3 ms·
Not using a set of axioms written in the original signature, of equations using the three operations and the constant 1. It's certainly possible to come up with
by LegionMammal978 2mo ago
Not using a set of axioms written in the original signature, of equations using the three operations and the constant 1. It's certainly possible to come up with a finite description of the true sentences in the theory, but only by extending the signature or using some method of description other than a set of axioms.
- cubefox 2mo agoUsually when people say "not finitely axiomatizable" they just mean first-order logic. For example, Peano arithmetic is finitely axiomatizable using second-order logic, which allows quantification of predicate variables, which is required for the axiom of induction.