Re:Real Not Complex: Free Math Resources
by thaumasiotes on Saturday March 28, 2020 @12:02PM
(#22711923)
Yes, I have no problem with the fact that the concept "sup A" is defined before the statement of the axiom and then not repeated in it. (It's defined before the statement of theorem 3.1.1 in Strichartz too, it's just also repeated there.) I'm objecting to the fact that they present a theorem as if it were an axiom.
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One model's axiom is another model's theorem.
Re:Real Not Complex: Free Math Resources
by thaumasiotes on Saturday March 28, 2020 @02:01PM
(#22712940)
Sure, the proof in Strichartz is constructive, defining a real number that is the supremum of the set.
LLN (I'm going to assume that the family name is Nguyen) can't do that, because they have no model of the real numbers, so it isn't possible to say that some construct is or isn't a real number.
So this is more a case of "what is a theorem if you have a model is only an axiom if you don't".
If you changed the statement of the completeness axiom from "every nonempty subset A of the real numbers that is bounded above has a unique real least upper bound sup A" to "every nonempty subset A of the rational numbers that is bounded above has a unique real least upper bound r", you'd have the Dedekind cut construction of the real numbers. That's not generally presented as an axiom either.
There are sometimes good reasons to define a mathematical object under study (vector space, real numbers, ...) by an axiomatic list of properties it satisfies instead of constructing it from simpler objects. John Conway's ONAG has an interesting mini-chapter about this.