CHAPTER III
A CRITIQUE OF MATHEMATICAL REASONING
§ 11. The paradoxes. This chapter is intended to present the problemsituation out of which the investigations to be reported in the rest of thebook arose, i.e. the situation preceding those investigations (but nothow it has since changed).- P36
The Epimenides paradox, known also as the liar, appears in stark form,
if a person says simply, "This statement I am now making is a lie." Thequoted statement can neither be true nor false without entailing a con-tradiction. This version of the paradox is attributed to Eubulides (fourthcentury B.C.), and was well known in ancient times. (Cf. Rüstow 1910.
If the statement "Cretans are always liars ..." is not authenticallyEpimenides‘, or was not originally recognized as paradoxical, the Eubuli-of the Liar may then be older than the "lying Cretan" version.)- P39
§ 12. First inferences from the paradoxes. The reader may tryhis hand at solving the paradoxes. In the half century since the problem hasbeen open, no solution has been found which is universally agreed upon.
The simplest kind of solution would be to locate a specific fallacy, likea mistake in a student‘s algebra exercise or geometry proof, with nothingelse needing to be changed.- P40
AXIOMATIC SET THEORY. Reconstructions of set theory can be given,
placing around the notion of set as few restrictions to exclude too largesets as appear to be required to forestall the known antinomies. Sincethe free use of our conceptions in constructing sets under Cantor‘s def-inition led to disaster, the notions of set theory are governed by axioms,
like those governing ‘point‘ and ‘line‘ in Euclidean plane geometry. Thefirst system of axiomatic set theory was Zermelo‘s (1908). Refinementsin the axiomatic treatment of sets are due to Fraenkel (1922, 1925),
Skolem (1922-3, 1929), von Neumann (1925, 1928), Bernays (1937-48),
and others. Analysis can be founded on the basis of axiomatic set theory, which perhaps is the simplest basis set up since the paradoxes for thededuction of existing mathematics. Some very interesting discoverieshave been made in connection with axiomatic set theory, notably bySkolem (1922-3; cf. § 75 below) and Gödel (1938, 1939, 1940).- P40
But in thr case of arithmetic and analysis, theories culminating inset theory, mathematicians prior to the current epoch of criticism general-ly supposed that they were dealing with systems of objects, set upgenetically, by definitions purporting to establish their structure com-pletely. The theorems were thought of as expressing truths about thesesystems, rather than as propositions applying hypothetically to whateversystems of objects (if any) satisfy the axioms. But then how could con-tradictions have arisen in these subjects, unless there is some defect inthe logic, some error in the methods of constructing and reasoning aboutmathematical objects, which we had hitherto trusted?- P41
IMPREDICATIVE DEFINITION. When a set M and a particular object mare so defined that on the one hand m is a member of M, and on theother hand the definition of m depends on M, we say that the procedure(or the definition of m, or the definition of M) is impredicative. Similarly,
when a property P is possessed by an object m whose definition dependson P (here M is the set of the objects which possess the property P).
An impredicative definition is circular, at least on its face, as what is defined participates in its own definition.- P42
