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  1. 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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  2. 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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  3. 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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  4. 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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  5. 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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  6. 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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  7. 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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  8. 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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  9. 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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  10. 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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  11. 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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  12. 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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  13. 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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  14. 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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  15. 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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  16. 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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  17. 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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  18. 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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  19. 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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  20. 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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  21. 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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  22. 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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  23. 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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  24. 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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  25. 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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  26. 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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  27. 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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  28. 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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  29. 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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  30. 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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  31. 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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  32. 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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  33. 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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  34. 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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  35. Harvey M. Friedman, Boolean Relation Theory.
    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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  36. 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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  37. Harvey M. Friedman, Concrete Incompleteness From Efa Through Large Cardinals.
    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. Harvey M. Friedman, Decision Procedures for Verification.
    We focus on two formal methods contexts which generate investigations into decision problems for finite strings.
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  39. Harvey M. Friedman, Decision Problems in Strings and Formal Methods.
    We focus on two formal methods contexts which generate investigations into decision problems for finite strings.
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  40. Harvey M. Friedman, Foundational Adventures for the Future.
    • 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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  41. Harvey M. Friedman, 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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  42. Harvey M. Friedman, Shocking(?) Unprovability.
    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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  43. Harvey M. Friedman, Unprovable Theorems.
    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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  44. Harvey M. Friedman, What Are These Three Aspects?
    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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  45. Harvey M. Friedman (2011). Part III. Technical Perspectives on Infinity From Advanced Mathematics : 4. The Realm of the Infinite / W. Hugh Woodin ; 5. A Potential Subtlety Concerning the Distinction Between Determinism and Nondeterminism / W. Hugh Woodin ; 6. Concept Calculus : Much Better Than. [REVIEW] In Michał Heller & W. H. Woodin (eds.), Infinity: New Research Frontiers. Cambridge University Press.
     
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  46. Harvey M. Friedman (2001). Subtle Cardinals and Linear Orderings. Annals of Pure and Applied Logic 107 (1-3):1-34.
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  47. 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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  48. Harvey M. Friedman & Jeffry L. Hirst (1991). Reverse Mathematics and Homeomorphic Embeddings. Annals of Pure and Applied Logic 54 (3):229-253.
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  49. Harvey M. Friedman & Jeffry L. Hirst (1990). Weak Comparability of Well Orderings and Reverse Mathematics. Annals of Pure and Applied Logic 47 (1):11-29.
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  50. Harvey M. Friedman & Andre Scedrov (1986). Intuitionistically Provable Recursive Well-Orderings. Annals of Pure and Applied Logic 30 (2):165-171.
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