Type

Whitehead and Russell (1927) devised a hierarchy of "types" in order to eliminate self-referential statements from Principia Mathematica, which purported to derive all of mathematics from logic. A set of the lowest type contained only objects (not sets), a set of the next higher type could contain only objects or sets of the lower type, and so on. Unfortunately, Gödel's first incompleteness theorem showed that both Principia Mathematica and all consistent formal systems must be incomplete.

REFERENCES

Curry, H. B. Foundations of Mathematical Logic. New York: Dover, pp. 21-22, 1977.

Ferreirós, J. "Russell's Theory of Types." §9.5 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkhäuser, pp. 325-333, 1999.

Gonseth, F. "La Théorie des types." §107 in Les mathématiques et la réalité: Essai sur la méthode axiomatique. Paris: Félix Alcan, pp. 257-259, 1936.

Hofstadter, D. R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 21-22, 1989.

Whitehead, A. N. and Russell, B. Principia Mathematica. New York: Cambridge University Press, 1927.