4 ms·
I mean the equivalence / transference proofs. Otherwise you're pulling a theorem out of the air and telling students to just accept it. And by "accessible" I me
by podperson 14y ago
I mean the equivalence / transference proofs. Otherwise you're pulling a theorem out of the air and telling students to just accept it. And by "accessible" I mean easy enough to understand that they don't become a huge distraction (or worse).
- aaronasterling 14y agoIf one starts with star-R to begin with, then why does one need the transference theorems?
- podperson 14y agoYou're then discussing math in terms of a set whose properties don't obviously match anything the student has dealt with. It needs to be established, one way or another, that adding infinitesimals etc. is not introducing new behavior.
- dllthomas 14y agoI don't see why that can't just be stated. Students need somewhere to start from, and there's plenty already that we say "trust us for now, we'll prove it later" - and it's not like the reals actually match anything the student has dealt with, at the corner cases, either...
- yummyfajitas 14y agoNo, it means you are starting with a different (but equivalent) set of axioms. Our use of R rather than star-R is merely a quirk of history - we managed to justify R to ourselves first, hence it's the "standard".
- podperson 14y agoYou're working with strictly _more_ axioms, and the extra axioms don't seem justified unless you use the equivalence proof to show you haven't added undesired behavior. (If you look at the other axioms, the hardest one to justify is the bounding axiom, and its necessity is reinforced by proof. Every other axiom fits in with the student's understanding of arithmetic.)