One can consider two complementary Peano arithmetics both staandard, but well-ordered oppositely to each other. For examplle, the one starts from "1" by the function of suucessor interpreted as "+1", and the other one, from the odinal of the countable set, "omega" by the function of successor interpreted as "-1". The former needs only the Peano axiioms, and the latter (or both consistently) needs them and (ZFC) set theory. Thus, those two complimentary Peano arithmetics are a tool for studyimg problems and paradoxes about the foundations of mathematics (e.g. Gödel's incopletenes/ inconsitency of Peano arithmetic to ZFC set theory).

Furthermore, one can admit a "nonstandard interpretation" of Peano arithmetic to reconcile the two complimentary Peano arithmetics to each other in order to be valid simultaneously: its first element can be interpreted both as "1" or as "omega", and the function of successor as "= successor". This implies the cyclicicity of that nonstandard interpretation, as well as a "topological" or ("pre-topological") meaning of the axiom of choice as the "topological cut" of the cyclicity or coherence of the whole into a well-ordering. "Choice", "(quantum) information", "bit", and "qubit" can asquire a relevant topological (or pre-topological) meaning.