Editing Set theory (section)

=== Early history ===
[[File:Arbor porphyrii (from Purchotius' Institutiones philosophicae I, 1730).png|thumb|236x236px|[[Porphyrian tree]] by [[:File:Arbor porphyrii (from Purchotius' Institutiones philosophicae I, 1730).png|Purchotius]] (1730), presenting [[Aristotle]]'s [[Categories (Aristotle)|Categories]].]]
The basic notion of grouping objects has existed since at least the [[Natural number#History|emergence of numbers]], and the notion of treating sets as their own objects has existed since at least the [[Tree of Porphyry]], 3rd-century AD. The simplicity and ubiquity of sets makes it hard to determine the origin of sets as now used in mathematics, however, [[Bernard Bolzano]]'s ''[[Paradoxes of the Infinite]]'' (''Paradoxien des Unendlichen'', 1851) is generally considered the first rigorous introduction of sets to mathematics. In his work, he (among other things) expanded on [[Galileo's paradox]], and introduced [[one-to-one correspondence]] of infinite sets, for example between the [[Interval (mathematics)|intervals]] <math>[0,5]</math> and <math>[0,12]</math> by the relation <math>5y =  12x</math>. However, he resisted saying these sets were [[equinumerous]], and his work is generally considered to have been uninfluential in mathematics of his time.<ref>{{Citation |last=Ferreirós |first=José |title=The Early Development of Set Theory |date=2024 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/settheory-early/ |access-date=2025-01-04 |edition=Winter 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri |archive-date=2023-03-20 |archive-url=https://archive.today/20230320205811/https://plato.stanford.edu/entries/settheory-early/ |url-status=live }}</ref><ref>{{Citation |last=Bolzano |first=Bernard |title=Einleitung zur Größenlehre und erste Begriffe der allgemeinen Größenlehre |volume=II, A, 7 |page=152 |year=1975 |editor-last=Berg |editor-first=Jan |series=Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al. |location=Stuttgart, Bad Cannstatt |publisher=Friedrich Frommann Verlag |isbn=3-7728-0466-7 |author-link=Bernard Bolzano}}</ref>

Before mathematical set theory, basic concepts of [[infinity]] were considered to be solidly in the domain of philosophy (see: ''[[Infinity (philosophy)]]'' and ''{{Section link|Infinity|History}}''). Since the 5th century BC, beginning with Greek philosopher [[Zeno of Elea]] in the West (and early [[Indian mathematics|Indian mathematicians]] in the East), mathematicians had struggled with the concept of infinity. With the [[History of calculus|development of calculus]] in the late 17th century, philosophers began to generally distinguish between [[Actual infinity|actual and potential infinity]], wherein mathematics was only considered in the latter.<ref>{{Citation |last=Zenkin |first=Alexander |title=Logic Of Actual Infinity And G. Cantor's Diagonal Proof Of The Uncountability Of The Continuum |periodical=The Review of Modern Logic |volume=9 |issue=30 |pages=27–80 |year=2004 |url=http://projecteuclid.org/euclid.rml/1203431978 |access-date=2025-01-04 |archive-date=2020-09-22 |archive-url=https://web.archive.org/web/20200922022622/https://projecteuclid.org/euclid.rml/1203431978 |url-status=live }}</ref> [[Carl Friedrich Gauss]] famously stated: "Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics."<ref>{{cite book |last1=Dunham |first1=William |url=https://archive.org/details/journeythroughge00dunh_359 |title=Journey through Genius: The Great Theorems of Mathematics |publisher=Penguin |year=1991 |isbn=9780140147391 |page=[https://archive.org/details/journeythroughge00dunh_359/page/n267 254] |url-access=limited}}</ref>

Development of mathematical set theory was motivated by several mathematicians. [[Bernhard Riemann]]'s lecture ''On the Hypotheses which lie at the Foundations of Geometry'' (1854) proposed new ideas about [[topology]], and about basing mathematics (especially geometry) in terms of sets or [[manifold]]s in the sense of a [[Class (set theory)|class]] (which he called ''Mannigfaltigkeit'') now called [[point-set topology]]. The lecture was published by [[Richard Dedekind]] in 1868, along with Riemann's paper on [[trigonometric series]] (which presented the [[Riemann integral]]), The latter was a starting point a movement in [[real analysis]] for the study of “seriously” [[discontinuous function]]s. A young [[Georg Cantor]] entered into this area, which led him to the study of [[Point set|point-sets]]. Around 1871, influenced by Riemann, Dedekind began working with sets in his publications, which dealt very clearly and precisely with [[equivalence relations]], [[Partition of a set|partitions of sets]], and [[homomorphisms]]. Thus, many of the usual set-theoretic procedures of twentieth-century mathematics go back to his work. However, he did not publish a formal explanation of his set theory until 1888.