scholarly journals DUMMETT ON INDEFINITE EXTENSIBILITY

2018 ◽  
Vol 28 (1) ◽  
pp. 196-220 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Indefinite Extensibility

scholarly journals “How did the serpent of inconsistency enter Frege’s paradise?”

2019 ◽  
pp. 411-436
Author(s):  
Crispin Wright
Keyword(s):  
Cardinal Number ◽  
Ordinal Number ◽  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Indefinite Extensibility ◽  
Basic Law ◽  
Per Se

The paper explores the alleged connection between indefinite extensibility and the classic paradoxes of Russell, Burali-Forti, and Cantor. It is argued that while indefinite extensibility is not per se a source of paradox, there is a degenerate subspecies—reflexive indefinite extensibility—which is. The result is a threefold distinction in the roles played by indefinite extensibility in generating paradoxes for the notions of ordinal number, cardinal number, and set respectively. Ordinal number, intuitively understood, is a reflexively indefinitely extensible concept. Cardinal number is not. And Set becomes so only in the setting of impredicative higher-order logic—so that Frege’s Basic Law V is guilty at worst of partnership in crime, rather than the sole offender.

Art, Open-Endedness, and Indefinite Extensibility

2013 ◽  
pp. 86-107
Author(s):  
Roy T. Cook
Keyword(s):  
Indefinite Extensibility

The Metasemantics of Indefinite Extensibility

2020 ◽  
pp. 1-18
Author(s):  
Vera Flocke
Keyword(s):  
Indefinite Extensibility

XIII-Categoricity and Indefinite Extensibility

2002 ◽  
Vol 102 (3) ◽  
pp. 217-235 ◽  
Author(s):  
James Walmsley
Keyword(s):  
Indefinite Extensibility

Incompleteness, Mechanism, and Optimism

1998 ◽  
Vol 4 (3) ◽  
pp. 273-302 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Mathematical Logic ◽  
Turing Machine ◽  
Digital Computer ◽  
Human Mind ◽  
Indefinite Extensibility ◽  
Incompleteness Theorems ◽  
The Devil ◽  
Gödel’S Incompleteness Theorems ◽  
Gödel’S Theorem ◽  
Different Parts

§1. Overview. Philosophers and mathematicians have drawn lots of conclusions from Gödel's incompleteness theorems, and related results from mathematical logic. Languages, minds, and machines figure prominently in the discussion. Gödel's theorems surely tell us something about these important matters. But what?A descriptive title for this paper would be “Gödel, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility”. Adding “God and the Devil” would probably be redundant. Despite the breath-taking, whirlwind tour, I have the modest aim of forging connections between different parts of this literature and clearing up some confusions, together with the less modest aim of not introducing any more confusions.I propose to focus on three spheres within the literature on incompleteness. The first, and primary, one concerns arguments that Gödel's theorem refutes the mechanistic thesis that the human mind is, or can be accurately modeled as, a digital computer or a Turing machine. The most famous instance is the much reprinted J. R. Lucas [18]. To summarize, suppose that a mechanist provides plans for a machine, M, and claims that the output of M consists of all and only the arithmetic truths that a human (like Lucas), or the totality of human mathematicians, will ever or can ever know. We assume that the output of M is consistent.

Grim’s arguments against omniscience and indefinite extensibility

2011 ◽  
Vol 72 (2) ◽  
pp. 89-101 ◽  
Author(s):  
Laureano Luna
Keyword(s):  
Indefinite Extensibility

Russell Reductio Redux

2019 ◽  
pp. 178-213
Author(s):  
J. P. Studd
Keyword(s):  
Set Theory ◽  
Ad Hoc ◽  
Indefinite Extensibility ◽  
Absolute Generality

By far and away the strongest argument against there being an absolutely comprehensive domain of quantification comes from the set-theoretic paradoxes. The argument from indefinite extensibility can be rigorously regimented with the help of schematic or modal resources. After dispensing with the charge that the argument relies on an incoherent conception of set, this chapter offers a defence of its premisses. Advocates of the orthodox absolutist means to defend absolute generality have yet to give a non-ad-hoc response to the paradoxes. A heterodox absolutist view, which seeks to give an absolutist-friendly account of indefinite extensibility, leads to severe problems with impure set theory. The chapter closes by considering a revenge problem for hybrid relativists, who take modalized quantifiers to achieve absolute generality.

The Question of Platonism

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Mathematical Beliefs ◽  
Mathematical Objects ◽  
Epistemology Of Mathematics ◽  
Indefinite Extensibility ◽  
Physical Objects

This book defends the existence of abstract mathematical objects. Should this be regarded as a defense of Platonism? Platonism involves an analogy between mathematical and physical objects. Although mathematical objects are counterfactually independent of us, just like paradigmatic physical objects, there are other respects in which mathematical objects are strikingly different from physical objects: by giving rise to the phenomenon of indefinite extensibility and by having a shallow nature. The view here is therefore not a full-blown form of Platonism. However, the shallow nature of mathematical objects has the advantage of enabling an epistemology of mathematics where our mathematical beliefs are appropriately sensitive to the truth of these beliefs.

Sign in / Sign up

Export Citation Format

Share Document