page_title: Axiom schema of replacement
section_title: History
hierarchical_section_title: Axiom schema of replacement / History
caption_attribution_description: English: Abraham Halevi (Adolf) Fraenkel, German-born Israeli mathematician, first Dean of Mathematics as the Hebrew University of Jerusalem and Rector of the University; 1956, recipient of the Israel Prize for exact sciences (1956); member of the Israel Academy of Sciences and Humanities; Zionist, member of the Jewish National Council and the Jewish Assembly of Representatives under the British mandate, belonged to the "Mizrachi" עברית: אברהם (אדולף) הלוי פרנקל (1891–1965), מתמטיקאי גרמני-ישראלי, ממניחי היסודות לתורת הקבוצות האקסיומטית; הדיקן הראשון של הפקולטה למתמטיקה ומדעי הטבע באוניברסיטה העברית בירושלים ורקטור האוניברסיטה; חתן פרס ישראל בתחום המדעים המדויקים (1956) וחבר האקדמיה הלאומית הישראלית למדעים; חבר הוועד הלאומי ואספת הנבחרים ופעיל בתנועת המזרחי.
context_page_description: In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF.
  The axiom schema is motivated by the idea that whether a class is a set depends only on the cardinality of the class, not on the rank of its elements. Thus, if one class is "small enough" to be a set, and there is a surjection from that class to a second class, the axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable surjections, which are identified with their defining formulas.
context_section_description: 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 Some informal approximation to it existed in Cantor's unpublished works, and it appeared again informally in Mirimanoff (1917).
   
  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. Although it is Skolem's first order version of the axiom list that we use today, 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.
  Zermelo and Fraenkel had corresponded heavily in 1921; the axiom of replacement was a major topic of this exchange. Fraenkel initiated correspondence with Zermelo sometime in March 1921. His letters before the one dated 6 May 1921 are lost though. 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.
  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 Mₐ, b ranges over all elements of a set Mb." 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.
  Zermelo himself never accepted Skolem's formulation of the axiom schema of replacement. At one point he called Skolem's approach “set theory of the impoverished”. Zermelo envisaged a system that would allow for large cardinals. He also objected strongly to the philosophical implications of countable models of set theory, which followed from Skolem's first-order axiomatization. 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.
caption: Abraham Fraenkel, between 1939 and 1949
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