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    $\begingroup$ How many generators do you need for $S_3$? $\endgroup$
    – Asaf Karagila ♦
    Commented Apr 8, 2020 at 18:17
  • 4
    $\begingroup$ @TarasBanakh I think Asaf's point is that the transposition axiom only includes one transposition (swapping second and third coordinates), and more generally a "one permutation per axiom" set-up will require two axioms. (And if we drop that setup, then we can just take the conjunction of the axioms.) $\endgroup$
    – Noah Schweber
    Commented Apr 8, 2020 at 18:29
  • 4
    $\begingroup$ Taras, any finite number of axioms can be replaced by one. Just take the conjunction... $\endgroup$
    – Asaf Karagila ♦
    Commented Apr 8, 2020 at 18:32
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    $\begingroup$ You need infinitely many transpositions to generate the group of finitary permutations of $\omega$, hence it is a minor miracle that you can make do with only finitely many axioms at all; it certainly does not give a reason to think that you can reduce the two premutation axioms to one. $\endgroup$
    – Emil Jeřábek
    Commented Apr 8, 2020 at 18:34
  • 4
    $\begingroup$ Taras, over-optimisation can be a bad thing. Redundancies exist to make things easier. We include Replacement and Separation as axioms in ZFC because it's easier, and we don't have to go through the proof that Replacement implies Separation. Likewise for Pairing, or Empty Set. $\endgroup$
    – Asaf Karagila ♦
    Commented Apr 9, 2020 at 8:35