Inductive Set

A set-theoretic term having a number of different meanings. Fraenkel (1953, p. 37) used the term as a synonym for "finite set." However, according to Russell's definition (Russell 1963, pp. 21-22), an inductive set is a nonempty partially ordered set in which every element has a successor. An example is the set of natural numbers , where 0 is the first element, and the others are produced by adding 1 successively.

Roitman (1990, p. 40) considers the same construction in a more abstract form: the elements are sets, 0 is replaced by the empty set , and the successor of every element  is the set . In particular, every inductive set contains a sequence of the form

For many other authors (e.g., Bourbaki 1970, pp. 20-21; Pinter 1971, p. 119), an inductive set is a partially ordered set in which every totally ordered subset has an upper bound, i.e., it is a set fulfilling the assumption of Zorn's lemma.

The versions of Lang (2002, p. 880) and Jacobson (1980, p. 2) contain slight variations; the former prefers the term "inductively ordered" and the latter replaces "upper bound" by "supremum."

Note that  is not an inductive set in this second meaning; however,  is.

REFERENCES

Bourbaki, N. "Ensembles Inductifs." Ch. 3, §2.4 in Théorie des Ensembles. Paris, France: Hermann, 1970.

Fraenkel, A. A. Abstract Set Theory. Amsterdam, Netherlands: North-Holland, 1953.

Jacobson, N. Basic Algebra II. San Francisco, CA: W. H. Freeman, 1980.

Lang, S. Algebra, rev. 3rd ed. New York: Springer-Verlag, 2002.

Pinter, C. C. Set Theory. Reading, MA: Addison-Wesley, 1971.

Roitman, J. Introduction to Modern Set Theory. New York: Wiley, 1990.

Russell, B. Introduction to Mathematical Philosophy, 11th ed. London: George Allen and Unwin, 1963.