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Did Georg Cantor ever think that set theory could serve as a foundational system for all of mathematics?

He died in 1918, but Zermelo set theory (just Z, no ZF or ZFC yet) was described in a paper by Zermelo in 1908, ten years before Cantor's death.

I haven't been able to find anything on the Web regarding this question. In particular, a long Wikipedia about Cantor does not provide an answer to my question — the only thing I could gather is that Cantor knew that his set theory needed to be axiomatized.

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  • $\begingroup$ See this post: "foundational" issues were not the sources of Cantor's theory. And see also The Early Development of Set Theory:"set theory has generally been identified with the branch of math.logic that studies transfinite sets, originating in Cantor’s result that R has a greater cardinality than N. But set theory was both effect and cause of the rise of modern mathematics: the traces of this origin are indelibly stamped on its axiomatic structure." $\endgroup$
    – Mauro ALLEGRANZA
    Commented Jul 25, 2023 at 6:29
  • $\begingroup$ Probably not. It took logicists such as Russell to make those connections. en.wikipedia.org/wiki/Russell%27s_paradox youtube.com/watch?v=ymGt7I4Yn3k $\endgroup$
    – DJohnson
    Commented Jul 25, 2023 at 13:49
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    $\begingroup$ I don't know the answer, but I suspect some of the other answers here might be a bit hasty. Cantor literally called his book the Grundlagen (foundation). Perhaps it meant something different in German, but it seems he was attempting to lay a pretty ambitious foundation. Maybe it was just for cardinality, and actually for "all of mathematics". $\endgroup$
    – combinatorist
    Commented Jul 25, 2023 at 22:18
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    $\begingroup$ Cantor did not think that mathematics needed foundations in the modern sense initiated by Frege and Hilbert. He was a classical platonist and saw it as 'founded' enough in something like objective platonic realm to be described, see Purkert. However, in his 1885 review of Frege's attempt at foundations he did write that it is "an inversion of what is correct" to ground cardinality on "concepts", as Frege attempted, see Ebert and Rossberg. $\endgroup$
    – Conifold
    Commented Aug 3, 2023 at 18:25
  • $\begingroup$ As late as the 1920s people considered general topology and measure theory as sub-topics of set theory. This is fairly clear in Hausdorff's foundational 1914 book on set theory en.wikipedia.org/wiki/Grundz%C3%BCge_der_Mengenlehre That ZFC and related systems became considered "the foundations" for all mathematics was not clear even when Gödel wrote about the relative consistency of CH and AC in the late 1930s, where the system of von Neumann/Bernays (now also named for Gödel) was adopted. As far as I understand it, vNBG was nearly standard until Cohen invented forcing. $\endgroup$
    – David Roberts
    Commented Aug 15, 2023 at 6:29

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Cantor was interested in, and developed, a theory of ordinal numbers and cardinalities. This was a significant accomplishment, in part because it overcame traditional resistance to working with actual infinity. Cantor's theory did not by any means become the foundation of mathematics (though it is certainly studied extensively in modern set theory). The axiomatisation provided by Zermelo and Fraenkel uses the language of sets, but the role of ordinals and cardinals is tangential. So not only did Cantor not think that his "theory could provide a foundation for all of mathematics", in point of fact it doesn't.

This recent article by Rowe on Cantor, Dedekind, and Hilbert suggests that both Cantor and Hilbert did think of set theory as foundation for real analysis. Cantor and Hilbert were both bothered by foundational problems in set theory (such as the fact that all aleph-numbers can't form a set without causing a contradiction), and hoped to sort this out as a way of providing a solid foundation for the set of real numbers.

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