Editing Georg Cantor (section)

====Absolute infinite, well-ordering theorem, and paradoxes====
In 1883, Cantor divided the infinite into the transfinite and the [[Absolute infinite|absolute]].<ref>{{harvnb|Cantor|1883|pp=587–588}}; English translation: [[#Ewald|Ewald 1996]], pp. 916&ndash;917.</ref>

The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α&nbsp;+&nbsp;1. On the other hand, the ordinals form an absolutely infinite sequence that cannot be increased in magnitude because there are no larger ordinals to add to it.<ref>[[#Hallett|Hallett 1986]], pp. 41–42.</ref> In 1883, Cantor also introduced the [[Well-ordering theorem|well-ordering principle]] "every set can be well-ordered" and stated that it is a "law of thought".<ref>[[#Moore1982|Moore 1982]], p. 42.</ref>

Cantor extended his work on the absolute infinite by using it in a proof. Around 1895, he began to regard his well-ordering principle as a theorem and attempted to prove it. In 1899, he sent Dedekind a proof of the equivalent aleph theorem: the cardinality of every infinite set is an [[aleph number|aleph]].<ref>[[#Moore1982|Moore 1982]], p. 51. Proof of equivalence: If a set is well-ordered, then its cardinality is an aleph since the alephs are the cardinals of well-ordered sets. If a set's cardinality is an aleph, then it can be well-ordered since there is a one-to-one correspondence between it and the well-ordered set defining the aleph.</ref> First, he defined two types of multiplicities: consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities). Next he assumed that the ordinals form a set, proved that this leads to a contradiction, and concluded that the ordinals form an inconsistent multiplicity. He used this inconsistent multiplicity to prove the aleph theorem.<ref>[[#Hallett|Hallett 1986]], pp. 166–169.</ref> In 1932, Zermelo criticized the construction in Cantor's proof.<ref>Cantor's proof, which is a [[proof by contradiction]], starts by assuming there is a set ''S'' whose cardinality is not an aleph. A function from the ordinals to ''S'' is constructed by successively choosing different elements of ''S'' for each ordinal. If this construction runs out of elements, then the function well-orders the set ''S''. This implies that the cardinality of ''S'' is an aleph, contradicting the assumption about ''S''. Therefore, the function maps all the ordinals one-to-one into ''S''. The function's [[Image (mathematics)|image]] is an inconsistent submultiplicity contained in ''S'', so the set ''S'' is an inconsistent multiplicity, which is a contradiction. Zermelo criticized Cantor's construction: "the intuition of time is applied here to a process that goes beyond all intuition, and a fictitious entity is posited of which it is assumed that it could make ''successive'' arbitrary choices." ([[#Hallett|Hallett 1986]], pp. 169–170.)</ref>

Cantor avoided [[paradox]]es by recognizing that there are two types of multiplicities. In his set theory, when it is assumed that the ordinals form a set, the resulting contradiction implies only that the ordinals form an inconsistent multiplicity. In contrast, [[Bertrand Russell]] treated all collections as sets, which leads to paradoxes. In Russell's set theory, the ordinals form a set, so the resulting contradiction implies that the theory is [[inconsistent]]. From 1901 to 1903, Russell discovered three paradoxes implying that his set theory is inconsistent: the [[Burali-Forti paradox]] (which was just mentioned), [[Cantor's paradox]], and [[Russell's paradox]].<ref>[[#Moore1988|Moore 1988]], pp. 52–53; [[#Moore1981|Moore and Garciadiego 1981]], pp. 330–331.</ref> Russell named paradoxes after [[Cesare Burali-Forti]] and Cantor even though neither of them believed that they had found paradoxes.<ref>[[#Moore1981|Moore and Garciadiego 1981]], pp. 331, 343; [[#Purkert1989|Purkert 1989]], p. 56.</ref>

In 1908, Zermelo published [[Zermelo set theory|his axiom system for set theory]]. He had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of the [[well-ordering theorem]].<ref>[[#Moore1982|Moore 1982]], pp. 158–160. Moore argues that the latter was his primary motivation.</ref> Zermelo had proved this theorem in 1904 using the [[axiom of choice]], but his proof was criticized for a variety of reasons.<ref>Moore devotes a chapter to this criticism: "Zermelo and His Critics (1904–1908)", [[#Moore1982|Moore 1982]], pp. 85–141.</ref> His response to the criticism included his axiom system and a new proof of the well-ordering theorem. His axioms support this new proof, and they eliminate the paradoxes by restricting the formation of sets.<ref>[[#Moore1982|Moore 1982]], pp. 158–160. [[#Zermelo1908|Zermelo 1908]], pp. 263–264; English translation: [[#Heijenoort|van Heijenoort 1967]], p. 202.</ref>

In 1923, [[John von Neumann]] developed an axiom system that eliminates the paradoxes by using an approach similar to Cantor's—namely, by identifying collections that are not sets and treating them differently. Von Neumann stated that a [[Class (set theory)|class]] is too big to be a set if it can be put into one-to-one correspondence with the class of all sets. He defined a set as a class that is a member of some class and stated the axiom: A class is not a set if and only if there is a one-to-one correspondence between it and the class of all sets. This axiom implies that these big classes are not sets, which eliminates the paradoxes since they cannot be members of any class.<ref>[[#Hallett|Hallett 1986]], pp. 288, 290–291. Cantor had pointed out that inconsistent multiplicities face the same restriction: they cannot be members of any multiplicity. ([[#Hallett|Hallett 1986]], p. 286.)</ref> Von Neumann also used his axiom to prove the well-ordering theorem: Like Cantor, he assumed that the ordinals form a set. The resulting contradiction implies that the class of all ordinals is not a set. Then his axiom provides a one-to-one correspondence between this class and the class of all sets. This correspondence well-orders the class of all sets, which implies the well-ordering theorem.<ref>[[#Hallett|Hallett 1986]], pp. 291–292.</ref> In 1930, Zermelo defined [[Zermelo's models and the axiom of limitation of size|models of set theory that satisfy von Neumann's axiom]].<ref>[[#Zermelo1930|Zermelo 1930]]; English translation: [[#Ewald|Ewald 1996]], pp. 1208–1233.</ref>