Is Pugh's Use of Geometry in Proving Uncountability Rigorous?

[Gyroid]

Hi PFers,

What I'm referring to is on p. 32 of Pugh's Real Mathematical Analysis. (trying to begin study early for this class as I've heard it's a toughie)

Basically, Pugh proves the uncountability of a closed interval [a,b] of real numbers using a geometric construction which pairs each point on the line segment [a,b] with a point on a unit circle which is then paired with a unique point on the real number line.

So, since the reals are uncountable, it follows that the reals on the interval [a,b] are uncountable as well (because a bijection has been defined between them.)

Now, my question is not about the content of the proof, per se. What troubles me is the reliance of this proof on geometry. Is this rigorous? The notion of the real numbers being represented as points on a line is certainly intuitive, and the proof definitely makes sense in that regard, but I always thought part of the purpose of this course was to clean up the fuzzy intuition regarding the real number line that is common to calculus students. Now here I see this fuzzy intuition being used as a tool for proof!

Now I am totally cool with the dedekind construction of the real numbers, but what I am not cool with is the (apparent) equivalence between the real numbers (in the sense of the dedekind construction) and the points on the geometric line. Again, it certainly makes sense, and perhaps I am being a bit pedantic here, but is the equivalence just so screamingly obvious that it can be used as a tool for proof with no comments, or am I missing something here? Thanks guys, awesome forum.

 

[HallsofIvy]

I'm not clear why you think it is "fuzzy intuition" that there is a one-to-one mapping from the set of real numbers to the line. That follows from the fact that set of real numbers is "complete". Perhaps the proof of that is what you want. (And it is not "screamingly obvious" but it is true.)

 

[SpaceTag]

Actually to my knowledge this is taken as an axiom; read

http://en.wikipedia.org/wiki/Cantor-Dedekind_axiom

so I can see your point.

 

[HallsofIvy]

Like any "axiom" it depends upon your starting point. If you take the real numbers as "given", then yes, completeness (in any of several equivalent forms) is taken as an axiom. If you define the real numbers, from the rationals, using Dedekind cuts, then the least upper bound property (one form of "completeness") can be proved easily. If you use "equivalence classes of increasing sequences of real numbers" instead, then it is easy to prove "monotone convergence", another form of "completeness". If you use "equivalence classes of Cauchy" sequences, then you can prove the Cauchy criterion.