ℑ𝔫𝔱𝔯𝔬𝔡𝔲𝔠𝔱𝔦𝔬𝔫

𝔗𝔥𝔢 story of set theory, a 'doctrine' that deals with the labyrinth of infinity and the continuum, is sometimes told in a way that resembles beautiful myths. The greek goddess 𝔄𝔱𝔥𝔢𝔫𝔞¹ sprang full-grown and armored from the forehead of ℨ𝔢𝔲𝔰, and was his favorite child. 𝔖𝔢𝔱 theory is generally taken to have been the work of a single man, Georg Cantor, who developed single‐handedly a basic discipline that has deeply affected the shape of modern mathematics. He loved his creature so much that his life became deeply intertwined with it, even suffering mental illness for its sake. The comparison between theory and goddess is interesting in other ways, too. 

ℨ𝔢𝔲𝔰 & 𝔄𝔱𝔥𝔢𝔫𝔞 illustrated

𝔄𝔱𝔥𝔢𝔫𝔞 was a virgin goddess, whereas set theory is comparable to number theory in its purity and abstractness. She was the goddess of wisdom and of the polis, while set theory plays an organizing role in the polis of mainstream modern mathematics and represents one of tbe highest achievements of mathematical wisdom. She was also the goddess of war, which brings to mind the polemics and disputes brought forwarded by the discipline that will occupy our attention in the present study. 

𝔗𝔥𝔢𝔯𝔢 certainly is wisdom in the search for the founding fathers or founding myths, especially when a new approach or a new discipline is fighting for recognition. Moreover, mathematicians tend to concentrate on advanced results and open problems, which often leads them to forget the ways in which the orientation that made their research possible actually emerged.² Both reasons help us understand why the tradition of ascribing the origins of set theory to Cantor alone goes back to the early 20th century. In [1914], Hausdorff dedicated his handbook, the first great manual of set theory, to its creator Cantor. One year later, the Deutsche MathematikerVereinigung sent a letter to the great mathematician, on the occasion of his 70th birthday, calling him "the creator of set theory."³ Later on Hilbert chose set theory as a key example of the abstract mathematics he so strongly advocated, and he constantly associated it with the name of Cantor (see, e.g., [Hilbert 1926]). This is natural, for Cantor turned the set-theocratical approach to mathematics into a true branch of the discipline, proving the earliest results in transfinite set theory and formulating its most famous problems. 
Georg Cantor 1845―1918
𝔅𝔲𝔱 this traditional view has also been contested from time to time. Some twenty years ago, Dugac wrote that the birthplace of set theory can be found in Dedekind's work on ideal theory [Dugac 1976, 29]. Presented this way, without further explanation, this assertion may seem confusing.⁴ In the introduction to Dugac's book, Jean Dieudonné wrote opinionatedly: 
𝔗𝔥𝔢 'paradise of Cantor,' that Hilbert believed to be entering, was in the end but an artificial paradise. Until further notice, what remains alive and fundamental in Cantor's work in his first treatises on the denumerable, the real numbers, and topology. But in these domains it is of justic to associate Dedekind to him, and to consider that both share equally the merit of having founded the set-theoratical basis of present mathematics.⁵ 
Richard Dedekind 1831―1916

𝔗𝔥𝔦𝔰 viewpoint is also in agreement with some older views, for instance with Zermelo's in the famous paper on the axionatization of the set theory, which he called the "theory created by Cantor and Dedekind" [1908, 200]. If one takes into account that Dedekind's a set-theoretical conceptions were very advanced by 1872, and that he and Cantor became – so we are told – good friends from that time (see Chapter VI), the customary story appears problematic. This uneasy state of affairs was actually the starting point for my own work, though its scope grew and changed substantially in time. 
𝔗𝔥𝔢 thrust of the argument most frequently used by those who attributed the authorship of set theory to Cantor is just the following. Cantor was the man who, in the latter half of the 19th century, introduced the infinite into mathematics; this, in turn, became one of the main nutrients in the spectacular flowering of modern mathematics.⁶ If this is the whole argument, one can simply say that is premise is historically inaccurate, so the conclusion does not follow. For at least another author, Dedekind, introduced the actual infinite unambiguously and influentially even before Cantor. On the other hand, Cantor did inaugurate transfinite set theory, after others had started to rely on actual infinity and while the theory of point-sets was being studied by several mathematicians. 
𝔗𝔥𝔢 above shows that, despite the large number of historical works which have dealt with set theory in one way or another, we still lack an adequate and balanced historical description of its emergence. The present study attempts to fill the gap with a general overview that synthesizes much previous work and at the same time tries to provide new insights.⁷ 
1. 𝓐𝓽𝓱𝓮𝓷𝓪, goddess of Parthenon in Athens was identified with the Roman 𝓜𝓲𝓷𝓮𝓻𝓿𝓪. 
2. On founding fathers, see [Bensaudi-Vincent 1983], Kuhn [1962, Chapter. 11] analyzed how scientists, in their systematic work, are continuously rewriting and hiding real history. 
3. [Purkert & Ilgauds 1987, 165—66]: “Schöpfer der Mengenlehre.”
4. A similar but subtler pronouncement can found in [Edwards 1980, 346]. 
5. [Dugac 1976, 11]; “le, 'paradise de Cantor' où Hilbert croyait entrer n'etait au fond qu'un paradis artificiel ”. Jusqu'a nouvel ordre, ce qui reste vivant et fondamental dans I'œuvre de Cantor, ce sont ses premiers travaux sur le dènombrable, les nombres réels et la Topologie. Mais dans ces domaines, il n'est que juste de lui associer Dedekind, et de considérer qu'iles partagent à titre égal le mérite d'avoir fondé les bases 'ensemblistes' de la mathématique d'ajourd'hui". 
6. I have paraphrased [Lavine 1996, 1], but see [Hausdorff 1914], [Fraenkel 1923; 1928]. 
7. Among previous historical contributions, the following, at least, stand out: [Jourdain 1906/14], [Cavaillès 1962](written in 1938), [Medvedev 1965], a number of books centering on Cantor ― above all [Dauben 1979] and [Hallett 1984], but also [Meschkowisky 1967], [Purkert & Ilgauds 1987], [Dugac 1976], and finally [Moore 1982]. 

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