Inductive Set
المؤلف:
Bourbaki, N.
المصدر:
"Ensembles Inductifs." Ch. 3, §2.4 in Théorie des Ensembles. Paris, France: Hermann, 1970.
الجزء والصفحة:
...
16-1-2022
2521
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 ![y union <span style=]()
{y}" src="https://mathworld.wolfram.com/images/equations/InductiveSet/Inline4.svg" style="height:22px; width:55px" />. In particular, every inductive set contains a sequence of the form
![emptyset,<span style=]() {emptyset},{emptyset,{emptyset}},.... " src="https://mathworld.wolfram.com/images/equations/InductiveSet/NumberedEquation1.svg" style="height:20px; width:134px" /> |
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, ![N union <span style=]()
{infty}" src="https://mathworld.wolfram.com/images/equations/InductiveSet/Inline6.svg" style="height:22px; width:61px" /> 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.
الاكثر قراءة في نظرية المجموعات
اخر الاخبار
اخبار العتبة العباسية المقدسة
