182 research outputs found
Fragments of Frege's Grundgesetze and G\"odel's Constructible Universe
Frege's Grundgesetze was one of the 19th century forerunners to contemporary
set theory which was plagued by the Russell paradox. In recent years, it has
been shown that subsystems of the Grundgesetze formed by restricting the
comprehension schema are consistent. One aim of this paper is to ascertain how
much set theory can be developed within these consistent fragments of the
Grundgesetze, and our main theorem shows that there is a model of a fragment of
the Grundgesetze which defines a model of all the axioms of Zermelo-Fraenkel
set theory with the exception of the power set axiom. The proof of this result
appeals to G\"odel's constructible universe of sets, which G\"odel famously
used to show the relative consistency of the continuum hypothesis. More
specifically, our proofs appeal to Kripke and Platek's idea of the projectum
within the constructible universe as well as to a weak version of
uniformization (which does not involve knowledge of Jensen's fine structure
theory). The axioms of the Grundgesetze are examples of abstraction principles,
and the other primary aim of this paper is to articulate a sufficient condition
for the consistency of abstraction principles with limited amounts of
comprehension. As an application, we resolve an analogue of the joint
consistency problem in the predicative setting.Comment: Forthcoming in The Journal of Symbolic Logi
The Julius Caesar objection
This paper argues that that Caesar problem had a technical aspect, namely, that it threatened to make it impossible to prove, in the way Frege wanted, that there are infinitely many numbers. It then offers a solution to the problem, one that shows Frege did not really need the claim that "numbers are objects", not if that claim is intended in a form that forces the Caesar problem upon us
Predicativity, the Russell-Myhill Paradox, and Church's Intensional Logic
This paper sets out a predicative response to the Russell-Myhill paradox of
propositions within the framework of Church's intensional logic. A predicative
response places restrictions on the full comprehension schema, which asserts
that every formula determines a higher-order entity. In addition to motivating
the restriction on the comprehension schema from intuitions about the stability
of reference, this paper contains a consistency proof for the predicative
response to the Russell-Myhill paradox. The models used to establish this
consistency also model other axioms of Church's intensional logic that have
been criticized by Parsons and Klement: this, it turns out, is due to resources
which also permit an interpretation of a fragment of Gallin's intensional
logic. Finally, the relation between the predicative response to the
Russell-Myhill paradox of propositions and the Russell paradox of sets is
discussed, and it is shown that the predicative conception of set induced by
this predicative intensional logic allows one to respond to the Wehmeier
problem of many non-extensions.Comment: Forthcoming in The Journal of Philosophical Logi
The Strength of Abstraction with Predicative Comprehension
Frege's theorem says that second-order Peano arithmetic is interpretable in
Hume's Principle and full impredicative comprehension. Hume's Principle is one
example of an abstraction principle, while another paradigmatic example is
Basic Law V from Frege's Grundgesetze. In this paper we study the strength of
abstraction principles in the presence of predicative restrictions on the
comprehension schema, and in particular we study a predicative Fregean theory
which contains all the abstraction principles whose underlying equivalence
relations can be proven to be equivalence relations in a weak background
second-order logic. We show that this predicative Fregean theory interprets
second-order Peano arithmetic.Comment: Forthcoming in Bulletin of Symbolic Logic. Slight change in title
from previous version, at request of referee
Introduction to Abstractionism
First paragraph: Abstractionism in philosophy of mathematics has its origins in Gottlob Frege’s logicism—a position Frege developed in the late nineteenth and early twentieth century. Frege’s main aim was to reduce arithmetic and analysis to logic in order to provide a secure foundation for mathematical knowledge. As is well known, Frege’s development of logicism failed. The infamous Basic Law V— one of the six basic laws of logic Frege proposed in his magnum opus Grundgesetze der Arithmetik—is subject to Russell’s Paradox. The striking feature of Frege’s Basic Law V is that it takes the form of an abstraction principle
The Finite and the Infinite in Frege's Grundgesetze der Arithmetik
Discusses Frege's formal definitions and characterizations of infinite and finite sets. Speculates that Frege might have discovered the "oddity" in Dedekind's famous proof that all infinite sets are Dedekind infinite and, in doing so, stumbled across an axiom of countable choice
The Convenience of the Typesetter; Notation and Typography in Frege’s Grundgesetze der Arithmetik
We discuss the typography of the notation used by Gottlob Frege in his Grundgesetze der Arithmetik
Minimal Logicism
PLV (Plural Basic Law V) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a first-order formulation of Frege's infamous Basic Law V. George Boolos' plural semantics is replaced with Enrico Martino's Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. ACS provides a form of logicism which is radically alternative to Frege's and which is grounded on the existence of individuals rather than on the existence of concepts.PLV (Plural Basic Law V) est un système de second ordre cohérent qui vise à dériver l'arithmétique de Peano du second ordre. Il emploie la notion de quantification plurielle et une formulation du premier ordre de la tristement célèbre Loi Fondamentale V de Frege. La sémantique plurielle de George Boolos est remplacée par la Acts of Choice Semantics (ACS) de Enrico Martino, qui est développée à partir de la notion de référence arbitraire en raisonnement mathématique. ACS fournit une forme de logicisme qui est radicalement alternative à celle de Frege et qui est fondée sur l'existence des individus plutôt que sur l'existence des concepts
- …