Confusion about Tao's construction of reals

Question

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?

Answer 1

This is a very formal definition of the real numbers (BTW there are others, look up "Dedekind cuts").

As to "what are they"? - well, they are exactly what he said: objects of the form ${\rm LIM}_{n\to\infty}a_n$. That is, they are nothing more nor less than a capital L, followed by a capital I, followed by a capital M, followed by... you get the point. And as this is the definition of real numbers, there is (at this point, and within the context of Tao's book) nothing else that we know about them.

Of course, Tao did not choose the letters L,I,M at random: he wants to help you make the connection between $${\rm LIM}_{n\to\infty}a_n$$ for rational $a_n$, which is the definition of a real number, and $$\lim\nolimits_{n\to\infty}a_n$$ for possibly real $a_n$, which is the definition of a limit (Tao 6.1.8). Note that here we have lowercase l,i,m because it's a different concept.

In other words, it is just as you stated in your question:

But Tao's definition seems to suggest that real numbers are limits of said sequences...

...he wants to suggest this before he has actually defined the concept of a limit. (So, whether deliberately or not, you used exactly the right word!!!)

You probably know lots about limits from previous courses: you should keep in mind all that you know and see how it fits in with what Tao is doing, but remember that "officially" you don't know what limits are because Tao hasn't defined them yet.

Answer 2

Tao is avoiding using the phrase "equivalence class", but what he describes is just that.

He says take a sequence and call it an "object". The object has the letters L-I-M which by some bizarre coincidence are the first three letters of "limit" but that is completely coincidental. (Looks at ceiling and whistles.) Two of these objects are declared to be "equal" if the sequence that the represent are equivalent (presumable two sequences were defined to be "equivalent" on the previous page; [$*$]).

So if you think of the two objects with different but equivalent sequences and "being the same thing" the all the objects with sequences equivalent to it are "the same thing" and this thing is a class of all the sequences that are equivalent

Hence... an equivalence class.

And that's it, a real number is one of these LIM objects that represent a Cauchy sequence and all the other LIM objects that represent Cauchy sequence that are equivalent to it.

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$[*]$ I assume "equivalent" was defined as something like $\{a_n\}$ and $\{b_n\}$ are equivalent if for any $\epsilon > 0$ there is an $N > 0$ so that for all $n > N$ we have $|a_n - b_n |< \epsilon$.

Answer 3

As mentioned in some of the other answers, Tao's notation is merely a substitute for talking about equivalence classes of Cauchy sequences of rational numbers. It is possible that he is motivated here by his notion of ultralimit, which can be used to define the hyperreals. Thus, a hyperreal will similarly be defined as $\mathbf{ULTRALIM} (a_n)$ where $(a_n)$ is a sequence of real numbers.

Answer 4

What are mathematical objects? The answer may surprise you. More on this story tonight.

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Mathematical objects are mathematical objects.¹ From a foundational point of view, we sometimes want to start with some atomic notion, and the argue that we can define the rest of the mathematical universe in terms of those objects.

These can be sets, as done in set theory, or it can be various types as done in type theory, and so on.

Ultimately, the goal is always to "reduce the existence to something more believable". Namely, if you believe that the rational numbers make sense, and that some basic constructions make sense (e.g. Cauchy sequences), then this is a proof why you should believe that the real numbers make sense.

Sure, you can now ask why do the rational numbers make sense. Then you can fallback to the integers, then to the natural numbers, and you can just accept that, or fall onto the empty set as done in the standard constructions in set theory.

But it's always something of the form:

1. If you agree with me on the validity of this object, and
2. you agree with me on the validity of that method, then
3. you agree with me on the validity of this new object.

So the real numbers can be equivalence classes of Cauchy sequences, because that is one way of building the real numbers. Or the real numbers can be Dedekind-cuts, or non-empty proper-initial segments. Or any other thing.

The important thing, however, is that we can prove they are all "the same". Namely, if you construct the real numbers using one method, and I construct the real numbers using a different method, then there is a structure-preserving way to identify the two versions of the real numbers.

So, are real numbers equivalence classes of Cauchy sequences of rational numbers? are they Dedekind cuts of rational numbers? Are they sets, or types, or some category? Maybe they are atomic to mathematics just like the natural numbers, and so the real numbers are just that, "the real numbers"?

The answer is that it doesn't matter. As long as they satisfy the properties we "expect" the real numbers to satisfy.

 

Let me just finish by pointing out that Tao doesn't really suggest that the real numbers are limits of these Cauchy sequences. Limits are only defined within a particular space (e.g. $0$ is not the limit of $\frac1n$ in the space $(0,1)$, simply because $0$ is not a point in that space).

But Tao is preparing the ground for proving that every real number is the limit of a Cauchy sequence of rational numbers. But at that point, this is just a notation of the real numbers which is defined from a certain Cauchy sequence.

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Footnotes.

1. I said it may surprise you. Not that it will surprise you.

Answer 5

But Tao's definition seems to suggest that real numbers are limits of said sequences so what are they?

Tao's definition suggests the real number associated with a rational sequence is a limit, because in real numbers $\mathbb R$ that real number actually is a limit of the sequence. However, before Tao defines real numbers it's impossible to say a real number is a limit of the sequence, because the limit does not necessarily exist in $\mathbb Q$.

So, formally, professor Tao defines some carefully chosen (but essentially arbitrary) symbol associated with a Cauchy rational sequence and then associates it with all 'equivalent Cauchy sequences'. That symbol is called a real number.

I understand this is a reversed way of saying we identify a 'real number' with an equivalence class of Cauchy rational sequences, where equivalence is usually defined in advance as vanishing difference between sequences' corresponding terms.

Answer 6

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.

Answer 7

Having just realised that real numbers are not limits of Cauchy sequences, I would like to share my point of view(which is kind of what CiaPan talks about as well).

A real number is merely defined to be an object of the form $L.I.M_{n\to\infty}a_n$ where $a_n$ is a Cauchy sequence of rational numbers because the reason that we don't define a real number to be the limit of a Cauchy sequence of rational numbers is because the definition of convergence requires the use of the number in the limit. And we so far have no idea what is the number in the limit.

But the reason that notation $L.I.M_{n\to\infty}a_n$ is chosen is because in chapter $6$, when Tao defines the convergence of a sequence of rational numbers(where the $L$ is taken to be a real number. So $L=L.I.M_{n\to\infty}a_n$ for some Cauchy sequence $a_n$ of rationals), there is an interesting result(proposition $6.1.15$) that: A Cauchy sequence $(a_n)_{n=1}^\infty$ of rational numbers converges to a real number of the form $L.I.M_{n\to\infty}a_n$.

This was the idea behind defining real numbers using Cauchy sequences of rational numbers. Namely, the underlying idea being that we identify a real number with the limit of a Cauchy sequence of real numbers. But we can't define a real number as the limit of a Cauchy sequence of real numbers as the definition of convergence requires a real number and furthermore requires the ability to do arithmetic operations on real numbers. Where all the operations of real numbers of the form $L.I.M_{n\to\infty}a_n$ are proved in chapter $5$.