Editing Axiom schema of replacement (section)

== History ==
The axiom schema of replacement was not part of [[Ernst Zermelo]]'s 1908 axiomatisation of set theory ('''Z'''). Some informal approximation to it existed in [[Georg Cantor|Cantor]]'s unpublished works, and it appeared again informally in [[Mirimanoff]] (1917).<ref>{{citation
 | last = Maddy | first = Penelope | author-link = Penelope Maddy
 | doi = 10.2307/2274520
 | jstor = 2274520
 | issue = 2
 | journal = [[Journal of Symbolic Logic]]
 | mr = 947855
 | pages = 481–511
 | quote =  Early hints of the Axiom of Replacement can be found in Cantor's letter to Dedekind [1899] and in Mirimanoff [1917]
 | title = Believing the axioms. I
 | volume = 53
 | year = 1988}}. Maddy cites two papers by Mirimanoff, "Les antinomies de Russell et de Burali-Forti et le problème fundamental de la théorie des ensembles" and "Remarques sur la théorie des ensembles et les antinomies Cantorienne", both in ''L'Enseignement Mathématique'' (1917).</ref>

[[File:Adolf Abraham Halevi Fraenkel.jpg|thumb|upright|alt=refer to caption|Abraham Fraenkel, between 1939 and 1949]] [[File:ThoralfSkolem-OB.F06426c.jpg|thumb|upright|alt=refer to caption|Thoralf Skolem, in the 1930s]]
Its publication by [[Abraham Fraenkel]] in 1922 is what makes modern set theory Zermelo-''Fraenkel'' set theory ('''ZFC'''). The axiom was independently discovered and announced by [[Thoralf Skolem]] later in the same year (and published in 1923). Zermelo himself incorporated Fraenkel's axiom in his revised system he published in 1930, which also included as a new axiom von Neumann's [[axiom of foundation]].<ref name="Ebbinghaus92">Ebbinghaus, p. 92.</ref> Although it is Skolem's first order version of the axiom list that we use today,<ref name="Ebbinghaus2"/> he usually gets no credit since each individual axiom was developed earlier by either Zermelo or Fraenkel. The phrase “Zermelo-Fraenkel set theory” was first used in print by von Neumann in 1928.<ref name="Ebbinghaus189">Ebbinghaus, p. 189.</ref>

Zermelo and Fraenkel had corresponded heavily in 1921; the axiom of replacement was a major topic of this exchange.<ref name="Ebbinghaus2">Ebbinghaus, pp. 135-138.</ref> Fraenkel initiated correspondence with Zermelo sometime in March 1921. However, his letters before the one dated 6 May 1921 are lost. Zermelo first admitted to a gap in his system in a reply to Fraenkel dated 9 May 1921. On 10 July 1921, Fraenkel completed and submitted for publication a paper (published in 1922) that described his axiom as allowing arbitrary replacements: "If ''M'' is a set and each element of ''M'' is replaced by [a set or an urelement] then ''M'' turns into a set again" (parenthetical completion and translation by Ebbinghaus). Fraenkel's 1922 publication thanked Zermelo for helpful arguments. Prior to this publication, Fraenkel publicly announced his new axiom at a meeting of the [[German Mathematical Society]] held in [[Jena]] on 22 September 1921. Zermelo was present at this meeting; in the discussion following Fraenkel's talk he accepted the axiom of replacement in general terms, but expressed reservations regarding its extent.<ref name="Ebbinghaus2"/>

Thoralf Skolem made public his discovery of the gap in Zermelo's system (the same gap that Fraenkel had found) in a talk he gave on 6 July 1922 at the 5th [[Congress of Scandinavian Mathematicians]], which was held in [[Helsinki]]; the proceedings of this congress were published in 1923. Skolem presented a resolution in terms of first-order definable replacements: "Let ''U'' be a definite proposition that holds for certain pairs (''a'', ''b'') in the domain ''B''; assume further, that for every ''a'' there exists at most one ''b'' such that ''U'' is true. Then, as ''a'' ranges over the elements of a set ''M<sub>a</sub>'', ''b'' ranges over all elements of a set ''M<sub>b</sub>''." In the same year, Fraenkel wrote a review of Skolem's paper, in which Fraenkel simply stated that Skolem's considerations correspond to his own.<ref name="Ebbinghaus2"/>

Zermelo himself never accepted Skolem's formulation of the axiom schema of replacement.<ref name="Ebbinghaus2"/> At one point he called Skolem's approach “set theory of the impoverished”. Zermelo envisaged a system that would allow for [[large cardinal]]s.<ref name="Ebbinghaus3">Ebbinghaus, p. 184.</ref> He also objected strongly to the philosophical implications of [[Skolem's paradox#Reception by the mathematical community|countable models of set theory]], which followed from Skolem's first-order axiomatization.<ref name="Ebbinghaus189"/> According to the biography of Zermelo by [[Heinz-Dieter Ebbinghaus]], Zermelo's disapproval of Skolem's approach marked the end of Zermelo's influence on the developments of set theory and logic.<ref name="Ebbinghaus2"/>