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:
“The greatest horrors in the history of mankind are not due to the ambition of the Napoleons or the vengeance of the Agamemnons, but to the doctrinaire philosophers. The theories of the sentimentalist Rousseau inspired the integrity of the passionless Robespierre. The cold-blooded calculations of Karl Marx led to the judicial and business-like operations of the Cheka.”
—Aleister Crowley (1875–1947)
“There is one great fact, characteristic of this our nineteenth century, a fact which no party dares deny. On the one hand, there have started into life industrial and scientific forces which no epoch of former human history had ever suspected. On the other hand, there exist symptoms of decay, far surpassing the horrors recorded of the latter times of the Roman empire. In our days everything seems pregnant with its contrary.”
—Karl Marx (1818–1883)
“The best history is but like the art of Rembrandt; it casts a vivid light on certain selected causes, on those which were best and greatest; it leaves all the rest in shadow and unseen.”
—Walter Bagehot (1826–1877)