>>10982527> the reals, their practicalityWhat did you mean by this.
It's rare that any engineer even uses formal field extensions - only some physicists do when the work with mathematica. Most of the time they work with 16-digit approximations of rational numbers. Would be interested we we lived in a world where you could even calculate with the rational. Try doing Newton method without truncating at each and ever step. You can't even use the method using Q, let alone R.
The reals are "used" in algorithm descriptions to not having to track error terms of trigonometric functions and roots - that's about it.
>>10982557Axioms make a hoc judgement about boolean sentences. Definitions introduce symbols and their subsitution rules.
If you work within an axiomatic framework, often people introduce predicates that aim at specifying / pinning down a certain set of object (e.g. you formally thin of yourself workin in a set theoretic grounding and talk about groups - then you say what a group is and continue working with sets that, explicitly stated or not, are restricted to those fulfilling the group axioms). In that case, it looks like an axiom, but in reality you're just introducing a conditional for all statements that follow.
