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.
