The terms z-STATISTIC and z-DISTRIBUTION were introduced by R. A. Fisher in "On a distribution yielding the error functions of several well-known statistics," Proceedings of the International Mathematics Congress, Toronto (1924) [James A. Landau].
ZERMELO-FRAENKEL SET THEORY is found in the title "Ein axiomatisches System der Mengenlehre nach Zermelo und Fraenkel," by Ernst-Jochen Thiele, Z. Math. Logik Grundlagen Math. (1955).
The term is also found in R. Montague, "Zermelo-Fraenkel set theory is not a finite extension of Zermelo set theory," Bull. Amer Math. Soc. 62 (1956).
Attributions to Zermelo occur in A. Fraenkel, "Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre," Math. Annalen (1922) and "Über die Zermelosche Begründung der Mengenlehre," Jahresbericht der Deutschen Mathematiker-Vereinigung 30, 2nd section (1921) [James A. Landau].
ZERO. The Hindus called the symbol sunya, and the term passed over into Arabic as as-sifr or sifr (Smith vol. 2, page 71). Sunya, meaning "empty," was used around A. D. 400 to indicate the empty column on the abacus.
Dionysius Exiguus (died about 545) used the word nulla in his Easter tables. The first epact in each nineteen-year cycle is "nulla" (rather than thirty, as in those of his predecessors). A reference is Migne, Patrologiae Latinae, vol. 67, col. 493 [Christian Marinus Taisbak].
Abraham ben Meir ibn Ezra (1092-1167) used galgal for zero in a description he wrote of a decimal system of numeration.
Leonardo of Pisa (1180-1250) (or Fibonacci) used the word zephirum for this symbol in Liber Abaci: "...quod arabice zephirum appelatur."
According to Smith (vol. 2), some other old names for zero include sipos, tsiphron, tziphra, rota, omicron, circulus, theca, null, zeuero, ceuero, cifra, zepiro and figura nihili.
Cipher is found in English as early as 1399. Other old names for this symbol are aught and naught.
According to Cajori (1919, page 128), the word zero "is found in some fourteenth century manuscripts."
Cajori also states that the first printed treatise containing the word zero is De arithmetrica opusculum, by Filippo Calandri, which was printed in Florence in 1491. Cajori attributes this information to Eneström. [Calandri's name is also spelled Philippus Calender and Philippus Calandrus.]
The earliest citation for zero in English in the OED2 is from Edward Grimstone's 1604 translation of The Natural and Moral History of the Indies by José de Acosta: "They accompted their weekes by thirteene dayes, marking the dayes with a Zero or cipher."
A 1706 dictionary has: "Zero, a Word sometimes us'd especially among the French, for a Cipher or Nought (0)."
In 1882 Complete Graded Arithmetic by James B. Thomson has: "The last one is called Naught, because when standing alone it has no value. It is also called Cipher or Zero."
For the history of symbols for zero (as opposed to words for zero), see the companion math symbols web page, linked from the front page of this website.
ZERO (of a function) is found in 1893 in A Treatise on the Theory of Functions by James Harkness and Frank Morley: "By a zero of P(x) is meant a point at which P(x) vanishes."
ZERO-SUM GAME appears in 1944 in Theory of Games and Economic Behavior by J. von Neumann and O. Morgenstern (David, 1998).
ZEROTH is found in 1893 in An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics with applications to problems in mathematical physics by William Elwood Byerly: "It is a finite sum terminating with the first power of x if m is odd, and with the zeroth power of x if m is even" [University of Michigan Historic Math Collection].
The term ZETAIC MULTIPLICATION was coined by James Joseph Sylvester.
Zeta-ic multiplication was used in 1840 and is found in Sylvester's
Collected Mathematical Papers (1904) I. 47: "I use the Greek letter _files/zeta.gif)
to denote that the product of factors to which it is prefixed is to
be effected after a certain symbolical manner. This I shall distinguish as the
zeta-ic product. ... Rule for zeta-ic multiplication. Note. An analogous
interpretation may be extended to any zeta-ic function whatever" (OED2).
ZORN's LEMMA. Gregory Moore, in his definitive book Zermelo's Axiom of Choice: Its Origins, Development and Influence, says, "By late in 1934, Zorn's principle had found users in the United States who dubbed it Zorn's Lemma." [Bill Dubuque] Julio González Cabillón believes the term may have been coined by John W. Tukey.