55 found
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  1. Does Mathematics Need New Axioms.Solomon Feferman, Harvey M. Friedman, Penelope Maddy & John R. Steel - 1999 - 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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  2. 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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  3. 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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  4. 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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  5. 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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  6. 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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  7. 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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  8. 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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  9. 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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  10. A Complete Theory of Everything: Satisfiability in the Universal Domain.Harvey M. Friedman - unknown
    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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  11. Transfer Principles in Set Theory.Harvey M. Friedman - unknown
    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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  12. 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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  13. 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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  14. Sentential Reflection.Harvey M. Friedman - unknown
    We present two forms of “sentential reflection”, which are shown to be mutually interpretable with Z2 and ZFC, respectively.
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  15. Remarks On GÖDel Phenomena and the Field of Reals.Harvey M. Friedman - unknown
    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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  16.  27
    Weak Comparability of Well Orderings and Reverse Mathematics.Harvey M. Friedman & Jeffry L. Hirst - 1990 - Annals of Pure and Applied Logic 47 (1):11-29.
    Two countable well orderings are weakly comparable if there is an order preserving injection of one into the other. We say the well orderings are strongly comparable if the injection is an isomorphism between one ordering and an initial segment of the other. In [5], Friedman announced that the statement “any two countable well orderings are strongly comparable” is equivalent to ATR 0 . Simpson provides a detailed proof of this result in Chapter 5 of [13]. More recently, Friedman has (...)
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  17. Equational Representations.Harvey M. Friedman - unknown
    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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  18. Boolean Relation Theory.Harvey M. Friedman - unknown
    BRT is always based on a choice of BRT setting. A BRT setting is a pair (V,K), where V is an interesting family of multivariate functions. K is an interesting family of sets. In this talk, we will only consider V,K, where V is an interesting family of multivariate functions from N into N. K is an interesting family of subsets of N.
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  19. From Russell's Paradox To.Harvey M. Friedman - unknown
    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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  20. Adjacent Ramsey Theory.Harvey M. Friedman - unknown
    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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  21. Friedman@Math.Ohio-State.Edu.Harvey M. Friedman - unknown
    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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  22. Philosophy 532 Philosophical Problems in Logic Lecture 1 9/25/02.Harvey M. Friedman - unknown
    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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  23. Finite Phase Transitions.Harvey M. Friedman - unknown
    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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  24. What Are These Three Aspects?Harvey M. Friedman - unknown
    Provide a formal system that is a conservative extension of PA for Π02 sentences, and even a conservative extension of HA, that supports the worry free smooth development of constructive analysis in the style of Errett Bishop.
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  25. Decision Procedures for Verification.Harvey M. Friedman - unknown
    We focus on two formal methods contexts which generate investigations into decision problems for finite strings.
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  26. Clay Millenium Problem: P = Np.Harvey M. Friedman - unknown
    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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  27. Strict Reverse Mathematics Draft.Harvey M. Friedman - unknown
    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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  28. Philosophy 536 Philosophy of Mathematics Lecture 1 9/25/02.Harvey M. Friedman - unknown
    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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  29. What is o-Minimality?Harvey M. Friedman - 2008 - 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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  30. Introduction.Harvey M. Friedman - unknown
    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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  31. Primitive Independence Results.Harvey M. Friedman - 2003 - Journal of Mathematical Logic 3 (1):67-83.
    We present some new set and class theoretic independence results from ZFC and NBGC that are particularly simple and close to the primitives of membership and equality. They are shown to be equivalent to familiar small large cardinal hypotheses. We modify these independendent statements in order to give an example of a sentence in set theory with 5 quantifiers which is independent of ZFC. It is known that all 3 quantifier sentences are decided in a weak fragment of ZF without (...)
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  32. Decision Problems in Euclidean Geometry.Harvey M. Friedman - unknown
    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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  33. Shocking(?) Unprovability.Harvey M. Friedman - unknown
    Mathematical Logic had a glorious period in the 1930s, which was briefly rekindled in the 1960s. Any Shock Value, such as it is, has surrounded unprovability from ZFC.
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  34. Strict Reverse Mathematics.Harvey M. Friedman - unknown
    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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  35. Unprovable Theorems.Harvey M. Friedman - unknown
    I don’t remember if I got as high as 2-390, but I distinctly remember taking my first logic course - as a Freshman - with Hartley Rogers, in Fall 1964 - here in 2-190. Or was it in 2-290?
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  36. Agenda.Harvey M. Friedman - unknown
    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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  37. Concrete Incompleteness From Efa Through Large Cardinals.Harvey M. Friedman - unknown
    Normal mathematical culture is overwhelmingly concerned with finite structures, finitely generated structures, discrete structures (countably infinite), continuous and piecewise continuous functions between complete separable metric spaces, with lesser consideration of pointwise limits of sequences of such functions, and Borel measurable functions between complete separable metric spaces.
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  38. Issues in the Foundations of Mathematics.Harvey M. Friedman - unknown
    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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  39. Decision Problems in Strings and Formal Methods.Harvey M. Friedman - unknown
    We focus on two formal methods contexts which generate investigations into decision problems for finite strings.
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  40. P 1 INCOMPLETENESS: Finite Set Equations.Harvey M. Friedman - unknown
    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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  41. Geometry Axioms.Harvey M. Friedman - unknown
    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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  42.  16
    Subtle Cardinals and Linear Orderings.Harvey M. Friedman - 2000 - Annals of Pure and Applied Logic 107 (1-3):1-34.
    The subtle, almost ineffable, and ineffable cardinals were introduced in an unpublished 1971 manuscript of R. Jensen and K. Kunen. The concepts were extended to that of k-subtle, k-almost ineffable, and k-ineffable cardinals in 1975 by J. Baumgartner. In this paper we give a self contained treatment of the basic facts about this level of the large cardinal hierarchy, which were established by J. Baumgartner. In particular, we give a proof that the k-subtle, k-almost ineffable, and k-ineffable cardinals define three (...)
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  43. Foundational Adventures for the Future.Harvey M. Friedman - unknown
    • Wright Brothers made a two mile flight • Wright Brothers made a 42 mile flight • Want to ship goods • Want to move lots of passengers • Want reliability and safety • Want low cost • ... Modern aviation • Each major advance spawns reasonable demands for more and more • Excruciating difficulties overcome • Armies of people over decades or more • Same story for any practically any epoch breaking advance in anything..
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  44. Infinity: New Research Frontiers.Rudy Rucker, Wolfgang Achtner, Enrico Bombieri, Edward Nelson, W. Hugh Woodin & Harvey M. Friedman - 2011 - Cambridge University Press.
    This interdisciplinary study of infinity explores the concept through the prism of mathematics and then offers more expansive investigations in areas beyond mathematical boundaries to reflect the broader, deeper implications of infinity for human intellectual thought. More than a dozen world-renowned researchers in the fields of mathematics, physics, cosmology, philosophy, and theology offer a rich intellectual exchange among various current viewpoints, rather than a static picture of accepted views on infinity.The book starts with a historical examination of the transformation of (...)
     
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  45.  21
    Proofless Text.Harvey M. Friedman - unknown
    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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  46.  8
    Set Existence Property for Intuitionistic Theories with Dependent Choice.Harvey M. Friedman & Andrej Ščedrov - 1983 - Annals of Pure and Applied Logic 25 (2):129-140.
  47.  15
    Addendum to “Countable Algebra and Set Existence Axioms”.Harvey M. Friedman, Stephen G. Simpson & Rick L. Smith - 1984 - Annals of Pure and Applied Logic 28 (3):319-320.
  48.  10
    Reverse Mathematics and Homeomorphic Embeddings.Harvey M. Friedman & Jeffry L. Hirst - 1991 - Annals of Pure and Applied Logic 54 (3):229-253.
    Extrapolating from the work of Mahlo , one can prove that given any pair of countable closed totally bounded subsets of complete separable metric spaces, one subset can be homeomorphically embedded in the other. This sort of topological comparability is reminiscent of the statements concerning comparability of well orderings which Friedman has shown to be equivalent to ATR0 over the weak base system RCA0. The main result of this paper states that topological comparability is also equivalent to ATR0. In Section (...)
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  49.  7
    Intuitionistically Provable Recursive Well-Orderings.Harvey M. Friedman & Andre Scedrov - 1986 - Annals of Pure and Applied Logic 30 (2):165-171.
    We consider intuitionistic number theory with recursive infinitary rules . Any primitive recursive binary relation for which transfinite induction schema is provable is in fact well founded. Its ordinal is less than ε 0 if the transfinite induction schema is intuitionistically provable in elementary number theory. These results are provable intuitionistically. In fact, it suffices to consider transfinite induction with respect to one particular number-theoretic property.
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  50. The Interpretation of Set Theory in Mathematical Predication Theory.Harvey M. Friedman - unknown
    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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