“Every set of numbers has a least element” clearly does NOT define the natural numbers. Consider N U {-1}.
“Every set of numbers has a least element”—along with the other, non-inductive axioms of Peano arithmetic.
You might want to insert “non-empty”, though.
Done!
You should say “replace THEM”, in that case, to refer to the infinite set of axioms, as opposed to Peano Arithmetic.
Corrected
“Every set of numbers has a least element” clearly does NOT define the natural numbers. Consider N U {-1}.
“Every set of numbers has a least element”—along with the other, non-inductive axioms of Peano arithmetic.
You might want to insert “non-empty”, though.
Done!
You should say “replace THEM”, in that case, to refer to the infinite set of axioms, as opposed to Peano Arithmetic.
Corrected