friedman, Applications of Large Cardinals to Borel Functions.
    The space CS(R) has a unique “Borel structure” in the following sense. Note that there is a natural mapping from R¥ onto CS(R}; namely, taking ranges. We can combine this with any Borel bijection from R onto R¥ in order to get a “preferred” surjection F:R ® CS(R).
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  45. Harvey Friedman, Axiomatization of Set Theory by Extensionality, Separation, and Reducibility.
    We discuss several axiomatizations of set theory in first order predicate calculus with epsilon and a constant symbol W, starting with the simple system K(W) which has a strong equivalence with ZF without Foundation. The other systems correspond to various extensions of ZF by certain large cardinal hypotheses. These axiomatizations are unusually simple and uncluttered, and are highly suggestive of underlying philosophical principles that generate higher set theory.
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  46. Harvey Friedman, A Theory of Strong Indiscernibles.
    The Complete Theory of Everything (CTE) is based on certain axioms of indiscernibility. Such axioms of indiscernibility have been given a philosophical justification by Kit Fine. I want to report on an attempt to give strong indiscernibility axioms which might also be subject to such philosophical analysis, and which prove the consistency of set theory; i.e., ZFC or more. In this way, we might obtain a (new kind of) philosophical consistency proof for mathematics.
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  47. Harvey Friedman, A Way Out.
    We present a way out of Russell’s paradox for sets in the form of a direct weakening of the usual inconsistent full comprehension axiom scheme, which, with no additional axioms, interprets ZFC. In fact, the resulting axiomatic theory 1) is a subsystem of ZFC + “there exists arbitrarily large subtle cardinals”, and 2) is mutually interpretable with ZFC + the scheme of subtlety.
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  48. Harvey Friedman, Borel and Baire Reducibility.
    The Borel reducibility theory of Polish equivalence relations, at least in its present form, was initiated independently in [FS89] and [HKL90]. There is now an extensive literature on this topic, including fundamental work on the Glimm-Effros dichotomy in [HKL90], on countable Borel equivalence relations in [DJK94], and on Polish group actions in [BK96].
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  49. Harvey Friedman, Boolean Relation Theory Notes.
    We give a detailed extended abstract reflecting what we know about Boolean relation theory. We follow this by a proof sketch of the main instances of Boolean relation theory, from Mahlo cardinals of finite order, starting at section 19. The proof sketch has been used in lectures.
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  50. Harvey Friedman, Conservation.
    I. WKL0 is a conservative extension of PRA for ’-0-2 sentences. II. ACA0 is a conservative extension of PA for arithmetic sentences. III. ATR0 is a conservative extension of IR for arithmetic sentences. IV. ’-1-1-CA0 is a conservative extension of ID(
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