TITLE: Confusion about Tao's construction of reals
QUESTION [29 upvotes]: Background:
I am currently studying real analysis using Tao's Analysis Volume One and so far I am really enjoying myself though I seem to have run into some confusion regarding professor Tao's construction of the reals using rationals. The following is the definition of reals that he provides in the text:

$\DeclareMathOperator*{\LIM}{LIM}$Definition 5.3.1 (Real numbers). A real number is defined to be an object of the form $\LIM\limits_{n → ∞} a_n$, where $(a_n)_{n = 1}^∞$ is a Cauchy sequence of rational numbers. Two real numbers $\LIM\limits_{n → ∞} a_n$ and $\LIM\limits_{n→∞} b_n$ are said to be equal iff $(a_n)_{n = 1}^∞$ and $(b_n)_{n = 1}^∞$ are equivalent Cauchy sequences. The set of all real numbers is denoted $\mathbb{R}$.

Problem:
While snooping around the internet I have found that a real number is in fact an equivalence class of sequences of rationals whose corresponding terms can be arbitrary close to each other i.e $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ are equivalent if and only if 
$$\forall ε>0, \ \exists N \in \mathbb{N} \ \text{such that} \ \forall n \ge N, \ |a_n-b_n|\leq ε.$$
But Tao's definition seems to suggest that real numbers are limits of said sequences so what are they?

REPLY [2 votes]: Tao's definition is problematic. There is nothing confusing about identifying the real numbers directly as equivalence classes of Cauchy sequences, which is pretty standard.
Ultimately, the philosophical question of what mathematical objects actually are is unsettled. But the issue is at least better focused if we define all our objects as sets; and equivalence classes of Cauchy sequences are sets.
One issue with Tao's definition is that it identifies mathematical objects with notation. And what is notation? Is it ink? is it a geometrical form? Is it a prescritive rule for writing? Is it a cultural pattern? And so on.
Another problem with Tao's approach is that the idea of equivalence class, made mathematically explicit in the standard approach, is swept aside; and the challenge that ${\rm LIM}_{n\to\infty}a_n$ isn't the same as ${\rm LIM}_{n\to\infty}b_n$ as notation is dismissed as inconseqential. If we allow such informality, we are on the slippery slope of reverting to the commonsense notion of real numbers that nonmathematicians are happy with.
For completeness, one should mention also the separate issue of the arbitrariness of the Cauchy-sequence definition. The definition as Dedekind cuts (and there are others) is arguably just as good, and this too can be made purely set-theoretic. Some mathematicians point out that the mode of construction is a distraction, and that what really matters is the structure of the operations and relations within the reals. There is something to be said for defining the reals as a complete archimidean ordered field, with the appropriate embedding of the rationals, showing that any two such objects are isomorphic, and that Cauchy sequences (say) form a model of them (to prove that the definition isn't vacuous). But trying to avoid the selection of any particular construction opens up further issues—quite apart from that of the poor old student, trying to learn analysis, not wanting to carry any more philosophical baggage than necessary.