Editing Cardinality (section)

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Parallel to Cantor’s development, [[Richard Dedekind]] independently formulated [[Dedekind-infinite set|a definition of infinite set]] as one that can be placed in bijection with a proper subset of itself, which was shown to be equivalent with Cantor’s definition of cardinality (given the [[axiom of choice]]). Dedekind’s ''[[Was sind und was sollen die Zahlen?]]'' (1888) emphasized structural properties over extensional definitions, and supported the bijective formulation of size and number. Dedekind was in correspondence with Cantor during the development of set theory; he supplied Cantor with a proof of the countability of the [[algebraic numbers]], and gave feedback and modifications on Cantor's proofs before publishing.

After Cantor's 1883 proof that all finite-dimensional [[manifolds]] have the same cardinality,<ref>{{Cite journal |last=Cantor |first=Georg |date=1883-12-01 |title=Ueber unendliche, lineare Punktmannichfaltigkeiten |url=https://doi.org/10.1007/BF01446819 |journal=Mathematische Annalen |language=de |volume=21 |issue=4 |pages=545–591 |doi=10.1007/BF01446819 |issn=1432-1807}}</ref>{{clarify|reason=Cantor dis not know the modern notion of a manifols. Using "manifold" here seem a mistranslation.|date=June 2025}} in 1890, [[Giuseppe Peano]] introducted the [[Peano curve]], which  was a more visual proof that the [[unit interval]] <math>[0,1]</math> has the same cardinality as the [[unit square]] on <math>\R^2.</math><ref>{{Cite journal |last=Peano |first=G. |date=1890-03-01 |title=Sur une courbe, qui remplit toute une aire plane |url=https://doi.org/10.1007/BF01199438 |journal=Mathematische Annalen |language=fr |volume=36 |issue=1 |pages=157–160 |doi=10.1007/BF01199438 |issn=1432-1807 |archive-url=https://archive.org/details/PeanoSurUneCurve |archive-date=2018-07-22}}</ref> This created a new area of mathematical analysis studying what is now called [[space-filling curves]].<ref>{{citation |last=Gugenheimer |first=Heinrich Walter |title=Differential Geometry |page=3 |year=1963 |url=https://books.google.com/books?id=CSYtkV4NTioC&pg=PA |publisher=Courier Dover Publications |isbn=9780486157207}}.</ref> 

German logician [[Gottlob Frege]] sought to ground the concept of number in logic, defining numbers using Cantor's theory of cardinality, connecting the notion to [[Hume's principle]]. In ''[[Die Grundlagen der Arithmetik]]'' (1884) and the subsequent ''Grundgesetze der Arithmetik'' (1893, 1903), Frege attempted to derive arithmetic from logical principles, treating cardinality and cardinal number as a [[primitive notion]]. However, Frege's approach to set theory was undermined by the discovery of [[Russell's paradox]] in 1901. The paradox played a crucial role in the [[foundational crisis in mathematics]] and especially the [[Logicism#History|logicist program]]. 

This was eventually resolved by [[Bertrand Russell]] himself in ''[[Principia Mathematica]]'' (1910{{En dash}}1913, vol. II),{{Sfn|Russell|Whitehead}} co-authored with [[Alfred North Whitehead]], which introduced a [[Type theory#History|theory of types]] to avoid such paradoxes, defining cardinal numbers at each level of the type hierarchy. Cardinal numbers were treated as [[equivalence classes]] of sets under equinumerosity, but only within a type-theoretic framework. Though Russell initially had difficulties understanding Cantor's and Frege’s intuitions of cardinality, shown in his 1905 manuscript ''On Some Difficulties in the Theory of Transfinite Numbers and Order Types.''<ref>{{Cite journal |last=Russell |first=B. |date=1907 |title=On Some Difficulties in the Theory of Transfinite Numbers and Order Types |url=https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s2-4.1.29?doi=10.1112%2Fplms%2Fs2-4.1.29 |journal=Proceedings of the London Mathematical Society |language=en |volume=s2-4 |issue=1 |pages=29–53 |doi=10.1112/plms/s2-4.1.29 |issn=1460-244X}}</ref>