History
The concept of well-foundedness and rank of a set were both introduced by Dmitry Mirimanoff (1917) cf. Lévy (2002, p. 68) and Hallett (1986, §4.4, esp. p. 186, 188). Mirimanoff called a set x "regular" (French: "ordinaire") if every descending chain x ∋ x1 ∋ x2 ∋ ... is finite. Mirimanoff however did not consider his notion of regularity (and well-foundedness) as an axiom to be observed by all sets (Halbeisen 2012, pp. 62–63); in later papers Mirimanoff also explored what are now called non-well-founded sets ("extraordinaire" in Mirimanoff's terminology) (Sangiorgi 2011, pp. 17–19, 26).
According to Adam Rieger, von Neumann (1925) describes non-well-founded sets as "superfluous" (on p. 404 in van Heijenoort 's translation) and in the same publication von Neumann gives an axiom (p. 412 in translation) which excludes some, but not all, non-well-founded sets (Rieger 2011, p. 179). In a subsequent publication, von Neumann (1928) gave the following axiom (rendered in modern notation by A. Rieger):
- .
Read more about this topic: Axiom Of Regularity
Famous quotes containing the word history:
“You treat world history as a mathematician does mathematics, in which nothing but laws and formulas exist, no reality, no good and evil, no time, no yesterday, no tomorrow, nothing but an eternal, shallow, mathematical present.”
—Hermann Hesse (1877–1962)
“The history of reform is always identical; it is the comparison of the idea with the fact. Our modes of living are not agreeable to our imagination. We suspect they are unworthy. We arraign our daily employments.”
—Ralph Waldo Emerson (1803–1882)
“Look through the whole history of countries professing the Romish religion, and you will uniformly find the leaven of this besetting and accursed principle of action—that the end will sanction any means.”
—Samuel Taylor Coleridge (1772–1834)