David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Constructivist Foundations 7 (2):112-115 (2012)
Context: The question of how to understand the epistemology of set theory has been a longstanding problem in the foundations of mathematics since Cantor formulated the theory in the 19th century, and particularly since Bertrand Russell articulated his paradox in the early twentieth century. The theory of types pioneered by Russell and Whitehead was simplified by mathematicians to a single distinction between sets and classes. The question of the meaning of this distinction and its necessity still remains open. Problem: I am concerned with the meaning of the set/class distinction and I wish to show that it arises naturally due to the nature of the sort of distinctions that sets create. Method: The method of the paper is to discuss first the Russell paradox and the arguments of Cantor that preceded it. Then we point out that the Russell set of all sets that are not members of themselves can be replaced by the Russell operator R, which is applied to a set S to form R(S), the set of all sets in S that are not members of themselves. Results: The key point about R(S) is that it is well-defined in terms of S, and R(S) cannot be a member of S. Thus any set, even the simplest one, is incomplete. This provides the solution to the problem that I have posed. It shows that the distinction between sets and classes is natural and necessary. Implications: While we have shown that the distinction between sets and classes is natural and necessary, this can only be the beginning from the point of view of epistemology. It is we who will create further distinctions. And it is up to us to maintain these distinctions, or to allow them to coalesce. Constructivist content: I argue in favor of a constructivist perspective for set theory, mathematics, and the way these structures fit into our natural language and constructed speech and worlds. That is the point of this paper. It is only in the reach for absolutes, ignoring the fact that we are the authors of these structures, that the paradoxes arise
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Kevin C. Klement, Russell's Paradox. Internet Encyclopedia of Philosophy.
G. Landini (2013). Zermelo and Russell's Paradox: Is There a Universal Set? Philosophia Mathematica 21 (2):180-199.
Scott Soames (2008). No Class: Russell on Contextual Definition and the Elimination of Sets. Philosophical Studies 139 (2):213 - 218.
Adam R. Day (2013). Indifferent Sets for Genericity. Journal of Symbolic Logic 78 (1):113-138.
John P. Burgess (1988). Sets and Point-Sets: Five Grades of Set-Theoretic Involvement in Geometry. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1988:456 - 463.
Athanassios Tzouvaras (2003). An Axiomatization of 'Very' Within Systiems of Set Theory. Studia Logica 73 (3):413 - 430.
Christopher Menzel (1986). On the Iterative Explanation of the Paradoxes. Philosophical Studies 49 (1):37 - 61.
A. Weir (1998). Naïve Set Theory is Innocent! Mind 107 (428):763-798.
F. A. Muller (2001). Sets, Classes, and Categories. British Journal for the Philosophy of Science 52 (3):539-573.
Wiebke Petersen (2004). A Mathematical Analysis of Pānini's Śivasūtras. Journal of Logic, Language and Information 13 (4):471-489.
Julián Garrido Garrido (2002). Las Paradojas De La Teoria De Conjuntos. Theoria 17 (1):35-62.
Bradley E. Wilson (1991). Are Species Sets? Biology and Philosophy 6 (4):413-431.
Added to index2012-03-14
Total downloads11 ( #143,899 of 1,101,812 )
Recent downloads (6 months)1 ( #306,516 of 1,101,812 )
How can I increase my downloads?