Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?
[Math] Text for an introductory Real Analysis course.
big-listbooksca.classical-analysis-and-odesreal-analysistextbook-recommendation
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As it happens, I just finished teaching a quarter of undergraduate real analysis. I am inclined to rephrase Pugh's statement into a form that I would agree with. If you view analysis broadly as both the theorems of analysis and methods of calculation (calculus), then obviously it has a ton of applications. However, I much prefer to teach undergraduate real analysis as pure mathematics, more particularly as an introduction to rigorous mathematics and proofs. This is partly as a corrective (or at least a complement) to the mostly applied and algorithmic interpretation of calculus that most American students see first.
Some mathematicians think, and I've often been tempted to think, that it's a bad thing to do analysis twice, first as algorithmic and applied calculus and second as rigorous analysis. It can seem wrong not to have the rigor up-front. Now that I have seen what BC Calculus is like in a high school, I no longer think that it is a bad thing. Obviously I still think that the pure interpretation is important. On the other hand, both interpretations together is also fine by me. I notice that in France, calculus courses and analysis courses are both called "analyse mathématique". I think that they might separate rigorous and non-rigorous calculus a bit less than in the US, and it could be partly because of the name.
In fact, it took me a long time to realize how certain non-rigorous explanations guide good rigorous analysis. For instance, the easy way to derive the Jacobian factor in a multivariate integral is to "draw" an infinitesimal parallelepiped and find its volume. That's not rigorous by itself, but it is related to an important rigorous construction, the exterior algebra of differential forms.
Finally, I agree that Pugh's book is great. As the saying goes, you shouldn't judge it by its cover. :-)
Let me address the updated version of your question.
There is a philosophical current running through parts of descriptive set theory, and this includes anything that might be described as classical real analysis, to the effect that the realm of Borel mathematics is comparatively immune to the chaos of independence. On this view, one regards the Borel functions, relations and objects as being the most explicitly given, and the land of the Borel is the land of explicit mathematics.
For example, an important emerging field is the theory of equivalence relations under Borel reducibility, arising out of the observation that many of the most natural equivalence relations arising in other parts of mathematics, such as isomorphism relations on classes of algebraic structures, turn out to be Borel equivalence relations on a standard Borel space. Set-theorists seek to understand the comparative difficulty of the corresponding classification problems for these relations by considering the relations under Borel reducibility. This concept provides us with a precise way to measure the comparative difficulty of two classification problems, which then assemble themselves into a complex hiearchy, increasingly revealed to us. To give one example, it falls out of this theory that there can be no Borel classification of the finitely generated groups up to isomorphism by means of countable objects (this relation is not "smooth").
This theory has been largely immune from the independence phenomenon, for several reasons. Perhaps the best explanation of this is the fact that Borel assertions have complexity $\Delta^1_1$, which lies below the Shoenfield absoluteness theorem.
Theorem(Shoenfield Absoluteness) Any statement of complexity $\Sigma^1_2$ is absolute between any two models of set theory with the same ordinals.
In particular, this implies that the method of forcing is completely unable to affect existence assertions about Borel objects, since such assertions would have complexity $\Sigma^1_1$, as well as more complex assertions. Because forcing is one of the principal tools by which set-theorists have come to exhibit independence, this means that Borel mathematics is completely immune from the forcing technology.
Furthermore, when there are sufficient large cardinals, then one can attain an even greater degree of absoluteness in various senses. For example, in the presence of large cardinals there are various strong senses in which the theory of $L(\mathbb{R})$ is invariant by forcing. Thus, even the realm of projective mathematics ($\Sigma^1_n$ for any $n$) is unaffected by forcing, when there are sufficient large cardinals.
At the same time, we know that it isn't strictly true even that Borel mathematics is immune from independence, since the $\Delta^1_1$ level of complexity includes all of arithmetic, which therefore admits the Gödel incompleteness phenomenon. But because the method of forcing is struck down, however, none of the more spectacular independence results in the realm of analysis, such as the independence results concerning CH and cardinal invariants, arise at the Borel level of complexity. Thus, I believe that the realm of Borel mathematics may be the best, although imperfect, answer to your updated question.
At the same time, it must be said that although the method of forcing is ruled out as a means of proving independence for Borel existence assertions, we have no meta-theorem that says that there will not be some future method that is able to establish independence for such assertions. Surely a major lesson of logic over the past century is the pervasiveness of the independence phenomenon, and I believe that it is only a matter of time for such methods to arise.
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Best Answer
Stephen Abbott, Understanding Analysis
Strongly recommended to students who are ony getting to grips with abstraction in mathematics. Find a review here.