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It seems to me that in the program Mathematica, things that look like sets are actually lists ands what are called lists are actually functions. (is this so?) So that's what prompts the question. If there is such a theory, what is it called and where can one read about it, say at the level of Halmos's Naive Set Theory?

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    $\begingroup$ Yes Von Neumann's (original 1925 formulation of) set theory. $\endgroup$ – Mauro ALLEGRANZA Jan 1 '17 at 14:58
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    $\begingroup$ See English translation of John von Neumann, (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240. $\endgroup$ – Mauro ALLEGRANZA Jan 1 '17 at 14:59
  • $\begingroup$ It's tricky to answer this because it's not clear if you really just mean "functions being fundamental in some way" (in which case your choices are probably along the lines of Mauros comment or category theory) or "something like what mathematica does" in which case the answer is probably "no that wouldn't suffice" (but maybe something similar in constructive logic?). $\endgroup$ – Mark S. Jan 1 '17 at 15:00
  • $\begingroup$ @Mark S. I had in mind a "set" theory in which functions are" fundamental in some way," and I did not know that von Neumann had already developed such a theory early on. It seems to me (naively), that in both mathematics and computer science, functions have gained ascendency over sets, so the question implicit, I think, in what I asked, is why use sets and not functions as the starting point. To use Henning Makholm's expressions, why have the attempts to base mathematics exclusively on the concept of function, rather than on the concept of set, not gathered any wide following? $\endgroup$ – Airymouse Jan 1 '17 at 16:46
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There has been attempts to base mathematics exclusively on the concept of function rather than on the concept of set. They don't seem to have gathered any wide following, though.

One example is the map theory of Klaus Grue.

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