 New Foundations for Mathematical Logic. Offprint from: The American Mathematical Monthly, Vol. 44.

[Washington: Mathematical Association of America], 1937.

The first attempt to resolve Russell’s and other similar paradoxes was made by Russell himself (1908) in his theory of types. In this theory, every set is assigned a type (a positive whole number); the bottom type is a type of atoms and sets of type n+1 are sets of things of type n. Every variable of the theory is constrained to range over one level only. This means that Russell’s paradox cannot even be formulated within type theory. However, the theory was found to have many drawbacks, as it prevented not only the formulation of the troublesome paradoxes, but also other apparently sensible statements. In addition, it necessarily introduces infinite multiplicities: for example, there has to be one empty set of each possible type, as well as a set of natural numbers of each type, etc.

NF is similar to Russell’s theory in that it involves types, but rather than assign a type to each set once and for all, it assigns a type to each variable in a given formula. If a variable x in a given formula is assigned type n, and if ‘x ε y’ appears in the formula, then y must be assigned type n+1. In addition, if ‘x = y’ appears in the formula, then x and y must have the same type. A formula is ‘stratified’ if there is an assignment of types to variables in the formula that meets these constraints. The axioms of NF are now simply stated: extensionality, together with a scheme that says that the extension of a stratified formula is a stratified formula. NF avoids the paradoxes of naive set theory because the formulas necessary to formulate the paradoxes are not stratified (for example, a set cannot be a member of itself because x ε x is obviously not a stratified formula). But it also avoids the multiplicities and other difficulties inherent in Russell’s type theory.

The approach taken in ZF is to restrict what objects can be called sets, rather than to impose restrictions on how sets are defined, as is done in NF. In ZF the empty set is a set, and any collection of sets is a set (and there are no other sets). ZF appears to resolve Russell’s paradox because the ‘set’ of all sets that are not members of themselves is not in fact a set. Although ZF appears to be able to accommodate the whole of mathematics, many mathematicians believe ZF fails to capture the informal concept of a set: in ZF there is no universal set (‘the set of all sets’), and the universe of sets is not closed under the elementary operations of the algebra of sets. NF is less restrictive: for example, in NF there is a universal set (‘the set of all sets’). On the other hand, it was proved by Specker that the Axiom of Choice is false in NF, which some mathematicians believe restricts its usefulness for mathematics. The debate NF vs. ZF is ongoing.

T. Forster, ‘Quine’s NF – 60 years on,’ The American Mathematical Monthly 104 (1997), pp. 838-845.

8vo (255 x 182 mm, pp. 70-80. Original grey printed wrappers. A fine copy.

Item #2930

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