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19
Tarski's System of Geometry
 Bulletin of Symbolic Logic
, 1999
"... . This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabh auser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the ..."
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. This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabh auser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions of independence of axioms and primitive notions, and versions of the system suitable for the development of 1dimensional geometry. In his 192627 lectures at the University of Warsaw, Alfred Tarski gave an axiomatic development of elementary Euclidean geometry, the part of plane Euclidean geometry that is not based upon settheoretical notions, or, in other words, the part that can be developed within the framework of firstorder logic. He proved, around 1930, that his system of geometry admits elimination of quantifiers: every formula is provably equivalent (on the basis of the axioms) to a Boolean combination of basic formulas. From this theorem he...
Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show t ..."
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Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1
Step By Recursive Step: Church's Analysis Of Effective Calculability
 BULLETIN OF SYMBOLIC LOGIC
, 1997
"... Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, wa ..."
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Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of G odel's general recursiveness, not his own #definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of e#ective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the char...
Frege versus Cantor and Dedekind: On the Concept of Number
, 1997
"... This paper is in honor of my colleague and friend, Leonard Linsky, on the occasion of his retirement. I presented the earliest version in the Spring of 1992 to a reading group, the other members of which were Leonard Linsky, Steve Awodey, Andre Carus and Mike Price. I presented later versions in the ..."
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This paper is in honor of my colleague and friend, Leonard Linsky, on the occasion of his retirement. I presented the earliest version in the Spring of 1992 to a reading group, the other members of which were Leonard Linsky, Steve Awodey, Andre Carus and Mike Price. I presented later versions in the autumn of 1992 to the philosophy colloquium at McGill University and in the autumn of 1993 to the philosophy colloquium at CarnegieMellon University. The discussions following these presentations were valuable to me and I would especially like to acknowledge Emily Carson (for comments on the earliest draft), Michael Hallett, Kenneth Manders, Stephen Menn, G.E. Reyes, Teddy Seidenfeld, and Wilfrid Sieg and the members of the reading group for helpful comments. But, most of all, I would like to thank Howard Stein and Richard Heck, who read the penultimate draft of the paper and made extensive comments and corrections. Naturally, none of these scholars, except possibly Howard Stein, is respon
The Dedekind reals in abstract Stone duality
 Mathematical Structures in Computer Science
, 2008
"... Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos. ASD reconciles mathematical and computational viewpoints, providing an inherentl ..."
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Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos. ASD reconciles mathematical and computational viewpoints, providing an inherently computable calculus that does not sacrifice key properties of real analysis such as compactness of the closed interval. Previous theories of recursive analysis failed to do this because they were based on points; ASD succeeds because, like locale theory and formal topology, it is founded on the algebra of open subspaces. ASD is presented as a lambdacalculus, of which we provide a selfcontained summary, as the foundational background has been investigated in earlier work. The core of the paper constructs the real line using twosided Dedekind cuts. We show that the closed interval is compact and overt, where these concepts are defined using quantifiers. Further topics, such as the Intermediate Value Theorem, are presented in a separate paper that builds on this one. The interval domain plays an important foundational role. However, we see intervals as generalised Dedekind cuts, which underly the construction of the real line, not as sets or pairs of real numbers. We make a thorough study of arithmetic, in which our operations are more complicated than Moore’s, because we work constructively, and we also consider backtofront (Kaucher) intervals. Finally, we compare ASD with other systems of constructive and computable topology and analysis.
Reverse mathematics and Peano categoricity
"... We investigate the reversemathematical status of several theorems to the effect that the natural number system is secondorder categorical. One of our results is as follows. Define a system to be a triple A,i,f such that A is a set and i ∈ A and f: A → A. A subset X ⊆ A is said to be inductive if i ..."
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We investigate the reversemathematical status of several theorems to the effect that the natural number system is secondorder categorical. One of our results is as follows. Define a system to be a triple A,i,f such that A is a set and i ∈ A and f: A → A. A subset X ⊆ A is said to be inductive if i ∈ X and ∀a(a ∈ X ⇒ f(a) ∈ X). The system A,i,f is said to be inductive if the only inductive subset of A is A itself. Define a Peano system to be an inductive system such that f is onetoone and i /∈ the range of f. The standard example of a Peano system is N,0,S where N = {0,1,2,...,n,...} = the set of natural numbers and S: N → N is given by S(n) = n+1 for all n ∈ N. Consider the statement that all Peano systems are isomorphic toN,0,S. We prove that this statement is logically equivalent to WKL0 over RCA ∗ 0. From this and similar equivalences we
Gödel on Intuition and on Hilbert’s finitism
"... There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the con ..."
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There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, Gödel’s writings represent a smooth evolution, with just one rather small doublereversal, of his view of finitism. He used the term “finit ” (in German) or “finitary ” or “finitistic ” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [Gödel, 1938a] and the lecture notes for a lecture at Yale [Gödel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of firstorder number theory, P A; but starting in the Dialectica paper
Efficient Computation of Truncated Power Series: Direct Approach versus Newton’s Method
"... Newton’s method is used to approximate the zeros of a real function f�x�. In the generic case, Newton’s method is quadratically convergent, i. e. in each step the number of correct decimal digits roughly doubles. It is also wellknown that Newton’s method can be similarly used to compute the Taylor ..."
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Newton’s method is used to approximate the zeros of a real function f�x�. In the generic case, Newton’s method is quadratically convergent, i. e. in each step the number of correct decimal digits roughly doubles. It is also wellknown that Newton’s method can be similarly used to compute the Taylor coefficients of a function x�t�, given implicitly by f�t, x�t��⩵0, in an iterative way such that in each iteration step the number of correct coefficients doubles. In this paper, we present implementations of higher order iteration schemes generalizing Newton’s method which enable us to triple, quadruple, quintuple etc. the number of correct Taylor coefficients of an implicitly given function in each iteration step. These algorithms are of theoretical interest, but in practice they turn out to be not as efficient as the “usual ” Newton method since the expression swell generated by the complexity of the formulas is higher than the advantage by the higher order of the method. We give examples in Mathematica and in Maple showing that in certain cases Newton’s (implicit) method is faster than the direct (explicit) computation. Keywords: Implicit functions; computation of Taylor polynomials; NewtonRaphson method; quadratically convergent iteration; cubic and quartic iteration; generating functions; Catalan numbers; Lambert’s W function. 1 1 Newton’s Method A general assumption of this article is that the functions occurring are often enough differentiable or else they are members of a suitable differential field so that the statements make sense. In this section, the function f ∶�� � should be differentiable. Then to find a zeroΞ of f, one can approximate the function by its linear approximation f�x� � f�x 0 � � f ′ �x
Peano systems
, 2011
"... A set is a collection of objects. The objects in the collection are called the members of the set, or the elements of the set. The notation x ∈ A means that A is a set and x is one of the members of the set A. To describe sets, we sometimes use “setbuilder ” notation such as A = {...}. This means t ..."
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A set is a collection of objects. The objects in the collection are called the members of the set, or the elements of the set. The notation x ∈ A means that A is a set and x is one of the members of the set A. To describe sets, we sometimes use “setbuilder ” notation such as A = {...}. This means that A is a set and the members of A consist of all objects which have the property.... Here are some examples. 1. A = {x,y,z} means: A is the set consisting of x, y, and z. 2. E = {2,4,6,...} means: E is the set of even numbers. 3. [1,2) = {x  1 ≤ x < 2} means: [1,2) is the set consisting of all real numbers x such that x ≥ 1 and x < 2. Let A and B be sets. We say that A is a subset of B if every member of A is a member of B. Note that B itself is a subset of B. If A is a subset of B other than B itself, we say that A is a proper subset of B. For example, E and [1,2) are proper subsets of [1,∞), while [1,∞) is an “improper ” subset of [1,∞). We write A ⊆ B to mean that A is a subset of B. We write A ⊂ B to mean that A is a proper subset of B. This kind of notation is analogous to the standard algebra notation x ≤ y (x is less than or equal to y) and x < y (x is less than y). The extensionality principle is a basic principle concerning sets. Namely, if A ⊆ B and B ⊆ A, then A = B. In other words, if A and B are sets and every member of A is a member of B and vice versa, then A and B are the same set. 1 For example {1,2,3} = {3,2,1}. Thus, a set does not depend on the order in which the elements of the set are listed. 2