[FOM] About Paradox Theory

[Irving Anellis]

Charles Silver wrote: ''R's Paradox as having nothing to do with sets
or anything else in particular'' and asks for ''good reason not to
trivialize it so. ''

There is both the function-theoretic version of Russell's paradox, to
which you elude in setting forth the proposition ~EyAx[F(xy) <-->
~F(xx)], and which Russell  formulated in connection with sect. 9 of
the Begriffsschrift (1879) allowing a function to serve as an
indeterminate element of another (higher-order) function, and with
Basic Rule V in the first volume of the Grundgesetze der Arithmetik
(1893), and the set-theoretic version, involving the Russell set (or
class), defined as the class none of whose members are members of
themselves. What the various versions of the Russell paradox have in
common is what came, due to Poincare, to be called the Vicious Circle
Principle, and which Russell generalized in terms of impredicability.
Historian of logic Jean van Heijenoort (1912-1986), in his account of
the paradoxes (''Logical Paradoxes'', in Paul Edwards (ed.),
Encyclopedia of Philosophy (New York: Macmillan, 1963), vol. 5,
45-51), held that the Russell paradox was much more fundamental than
the other set-theoretic paradoxes, including in particular the Cantor
(greatest cardinal) and Burali-Forti (greatest ordinal) because it
involved the very notions of 'set' and 'elementhood'. Something of the
connection between the Russell paradox and the Cantor and Burali-Fort
paradoxes, and of the history of the origin of the Russell paradox, is
given in my ''The First Russell Paradox'', in Thomas Drucker (ed.),
Perspectives on the History of Mathematical Logic
(Boston/Basel/Berlin: Birkhauser, 1991), 33-46. An indication of some
of the manifold results in logic, set theory, and philosophy that
arose as a consequence of or in response to the Russell paradox can be
found in Godehard Link (ed.), One Hundred Years of the Russell
Paradox: Mathematics, Logic, Philosophy (Berlin & New York: de
Gruyter, 2004).

Sticklers in terminological precision may prefer 'Russell antinomy' to
'Russell paradox'.


Irving H. Anellis
Visiting Research Associate
Peirce Edition Project
Indiana University-Purdue University at Indianapolis