| | It certainly seems reasonable that not using AoC would be desirable for calling a proof elementary. However, in my experience, the notion of an elementary proof is closer to "not using mathematics from unrelated areas" than it is in avoiding specific techniques or axioms. In number theory, this usually entails avoiding the methods of complex analysis. But to me, it is perfectly natural to say "To avoid this, we next give an elementary proof, which uses Zorn's lemma in a typical way" in a paper on algebra, where the application is indeed typical. That excerpt is from "Some ring theoretic equivalents to the Axiom of Choice", easily googleable, and I find several other examples of authors citing AoC in "elementary" proofs. It's easy to ignore this as the work of sloppy authors and/or liberals, but clearly the usage defined in this CP is at the very least not universal, and indeed I have never encountered it. Might I modify the page to indicate that some authors use a different characterization of elementary proofs, and can we find a source that actually uses a definition resembling that here? --[[User:MarkGall|MarkGall]] 01:23, 25 August 2009 (EDT) | | It certainly seems reasonable that not using AoC would be desirable for calling a proof elementary. However, in my experience, the notion of an elementary proof is closer to "not using mathematics from unrelated areas" than it is in avoiding specific techniques or axioms. In number theory, this usually entails avoiding the methods of complex analysis. But to me, it is perfectly natural to say "To avoid this, we next give an elementary proof, which uses Zorn's lemma in a typical way" in a paper on algebra, where the application is indeed typical. That excerpt is from "Some ring theoretic equivalents to the Axiom of Choice", easily googleable, and I find several other examples of authors citing AoC in "elementary" proofs. It's easy to ignore this as the work of sloppy authors and/or liberals, but clearly the usage defined in this CP is at the very least not universal, and indeed I have never encountered it. Might I modify the page to indicate that some authors use a different characterization of elementary proofs, and can we find a source that actually uses a definition resembling that here? --[[User:MarkGall|MarkGall]] 01:23, 25 August 2009 (EDT) |
| | + | : The real objection is that non-elementary proofs rely on additional assumptions. For complex analysis, that additional assumption is that there is a unique square root of negative 1, which can be manipulated algebraically. Maybe true, maybe false, and an elementary proof is desirable because it does not rely on that assumption. The Axiom of Choice also includes a questionable assumption.--[[User:Aschlafly|Andy Schlafly]] 09:50, 25 August 2009 (EDT) |