153 found
Order:
Disambiguations
Harvey Friedman [101]Harvey M. Friedman [57]
  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 (...)
    Direct download (9 more)  
     
    Export citation  
     
    Bookmark   54 citations  
  2.  56
    An Axiomatic Approach to Self-Referential Truth.Harvey Friedman & Michael Sheard - 1987 - Annals of Pure and Applied Logic 33 (1):1--21.
  3.  24
    Countable Algebra and Set Existence Axioms.Harvey M. Friedman - 1983 - Annals of Pure and Applied Logic 25 (2):141.
  4.  57
    One Hundred and Two Problems in Mathematical Logic.Harvey Friedman - 1975 - Journal of Symbolic Logic 40 (2):113-129.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   51 citations  
  5.  38
    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 (...)
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   40 citations  
  6. 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 (...)
     
    Export citation  
     
    Bookmark   11 citations  
  7.  35
    Countable Models of Set Theories.Harvey Friedman - 1973 - In A. R. D. Mathias & H. Rogers (eds.), Cambridge Summer School in Mathematical Logic. New York: Springer Verlag. pp. 539--573.
    Direct download  
     
    Export citation  
     
    Bookmark   24 citations  
  8.  78
    The Consistency of Classical Set Theory Relative to a Set Theory with Intuitionistic Logic.Harvey Friedman - 1973 - Journal of Symbolic Logic 38 (2):315-319.
  9.  11
    Some Applications of Kleene's Methods for Intuitionistic Systems.Harvey Friedman - 1973 - In A. R. D. Mathias & H. Rogers (eds.), Cambridge Summer School in Mathematical Logic. New York: Springer Verlag. pp. 113--170.
    Direct download  
     
    Export citation  
     
    Bookmark   24 citations  
  10.  28
    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 (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   17 citations  
  11.  17
    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.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   17 citations  
  12. Uniformly Defined Descending Sequences of Degrees.Harvey Friedman - 1976 - Journal of Symbolic Logic 41 (2):363-367.
  13.  61
    Whither Relevant Arithmetic?Harvey Friedman & Robert K. Meyer - 1992 - Journal of Symbolic Logic 57 (3):824-831.
    Based on the relevant logic R, the system R# was proposed as a relevant Peano arithmetic. R# has many nice properties: the most conspicuous theorems of classical Peano arithmetic PA are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that R# is properly weaker than PA, in the sense that there is a strictly positive theorem QRF of PA which is unprovable in R#. (...)
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   12 citations  
  14. 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.
    Translate
     
     
    Export citation  
     
    Bookmark   2 citations  
  15. Interpretations, According to Tarski.Harvey Friedman - unknown
    The notion of interpretation is absolutely fundamental to mathematical logic and the foundations of mathematics. It is also crucial for the foundations and philosophy of science - although here some crucial conditions generally need to be imposed; e.g., “the interpretation leaves the mathematical concepts unchanged”.
     
    Export citation  
     
    Bookmark   3 citations  
  16. 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.
    Translate
     
     
    Export citation  
     
    Bookmark   2 citations  
  17.  14
    The Disjunction and Existence Properties for Axiomatic Systems of Truth.Harvey Friedman & Michael Sheard - 1987 - Annals of Pure and Applied Logic 40 (1):1--10.
    In a language for arithmetic with a predicate T, intended to mean “ x is the Gödel number of a true sentence”, a set S of axioms and rules of inference has the truth disjunction property if whenever S ⊢ T ∨ T, either S ⊢ T or S ⊢ T. Similarly, S has the truth existence property if whenever S ⊢ ∃χ T ), there is some n such that S ⊢ T ). Continuing previous work, we establish whether (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   7 citations  
  18. 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, +, -.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  19. Normal Mathematics Will Need New Axioms.Harvey Friedman - 2000 - Bulletin of Symbolic Logic 6 (4):434-446.
  20. 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).
    Translate
     
     
    Export citation  
     
    Bookmark   2 citations  
  21.  24
    Periodic Points and Subsystems of Second-Order Arithmetic.Harvey Friedman, Stephen G. Simpson & Xiaokang Yu - 1993 - Annals of Pure and Applied Logic 62 (1):51-64.
    We study the formalization within sybsystems of second-order arithmetic of theorems concerning periodic points in dynamical systems on the real line. We show that Sharkovsky's theorem is provable in WKL0. We show that, with an additional assumption, Sharkovsky's theorem is provable in RCA0. We show that the existence for all n of n-fold iterates of continuous mappings of the closed unit interval into itself is equivalent to the disjunction of Σ02 induction and weak König's lemma.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  22. Higher Set Theory.Harvey Friedman - manuscript
    Russell’s way out of his paradox via the impre-dicative 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.
     
    Export citation  
     
    Bookmark   2 citations  
  23.  9
    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.
  24. 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 (...)
    No categories
    Translate
     
     
    Export citation  
     
    Bookmark   1 citation  
  25. 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 (...))
    Translate
     
     
    Export citation  
     
    Bookmark   1 citation  
  26. 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.
    No categories
    Translate
     
     
    Export citation  
     
    Bookmark   1 citation  
  27. 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.
     
    Export citation  
     
    Bookmark   1 citation  
  28. 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.
     
    Export citation  
     
    Bookmark   1 citation  
  29.  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 (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  30. 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.
     
    Export citation  
     
    Bookmark   1 citation  
  31. 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 (...)
     
    Export citation  
     
    Bookmark   1 citation  
  32.  10
    Expansions of o-Minimal Structures by Fast Sequences.Harvey Friedman & Chris Miller - 2005 - Journal of Symbolic Logic 70 (2):410-418.
    Let ℜ be an o-minimal expansion of (ℝ, <+) and (φk)k∈ℕ be a sequence of positive real numbers such that limk→+∞f(φk)/φk+1=0 for every f:ℝ→ ℝ definable in ℜ. (Such sequences always exist under some reasonable extra assumptions on ℜ, in particular, if ℜ is exponentially bounded or if the language is countable.) Then (ℜ, (S)) is d-minimal, where S ranges over all subsets of cartesian powers of the range of φ.
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  33. 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.
    No categories
    Translate
     
     
    Export citation  
     
    Bookmark   1 citation  
  34.  17
    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.
  35. A Language for Mathematical Knowledge Management.Jeremy Avigad, Steven Kieffer & Harvey Friedman - manuscript
    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 (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  36.  9
    Lindenbaum Algebras of Intuitionistic Theories and Free Categories.Peter Freyd, Harvey Friedman & Andre Scedrov - 1987 - Annals of Pure and Applied Logic 35 (2):167-172.
    We consider formal theories synonymous with various free categories . Their Lindenbaum algebras may be described as the lattices of subobjects of a terminator. These theories have intuitionistic logic. We show that the Lindenbaum algebras of second order and higher order arithmetic , and set theory are not isomorphic to the Lindenbaum algebras of first order theories such as arithmetic . We also show that there are only five kernels of representations of the free Heyting algebra on one generator in (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  37.  14
    The Equivalence of the Disjunction and Existence Properties for Modal Arithmetic.Harvey Friedman & Michael Sheard - 1989 - Journal of Symbolic Logic 54 (4):1456-1459.
    In a modal system of arithmetic, a theory S has the modal disjunction property if whenever $S \vdash \square\varphi \vee \square\psi$ , either $S \vdash \square\varphi$ or $S \vdash \square\psi. S$ has the modal numerical existence property if whenever $S \vdash \exists x\square\varphi(x)$ , there is some natural number n such that $S \vdash \square\varphi(\mathbf{n})$ . Under certain broadly applicable assumptions, these two properties are equivalent.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  38.  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.
    Direct download  
    Translate
     
     
    Export citation  
     
    Bookmark   1 citation  
  39.  11
    On Existence Proofs of Hanf Numbers.Harvey Friedman - 1974 - Journal of Symbolic Logic 39 (2):318-324.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  40.  9
    Bar Induction and Π11-CA.Harvey Friedman - 1969 - Journal of Symbolic Logic 34 (3):353 - 362.
  41.  3
    Bar Induction and $Pi^1_1-CA^1$.Harvey Friedman - 1969 - Journal of Symbolic Logic 34 (3):353-362.
  42.  30
    PCA Well-Orderings of the Line.Harvey Friedman - 1974 - Journal of Symbolic Logic 39 (1):79-80.
    There is a PCA well-ordering of the real line if and only if there is a real from which every real is constructible.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  43.  20
    Large Sets in Intuitionistic Set Theory.Harvey Friedman & Andrej Ščedrov - 1984 - Annals of Pure and Applied Logic 27 (1):1-24.
    We consider properties of sets in an intuitionistic setting corresponding to large cardinals in classical set theory. Adding such ‘large set axioms’ to intuitionistic ZF set theory does not violate well-know metamathematical properties of intuitionistic systems. Moreover, we consider statements in constructive analysis equivalent to the consistency of such ‘large set axioms’.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  44.  26
    On the Derivability of Instantiation Properties.Harvey Friedman - 1977 - Journal of Symbolic Logic 42 (4):506-514.
    Every recursively enumerable extension of arithmetic which obeys the disjunction property obeys the numerical existence property [Fr, 1]. The requirement of recursive enumerability is essential. For extensions of intuitionistic second order arithmetic by means of sentences (in its language) with no existential set quantifiers, the numerical existence property implies the set existence property. The restriction on existential set quantifiers is essential. The numerical existence property cannot be eliminated, but in the case of finite extensions of HAS, can be replaced by (...)
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  45.  11
    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 (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  46.  10
    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.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  47.  10
    1995–1996 Annual Meeting of the Association for Symbolic Logic.Tomek Bartoszynski, Harvey Friedman, Geoffrey Hellman, Bakhadyr Khoussainov, Phokion G. Kolaitis, Richard Shore, Charles Steinhorn, Mirna Dzamonja, Itay Neeman & Slawomir Solecki - 1996 - Bulletin of Symbolic Logic 2 (4):448-472.
  48.  4
    Nijmegen, The Netherlands July 27–August 2, 2006.Rodney Downey, Ieke Moerdijk, Boban Velickovic, Samson Abramsky, Marat Arslanov, Harvey Friedman, Martin Goldstern, Ehud Hrushovski, Jochen Koenigsmann & Andy Lewis - 2007 - Bulletin of Symbolic Logic 13 (2).
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  49. 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.
    Translate
     
     
    Export citation  
     
    Bookmark  
  50. A Big Difference Between Interpretability and Definability in an Expansion of the Real Field.Harvey Friedman & Chris Miller - unknown
    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 (...)
    Translate
     
     
    Export citation  
     
    Bookmark  
1 — 50 / 153