Wikipedia:Reference desk/Archives/Mathematics/2019 March 12

= March 12 =

(Non trivial) theorems, equivalent to (trivial) axioms.
The best known example is probably the Pythagorean theorem, equivalent to the Parallel axiom. Another well-known example is the Well-ordering theorem, equivalent to the Axiom of choice.

I'm looking for other such equivalences (valid at least under reasonable conditions defined in advance), mainly with respect to the axiom of distributivity of multiplication over addition (whether with respect to the natural numbers, or with respect to the real numbers - under the condition of distributivity of natural numbers, or with respect to the complex numbers - under the condition of distributivity of real numbers, and likewise). HOTmag (talk) 23:47, 12 March 2019 (UTC)
 * Reverse mathematics, a subject in proof theory about proving axioms from theorems, might be what you are looking for. But it mostly studies theorems from real analysis and axioms of second-order arithmetic (basically a first-order theory of the reals). 173.228.123.166 (talk) 06:17, 15 March 2019 (UTC)