Results 1  10
of
13
Weyl’s predicative classical mathematics as a logicenriched type theory
, 2006
"... Abstract. In Das Kontinuum, Weyl showed how a large body of classical mathematics could be developed on a purely predicative foundation. We present a logicenriched type theory that corresponds to Weyl’s foundational system. A large part of the mathematics in Weyl’s book — including Weyl’s definitio ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
Abstract. In Das Kontinuum, Weyl showed how a large body of classical mathematics could be developed on a purely predicative foundation. We present a logicenriched type theory that corresponds to Weyl’s foundational system. A large part of the mathematics in Weyl’s book — including Weyl’s definition of the cardinality of a set and several results from real analysis — has been formalised, using the proof assistant Plastic that implements a logical framework. This case study shows how type theory can be used to represent a nonconstructive foundation for mathematics. Key words: logicenriched type theory, predicativism, formalisation 1
Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
Complexity and “Closure to Efficient Cause”
"... This paper has two main purposes. First, it will provide an introductory discussion of hyperset theory, and show that it is useful for modeling complex systems. Second, it will use hyperset theory to analyze Robert Rosen’s metabolismrepair systems and his claim that living things are closed to effic ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
This paper has two main purposes. First, it will provide an introductory discussion of hyperset theory, and show that it is useful for modeling complex systems. Second, it will use hyperset theory to analyze Robert Rosen’s metabolismrepair systems and his claim that living things are closed to efficient cause. It will also briefly compare closure to efficient cause to two other understandings of autonomy, operational closure and catalytic closure.
A Predicative and Decidable Characterization of the Polynomial Classes of Languages
"... The definition of a class C of functions is predicative if it doesn't use a class containing C; and is decidable if membership to C can be decided syntactically, from the construction of its elements. Decidable and predicative characterizations of Polytime functions are known. We present here such ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
The definition of a class C of functions is predicative if it doesn't use a class containing C; and is decidable if membership to C can be decided syntactically, from the construction of its elements. Decidable and predicative characterizations of Polytime functions are known. We present here such a characterization for the following classes of languages : P, \Sigma p n ; \Delta p n , PH and PSPACE. It is obtained by means of a progressive sequence of restrictions to recursion in a dialect of Lisp. 1 Introduction The impredicativity [ 12 ] of defining a complexity class C by means of TM's plus clocks or meters has been pointedout by Leivant [ 8 ]. Moreover, studying C in terms of operators, instead of resources, may help understanding its inner nature. To this purpose, several predicative definitions by closure under different kinds of limited recursion have been suggested. Somehow in the spirit of the Grzegorczyk classes, these partial operators are restrictions R of other...
Syntactic Characterization In Lisp Of The Polynomial Complexity Classes And Hierarchy
, 1997
"... . The definition of a class C of functions is syntactic if membership to C can be decided from the construction of its elements. Syntactic characterizations of PTIMEF, of PSPACEF, of the polynomial hierarchy PH, and of its subclasses \Delta p n are presented. They are obtained by progressive restri ..."
Abstract
 Add to MetaCart
. The definition of a class C of functions is syntactic if membership to C can be decided from the construction of its elements. Syntactic characterizations of PTIMEF, of PSPACEF, of the polynomial hierarchy PH, and of its subclasses \Delta p n are presented. They are obtained by progressive restrictions of recursion in Lisp, and may be regarded as predicative according to a foundational point raised by Leivant. 1 Introduction At least since 1965 [6] people think to complexity in terms of TM's plus clock or meter. However, understanding a complexity class may be easier if we define it by means of operators instead of resources. Different forms of limited recursion have been used to this purpose. After the wellknown characterizations of Linspacef [15] and Ptimef [5], further work in this direction has been produced (see, for example, [11], [8], [4]). Both approaches (resources and limited operators) are not syntactic, in the sense that membership to a given class cannot be decided ...
Are Other Logics Possible? Maccoll's Logic And Some English Reactions, 1905 1912
"... this paper I comment on some main features of the book and note reactions to it at the time, especially by Russell ..."
Abstract
 Add to MetaCart
this paper I comment on some main features of the book and note reactions to it at the time, especially by Russell
Chapter 0: The Easy Way to Gödel’s Proof and Related Topics ∗
"... This short sketch of Gödel’s incompleteness proof shows how it arises naturally from Cantor’s diagonalization method [1891]. It renders Gödel’s proof and its relation to the semantic paradoxes transparent. Some historical details, which are often ignored, are pointed out. We also make some observati ..."
Abstract
 Add to MetaCart
This short sketch of Gödel’s incompleteness proof shows how it arises naturally from Cantor’s diagonalization method [1891]. It renders Gödel’s proof and its relation to the semantic paradoxes transparent. Some historical details, which are often ignored, are pointed out. We also make some observations on circularity and draw brief comparisons with natural language. The sketch does not include the messy details of the arithmetization of the language, but the motives for it are made obvious. We suggest this as a more efficient way to teach the topic than what is found in the standard textbooks. For the sake of self–containment Cantor’s original diagonalization is included. A broader and more technical perspective on diagonalization is given in [Gaifman 2005]. In [1891] Cantor presented a new type of argument that shows that the set of all binary sequences (sequences of the form a0, a1,…,an,…, where each ai is either 0 or 1) is not denumerable ─ that is, cannot be arranged in a sequence, where the index ranges over the natural numbers. Let A0, A2,…An, … be a sequence of binary sequences. Say An = an,0, an,1, …, an,i, …. Define a new sequence A * = b0, b1,…,bn, … , by putting: bn = 1, if an,n = 0, bn = 0, if an,n = 1 Then, for each n, A * ≠ An, since the n th member of A * differs from the the n th member of An. Hence, A * does not appear among the Ai’s. A diagram of the following form, which appears already in Cantor’s original paper, illustrate the idea. The new sequence A * is obtained from the diagonal, by changing each of its values. The method came to be known as diagonalization. A0 = a0,0 a0,1... a0,n... A1 = a1,0 a1,1... a1,n...
Autonomy and hypersets
, 2007
"... This article was published in an Elsevier journal. The attached copy is furnished to the author for noncommercial research and education use, including for instruction at the author’s institution, sharing with colleagues and providing to institution administration. Other uses, including reproductio ..."
Abstract
 Add to MetaCart
This article was published in an Elsevier journal. The attached copy is furnished to the author for noncommercial research and education use, including for instruction at the author’s institution, sharing with colleagues and providing to institution administration. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit:
In Defense of the Ideal 2nd DRAFT
"... This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size of α. ..."
Abstract
 Add to MetaCart
This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size of α. The question of the grounds for doing so is perhaps the central question of the workshop. But I want to discuss another question which, as I said, arises at the periphery: How do we know that there are structures 〈Rα, ∈ ∩R2 α〉? How do we know that there exist such things as sets and how do we know that, given such things, the axioms we write down are true of them? These seem very primitive questions, but the skepticism implicit in them has deep (and ancient) roots. In particular, they are questions about ideal objects in general, and not just about the actual infinite. I want to explain why I think the questions (as intended) are empty and the skepticism unfounded. 1 I will be expanding the argument of the first part of my paper “Proof and truth: the Platonism of mathematics”[1986a]. 2 The argument in question