In his paper Two notes on the foundations of set-theory, Kreisel begins:

Recall that, naively, sets present themselves in a number of distinct contexts. [...] One may therefore doubt whether any definite general notion (of set) is involved here; it look[s] much more like a mixture of notions. As a matter of historical fact this was the common feeling among Cantor's contemporaries.

(Emphasis mine.)

Kreisel does not give a source for the emphasized claim. My question is whether in fact this is accurate.

One possible difficulty, at least in looking for sources slightly pre-Cantor, is likely to be that insofar as that was the common feeling there may not be much positive evidence of that - if nobody sees a particularly unified conception of X, there's no obvious impetus to say so. However, I'm still hopeful.

  • $\begingroup$ Another issue is that there was no clear distinction between sets and classes (as in extensions/intensions), Kanamori says that this did not clearly emerge until Hausdorff. You may also find Mancuso's Measuring the Size of Infinite Collections on diverging intuitions in this regard interesting. For example, Bolzano rejected Hume's principle that bijective correspondence entails equal size. $\endgroup$
    – Conifold
    Commented May 1, 2020 at 21:44
  • $\begingroup$ @Conifold Indeed. Basically, I'm somewhat skeptical of Kreisel's claim, and I'm interested in what sources he was basing it on (and whether we'd read them as such). $\endgroup$ Commented May 1, 2020 at 21:45
  • $\begingroup$ See The Early Development of Set Theory: "The concept of a set appears deceivingly simple. The notion of a collection is as old as counting, and logical ideas about classes have existed since at least the “tree of Porphyry” (3rd century CE)." We can at least consider the concept of set/class as the extension of a concept (from which the "obvious" Compehension principle) and the concept of set as collection of objects, more natural in mathematics context: the line is a collection of points, the set of all natural numbers,... $\endgroup$ Commented May 2, 2020 at 10:15
  • $\begingroup$ See also José Ferreiros, On Arbitrary sets and ZFC, BSL (2011) $\endgroup$ Commented May 2, 2020 at 13:21
  • $\begingroup$ And see Luca Incurvati, Conceptions of Set and the Foundations of Mathematics (CUP 2020), page 73, footnote 4. $\endgroup$ Commented May 13, 2020 at 9:50


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