Re: [math-fun] good introductory real-analysis text?
I think constructing the reals in some natural way (probably Cauchy sequences is most intuitive) is pedagogically superior to presenting the reals as a set satisfying certain axioms. First of all, when something is defined by axioms, its mere existence isn't obvious. When it's constructed (essentially assuming no more than set theory, whose axioms are too abstruse to go into in a course like this), then it must exist unless set theory is self-contradictory. Second, when something is defined by axioms, it's easy to feel like you're flying blind. Someone has set you down in front of an instrument panel (the axioms), and your only options for staying aloft is to fiddle with the buttons and dials. It's as though you can't see out the windows. But when something is constructed in a simple and natural construction (which should of course come sometime after a demonstration that there are irrational numbers that "ought to" exist), then you can in a sense see and feel what you have wrought. Finally, the whole process of rigorously constructing a mathematical object is a core concept in math, and I can't think of a better place to have it introduced than with the reals. --Dan Jim wrote: << . . . I should also say that I'm NOT looking for a book that constructs the reals (via Dedekind cuts or Cantor sequences or Conway games). My view is that a student's first exposure to real analysis should be based on an axiomatic DESCRIPTION of the reals, not a formal CONSTRUCTION of them. My point of view might be expressed succinctly, with only a small amount of distortion, in the slogan "Why construct the reals when they already exist?" . . . On the other hand, I love a good (friendly) argument, so if any of you think I'm going in the wrong direction, please offer your counter-arguments!
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
My favorite construction for the reals is to match the implicit "infinte decimals" that we grew up with. It's the ground state, hence "intuitive". There's no harm in mentioning the alternatives, nor in emphasizing there are a bunch of equivalents. Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] On Behalf Of Dan Asimov [dasimov@earthlink.net] Sent: Thursday, May 28, 2009 9:06 PM To: math-fun Subject: Re: [math-fun] good introductory real-analysis text? I think constructing the reals in some natural way (probably Cauchy sequences is most intuitive) is pedagogically superior to presenting the reals as a set satisfying certain axioms. First of all, when something is defined by axioms, its mere existence isn't obvious. When it's constructed (essentially assuming no more than set theory, whose axioms are too abstruse to go into in a course like this), then it must exist unless set theory is self-contradictory. Second, when something is defined by axioms, it's easy to feel like you're flying blind. Someone has set you down in front of an instrument panel (the axioms), and your only options for staying aloft is to fiddle with the buttons and dials. It's as though you can't see out the windows. But when something is constructed in a simple and natural construction (which should of course come sometime after a demonstration that there are irrational numbers that "ought to" exist), then you can in a sense see and feel what you have wrought. Finally, the whole process of rigorously constructing a mathematical object is a core concept in math, and I can't think of a better place to have it introduced than with the reals. --Dan Jim wrote: << . . . I should also say that I'm NOT looking for a book that constructs the reals (via Dedekind cuts or Cantor sequences or Conway games). My view is that a student's first exposure to real analysis should be based on an axiomatic DESCRIPTION of the reals, not a formal CONSTRUCTION of them. My point of view might be expressed succinctly, with only a small amount of distortion, in the slogan "Why construct the reals when they already exist?" . . . On the other hand, I love a good (friendly) argument, so if any of you think I'm going in the wrong direction, please offer your counter-arguments!
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (2)
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Dan Asimov -
Schroeppel, Richard