I am reading Bishop's "Constructive Analysis" and he says that defining a real number to just be an equivalence class of Cauchy sequences of rationals would be wrong. Why is that?
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Bishop refers here to a constructive approach to the reals. Classically, a Cauchy sequence of rationals is a sequence such that for any specified rational distance $\varepsilon >0$, there exists a modulus $N\in \mathbb N$ such that from that index onwards all elements are at distance at most $\varepsilon $ from each other. Constructively, this is unacceptable since the modulus $N$ needs to be constructed from the given sequence and the $\varepsilon >0$. In other words, a Cauchy sequences needs to be a sequence together with a function $\mathbb Q_+\to \mathbb N$ which is a constructive function giving an appropriate modulus for every $\varepsilon >0$. Now, exactly what a constructive function from one infinite countable set to another means can be answered in different ways (in the constructive world, classically there is no doubt how to continue). This is where it gets complicated.
Bishop is the prototypical constructivist-which means he won't accept any mathematical construction,procedure or definition that doesn't result in a precise and clear method of giving at least one example. Since Cauchy's definition is in terms of limits of a sequence of rationals and doesn't explicitly establish a bound for the limit, Bishop considers it mathematically suspect.
It's really a philosophical problem he has with the definition and nothing more-mathematically,the 2 are really equivalent to each other.I really should point out whether or not they're equivalent depends on what version of constructivism you're working in. As Steven Stadnicki points out in his objection-there are countably infinite"Bishop reals",which form "equivalence classes" under Bishop's definition where an countable constructive bijection algorithm cannot be devised (the Halting problem). As a result, by the absence of such a map, the Bishop reals are trivially uncountable! In intuitionism-Brouwer 's version of constructivism-there's a diagonalization arguement similar to Cantor's that proves the Bishop reals are uncountable.
So the truth is this matter is really subtle and sticky. But for our purposes,it certainly would be fair to say logically ,the Bishop reals are uncountable. You really need to ask a logician or set theorist this question,though-as I said,it's subtle and most people outside of research in constructive mathematics really understand it.