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Harvey Friedman [101]Harvey M. Friedman [59]
  1. The Upper Shift Kernel Theorems.Harvey M. Friedman - unknown
    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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  2. Similar Subclasses.Harvey M. Friedman - unknown
    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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  3. Does Mathematics Need New Axioms?Solomon Feferman, Harvey M. Friedman, Penelope Maddy & John R. Steel - 2000 - 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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  4. Maximal Nonfinitely Generated Subalgebras.Harvey Friedman - manuscript
    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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  5. Restrictions and Extensions.Harvey Friedman - manuscript
    We consider a number of statements involving restrictions and extensions of algebras, and derive connections with large cardinal axioms.
     
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  6. Decreasing Chains of Algebraic Sets.Harvey Friedman - manuscript
    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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  7. Remarks On the Unknowable.Harvey M. Friedman - unknown
    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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  8. Finite Trees and the Necessary Use of Large Cardinals.Harvey Friedman - manuscript
    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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  9. P01 INCOMPLETENESS: Finite Set Equations.Harvey M. Friedman - unknown
    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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  10.  16
    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]Harvey M. Friedman, Stephen G. Simpson & Rick L. Smith - 1983 - Annals of Pure and Applied Logic 25 (2):141-181.
  11. What You Cannot Prove 1: Before 2000.Harvey Friedman - manuscript
    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. Equational Boolean Relation Theory.Harvey Friedman - manuscript
    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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  13. Concept Calculus.Harvey Friedman - manuscript
    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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  14. A Consistency Proof for Elementary Algebra and Geometry.Harvey Friedman - manuscript
    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.  34
    One Hundred and Two Problems in Mathematical Logic.Harvey Friedman - 1975 - Journal of Symbolic Logic 40 (2):113-129.
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  16. Borel and Baire Reducibility.Harvey Friedman - manuscript
    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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  17. A Theory of Strong Indiscernibles.Harvey Friedman - manuscript
    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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  18. Uniformly Defined Descending Sequences of Degrees.Harvey Friedman - 1976 - Journal of Symbolic Logic 41 (2):363-367.
  19. Elemental Sentential Reflection.Harvey Friedman - manuscript
    “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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  20. Boolean Relation Theory Notes.Harvey Friedman - manuscript
    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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  21. Concept Calculus: Much Better Than.Harvey M. Friedman - unknown
    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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  22. Limitations on Our Understanding of the Behavior of Simplified Physical Systems.Harvey Friedman - manuscript
    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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  23. The Number of Certain Integral Polynomials and Nonrecursive Sets of Integers, Part.Harvey Friedman - manuscript
    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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  24. Computer Assisted Certainty.Harvey Friedman - manuscript
    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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  25. Unprovable Theorems in Discrete Mathematics.Harvey Friedman - manuscript
    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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  26. The Inevitability of Logical Strength: Strict Reverse Mathematics.Harvey Friedman - manuscript
    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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  27. 1 the Formalization of Mathematics.Harvey Friedman - manuscript
    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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  28. Finite Reverse Mathematics.Harvey Friedman - manuscript
    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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  29. Lecture Notes on Enormous Integers.Harvey Friedman - manuscript
    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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  30. Foundations of Mathematics: Past, Present, and Future.Harvey M. Friedman - unknown
    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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  31. Combining Decision Procedures for the Reals.Harvey Friedman & J. Avigad - manuscript
    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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  32.  21
    A Borel Reducibility Theory for Classes of Countable Structures.Harvey Friedman & Lee Stanley - 1989 - Journal of Symbolic Logic 54 (3):894-914.
    We introduce a reducibility preordering between classes of countable structures, each class containing only structures of a given similarity type (which is allowed to vary from class to class). Though we sometimes work in a slightly larger context, we are principally concerned with the case where each class is an invariant Borel class (i.e. the class of all models, with underlying set $= \omega$, of an $L_{\omega_1\omega}$ sentence; from this point of view, the reducibility can be thought of as a (...)
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  33. A Way Out.Harvey Friedman - manuscript
    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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  34. Quadratic Axioms.Harvey M. Friedman - unknown
    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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  35. Discrete Independence Results.Harvey Friedman - manuscript
    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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  36. The Ackermann Function in Elementary Algebraic Geometry.Harvey Friedman - manuscript
    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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  37. Axiomatization of Set Theory by Extensionality, Separation, and Reducibility.Harvey Friedman - manuscript
    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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  38. Metamathematics of Comparability.Harvey Friedman - manuscript
    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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  39. Vigre Lectures.Harvey M. Friedman - unknown
    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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  40. Concrete Mathematical Incompleteness.Harvey M. Friedman - unknown
    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. Working with Nonstandard Models.Harvey Friedman - manuscript
    Most of the research in foundations of mathematics that I do in some way or another involves the use of nonstandard models. I will give a few examples, and indicate what is involved.
     
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  42. The Mathematical Meaning of Mathematical Logic.Harvey Friedman - manuscript
    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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  43.  9
    Elementary Descent Recursion and Proof Theory.Harvey Friedman & Michael Sheard - 1995 - Annals of Pure and Applied Logic 71 (1):1-45.
    We define a class of functions, the descent recursive functions, relative to an arbitrary elementary recursive system of ordinal notations. By means of these functions, we provide a general technique for measuring the proof-theoretic strength of a variety of systems of first-order arithmetic. We characterize the provable well-orderings and provably recursive functions of these systems, and derive various conservation and equiconsistency results.
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  44. Selection for Borel Relations.Harvey M. Friedman - unknown
    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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  45. Ramsey Theory and Enormous Lower Bounds.Harvey Friedman - manuscript
    by Harvey M. Friedman Department of Mathematics Ohio State University friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman/ April 5, 1997..
     
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  46. Lecture Notes on Baby Boolean Relation Theory.Harvey Friedman - manuscript
    This is an introduction to the most primitive form of the new Boolean relation theory, where we work with only one function and one set. We give eight complete classifications. The thin set theorem (along with a slight variant), and the complementation theorem are the only substantial cases that arise in these classifications.
     
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  47. Kernel Structure Theory.Harvey M. Friedman - unknown
    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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  48. Fromal Statements of Godel's Second Incompleteness Theorem.Harvey Friedman - manuscript
    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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  49. Three Quantifier Sentences.Harvey Friedman - manuscript
    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. Interpreting Set Theory in Discrete Mathematics: Boolean Relation Theory.Harvey Friedman - manuscript
     
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