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Harvey Friedman [101]Harvey M. Friedman [56]
  1. Harvey Friedman (1976). Uniformly Defined Descending Sequences of Degrees. Journal of Symbolic Logic 41 (2):363-367.
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  2. 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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  3. 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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  4. Harvey Friedman, Decreasing Chains of Algebraic Sets.
    An ideal in a commutative ring R with unit is a nonempty I Õ R such that for all x,y Œ I, z Œ R, we have x+y and xz Œ I. A set of generators for I is a subset of I such that I is the least ideal containing that subset.
     
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  5. Harvey Friedman, Maximal Nonfinitely Generated Subalgebras.
    We show that “every countable algebra with a nonfinitely generated subalgebra has a maximal nonfinitely generated subalgebra” is provably equivalent to ’11-CA0 over..
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  6. Solomon Feferman, Harvey M. Friedman, Penelope Maddy & John R. Steel (2000). Does Mathematics Need New Axioms? Bulletin of Symbolic Logic 6 (4):401-446.
    Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest di erence lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Now some of my fellow mathematical logicians might protest this distinction, since they consider themselves to be just more of those \working mathematicians". Certainly, modern logic has established itself as a very respectable branch of mathematics, and there (...)
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  7. Harvey Friedman, Finite Trees and the Necessary Use of Large Cardinals.
    We introduce insertion domains that support the placement of new, higher, vertices into finite trees. We prove that every nonincreasing insertion domain has an element with simple structural properties in the style of classical Ramsey theory. This result is proved using standard large cardinal axioms that go well beyond the usual axioms for mathematics. We also establish that this result cannot be proved without these large cardinal axioms. We also introduce insertion rules that specify the placement of new, higher, vertices (...)
     
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  8. 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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  9. Harvey Friedman, Restrictions and Extensions.
    We consider a number of statements involving restrictions and extensions of algebras, and derive connections with large cardinal axioms.
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  10. 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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  11. Harvey Friedman, What You Cannot Prove 1: Before 2000.
    Most of my intellectual efforts have focused around a single general question in the foundations of mathematics (f.o.m.). I became keenly aware of this question as a student at MIT around 40 years ago, and readily adopted it as the principal driving force behind my research.
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  12. 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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  13. Harvey Friedman, Unprovable Theorems in Discrete Mathematics.
    An unprovable theorem is a mathematical result that can-not be proved using the com-monly accepted axioms for mathematics (Zermelo-Frankel plus the axiom of choice), but can be proved by using the higher infinities known as large cardinals. Large car-dinal axioms have been the main proposal for new axioms originating with Gödel.
     
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  14. 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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  15. Harvey Friedman, The Number of Certain Integral Polynomials and Nonrecursive Sets of Integers, Part.
    We present some examples of mathematically natural nonrecursive sets of integers and relations on integers by combining results from Part 1, recursion theory, and from the negative solution to Hilbert’s 10th Problem ([3], [1], and [2]).
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  16. Harvey Friedman, Equational Boolean Relation Theory.
    Equational Boolean Relation Theory concerns the Boolean equations between sets and their forward images under multivariate functions. We study a particular instance of equational BRT involving two multivariate functions on the natural numbers and three infinite sets of natural numbers. We prove this instance from certain large cardinal axioms going far beyond the usual axioms of mathematics as formalized by ZFC. We show that this particular instance cannot be proved in ZFC, even with the addition of slightly weaker large cardinal (...)
     
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  17. Harvey Friedman, The Inevitability of Logical Strength: Strict Reverse Mathematics.
    An extreme kind of logic skeptic claims that "the present formal systems used for the foundations of mathematics are artificially strong, thereby causing unnecessary headaches such as the Gödel incompleteness phenomena". The skeptic continues by claiming that "logician's systems always contain overly general assertions, and/or assertions about overly general notions, that are not used in any significant way in normal mathematics. For example, induction for all statements, or even all statements of certain restricted forms, is far too general - mathematicians (...)
     
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  18. 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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  19. Harvey Friedman, Lecture Notes on Enormous Integers.
    We discuss enormous integers and rates of growth after [PH77]. This breakthrough was based on a variant of the classical finite Ramsey theorem. Since then, examples have been given of greater relevance to a number of standard mathematical and computer science contexts, often involving even more enormous integers and rates of growth.
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  20. Harvey M. Friedman, Concept Calculus: Much Better Than.
    This is the initial publication on Concept Calculus, which establishes mutual interpretability between formal systems based on informal commonsense concepts and formal systems for mathematics through abstract set theory. Here we work with axioms for "better than" and "much better than", and the Zermelo and Zermelo Frankel axioms for set theory.
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  21. Harvey Friedman, Limitations on Our Understanding of the Behavior of Simplified Physical Systems.
    There are two kinds of such limiting results that must be carefully distinguished. Results of the first kind state the nonexistence of any algorithm for determining whether any statement among a given set of statements is true or false.
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  22. Harvey Friedman (2008). What is o-Minimality? Annals of Pure and Applied Logic 156 (1):59-67.
    We characterize the o-minimal expansions of the ring of real numbers, in mathematically transparent terms. This should help bridge the gap between investigators in o-minimality and mathematicians unfamiliar with model theory, who are concerned with such notions as non oscillatory behavior, tame topology, and analyzable functions. We adapt the characterization to the case of o-minimal expansions of an arbitrary ordered ring.
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  23. 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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  24. Harvey Friedman (2000). Does Mathematics Need New Axioms? Bulletin of Symbolic Logic 6 (4):401 - 446.
    Since about 1925, the standard formalization of mathematics has been the ZFC axiom system (Zermelo Frankel set theory with the axiom of choice), about which the audience needs to know nothing. The axiom of choice was controversial for a while, but the controversy subsided decades ago.
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  25. Harvey Friedman, Computer Assisted Certainty.
    Certainty (and the lack thereof) is a major issue in mathematics and computer science. Mathematicians strongly believe in a special kind of certainty for their theorems.
     
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  26. Harvey Friedman, Concept Calculus.
    PREFACE. We present a variety of basic theories involving fundamental concepts of naive thinking, of the sort that were common in "natural philosophy" before the dawn of physical science. The most extreme forms of infinity ever formulated are embodied in the branch of mathematics known as abstract set theory, which forms the accepted foundation for all of mathematics. Each of these theories embodies the most extreme forms of infinity ever formulated, in the following sense. ZFC, and even extensions of ZFC (...)
     
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  27. Harvey Friedman, Metamathematics of Comparability.
    A number of comparability theorems have been investigated from the viewpoint of reverse mathematics. Among these are various comparability theorems between countable well orderings ([2],[8]), and between closed sets in metric spaces ([3],[5]). Here we investigate the reverse mathematics of a comparability theorem for countable metric spaces, countable linear orderings, and sets of rationals. The previous work on closed sets used a strengthened notion of continuous embedding. The usual weaker notion of continuous embedding is used here. As a byproduct, we (...)
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  28. Harvey Friedman, Interpreting Set Theory in Discrete Mathematics: Boolean Relation Theory.
     
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  29. Harvey Friedman, 1 the Formalization of Mathematics.
    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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  30.  18
    Harvey Friedman (1975). One Hundred and Two Problems in Mathematical Logic. Journal of Symbolic Logic 40 (2):113-129.
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  31. 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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  32. 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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  33. Harvey Friedman, The Ackermann Function in Elementary Algebraic Geometry.
    We can equivalently present this by the recursion equations f1(n) = 2n, fk+1(1) = fk(1), fk+1(n+1) = fk(fk+1(n)), where k,n ≥ 1. We define A(k,n) = fk(n).
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  34. Harvey Friedman, Godel's Legacy in Mathematical Philosophy.
    Gödel's definitive results and his essays leave us with a rich legacy of philosophical programs that promise to be subject to mathematical treatment. After surveying some of these, we focus attention on the program of circumventing his demonstrated impossibility of a consistency proof for mathematics by means of extramathematical concepts.
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  35. 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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  36.  2
    Harvey M. Friedman, Stephen G. Simpson & Rick L. Smith (1983). Countable Algebra and Set Existence axioms11Research Partially Supported by NSF Grants MCS-79-23743, MCS-78-02558, and MCS 8107867. Simpson's Research Was Also Supported by an Alfred P. Sloan Research Fellowship. [REVIEW] Annals of Pure and Applied Logic 25 (2):141-181.
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  37. Harvey Friedman, Finite Reverse Mathematics.
    We present some formal systems in the language of linearly ordered rings with finite sets whose nonlogical axioms are strictly mathematical, which correspond to polynomially bounded arithmetic. With an additional strictly mathematical axiom, the systems correspond to exponentially bounded arithmetic.
     
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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, Fromal Statements of Godel's Second Incompleteness Theorem.
    Informal statements of Gödel's Second Incompleteness Theorem, referred to here as Informal Second Incompleteness, are simple and dramatic. However, current versions of Formal Second Incompleteness are complicated and awkward. We present new versions of Formal Second Incompleteness that are simple, and informally imply Informal Second Incompleteness. These results rest on the isolation of simple formal properties shared by consistency statements. Here we do not address any issues concerning proofs of Second Incompleteness.
     
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  40. 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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  41. Harvey Friedman, Discrete Independence Results.
    A bi-infinite approximate fixed point of type (n,k) is an approximate fixed point of type (n,k) whose terms are biinfinite; i.e., contain infin-itely many positive and infinitely many negative elements.
     
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  42. 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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  43. 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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  44. 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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  45. 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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  46. Harvey Friedman, Elemental Sentential Reflection.
    “Sentential reflection” in the sense of [Fr03] is based on reflecting down from a category of classes. “Elemental sentential reflection” is based on reflecting down from a category of elemental classes. We present various forms of elemental sentential reflection, which are shown to interpret and be interpretable in certain set theories with large cardinal axioms.
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  47. 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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  48. Harvey Friedman, Completeness of Intuitionistic Propositional Calculus.
    An assignment is a function f that assigns subsets of N to some atoms. Then f is extended to f* which sends every formula A of HPC to a subset of S(A).
     
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  49. Harvey Friedman, Three Quantifier Sentences.
    We give a complete proof that all 3 quantifier sentences in the primitive notation of set theory (Œ,=), are decided in ZFC, and in fact in a weak fragment of ZF without the power set axiom. We obtain information concerning witnesses of 2 quantifier formulas with one free variable. There is a 5 quantifier sentence that is not decided in ZFC (see [Fr02]).
     
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  50. Harvey Friedman, The Mathematical Meaning of Mathematical Logic.
    Each of these theorems and concepts arose from very specific considerations of great general interest in the foundations of mathematics (f.o.m.). They each serve well defined purposes in f.o.m. Naturally, the preferred way to formulate them for mathe-matical logicians is in terms that are close to their roots in f.o.m.
     
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