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$\begingroup$ Thank you for your answer. Ok, let me ask a concrete question: can the axiom of transposition be simplied to the form: $\forall A\exists B\;\forall x\forall y \;((x,y)\in A\Leftrightarrow (y,x)\in B)$? As I understood the axiom of cyclic permutation composed with the axiom of domain is necessary for the proof of the existence of projections on each coordinate. And the axiom of transposition makes the main job generating all the permutations. Right? $\endgroup$
– Taras Banakh
Commented Apr 8, 2020 at 20:38
2

$\begingroup$ @TarasBanakh You might be interested to know that Godel's original set of operations has three different permutation rules. en.wikipedia.org/wiki/G%C3%B6del_operation $\endgroup$
– Pace Nielsen
Commented Apr 8, 2020 at 21:27
$\begingroup$ Thank you for this link. I mentions the book of Jech who defined 10 Godel's operations but in Exercise 13.4 Jech remarks that those 10 can be reduced to 8. The number of NBG axioms allowing to construct classes is also 8. Maybe 8 is this optimal number? What I will try to realize now is if the 9-th Godel's operation of Jech (en.wikipedia.org/wiki/G%C3%B6del_operation) is expressible via 8-th and other operations. $\endgroup$
– Taras Banakh
Commented Apr 9, 2020 at 4:25
$\begingroup$ Eight is not minimal, since one can always arrange for operations to be extremely complicated and do more than one job at once. $\endgroup$
– Pace Nielsen
Commented Apr 9, 2020 at 18:42
$\begingroup$ I have in mind the minimality of elementary operations (those that cannot be decomposed into simpler pieces). If you want, the minimality of the total number of symbols necessary for writing these axioms, so each artificial conjunction increases the number of such symbols. But let us leave this minimality discussion as fruitless. $\endgroup$
– Taras Banakh
Commented Apr 9, 2020 at 19:10

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