Help Finding the Correct Approach to this Proof (Intro Real Analysis)

  • #26
mathwonk
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Someone with my mathematicians' dna finds it incomprehensible that anyone could be happy studying a subject in which one did not know what all the words mean. I myself was never content until after sitting through, as a junior or sophomore, a course in which George Mackey laid out the foundations of analysis (for undergraduates), beginning with only the axiom that there exists at least one infinite set, proceeding to the Peano postulates, and definition of the natural numbers as I outlined above, and then the rationals and reals, fundamentals of metric spaces, and the proofs of the basic theorems of calculus. At the end I felt and thought: ahhh...., at last I can be at peace with these previously fuzzy statements! I still wonder how anyone can think they know anything about calculus who cannot define, or at least characterize by axioms, the real numbers. If you don't know what a real number is, why bother to state theorems about them, e.g. what do you mean when you invoke the intermediate value theorem? Of course the student is not at fault, since they have been taught by a teacher who did not bother to explain what was going on. The rubber hits the road with a student who thinks for instance that all real numbers are integers, or at best that they are all finite decimals that can be displayed on their calculator. With such an understanding, of course all the theorems of calculus are actually false!
 
  • #27
Infrared
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Someone with my mathematicians' dna finds it incomprehensible that anyone could be happy studying a subject in which one did not know what all the words mean.

I still wonder how anyone can think they know anything about calculus who cannot define, or at least characterize by axioms, the real numbers. If you don't know what a real number is, why bother to state theorems about them, e.g. what do you mean when you invoke the intermediate value theorem?
I agree with this to an extent, but there are some things I'm okay believing. Since you do algebraic geometry, surely you're aware of annoying set theoretical issues of when you need to have (locally) small categories for constructions to be safe, and what to do when you only have proper classes instead of sets, etc. I don't really ever run across this sort of things, but if I did, I think I'd err on the side of not worrying too much about it. I hope that you don't judge me too much for this!

Of course the student is not at fault, since they have been taught by a teacher who did not bother to explain what was going on. The rubber hits the road with a student who thinks for instance that all real numbers are integers, or at best that they are all finite decimals that can be displayed on their calculator. With such an understanding, of course all the theorems of calculus are actually false!
I've wondered a bit about the pedagogical value of constructing ##\mathbb{R}## with decimals, i.e. a real number is a sequence of natural numbers from 0 to 9, with a decimal point somewhere, modulo equivalences coming from trailing sequences of ##9##s. It would be a bit more annoying than in the usual constructions to carefully define arithmetic operations, but perhaps it would seem more familiar to students? I'm not sure, it's basically the Cauchy sequence construction, except that you're only working with rationals whose denominators are a power of ##10##. Maybe I shouldn't be let in front of an analysis classroom...

Edit: I've realized that you've had Mackey, Bott, and Tate as professors in undergrad. I've also had some well-known professors, but you were spoiled!
 
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  • #28
mathwonk
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Infrared, I presented exactly the decimal construction of real numbers you describe, from an exercise in the appendix of Spivak's Calculus, "high school real numbers", for a brilliant class of high schoolers some decades ago. It went really well, they composed a rap song about it, and we all learned a lot. One of those students was until recently chair of the math department of Brown, and now heads their data science initiative. That one exercise in Spivak occupied us for several weeks and was written up in a 22 page set of notes I still have in pdf. I attempt to attach the first half of them.
 

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"... a bijection mapping finite sets to finite sets and infinite sets to infinite sets is in the category of things that are too low level to need to prove."

I think that it's very healthy to prove this at least once. The proof is extremely easy, and it's well worth understanding why it is so easy.
 

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