Are Real Numbers Real?

Undoubtedly, real numbers are the most fundamental things to describe literally everything we know about our physical world, so questioning if real numbers are real may appear to be as silly as questioning if our world is real. But seriously, no one has ever actually seen irrational numbers, for example $\sqrt{2}$. All we have seen or we could ever possibly see is a floating point approximation of $\sqrt{2}$ like 1.414213562373095, which is a rational number, but not the irrational number $\sqrt{2}$ itself as a whole.  So how do we know $\sqrt{2}$ exists? One may point to the equation $$x^2=2$$ for that. However, writing the solution symbolically as $x=\pm\sqrt{2}$ does not prove anything about the existence of the irrational number $\sqrt{2}$, though if the existence of $\sqrt{2}$ is assumed, it can be shown that $\sqrt{2}$ cannot be a rational number. (See here for the proof.)

Traditionally, there are two ways to construct real (irrational) numbers in mathematics. One is by Dedekind cuts (named after Richard Dedekind) and the other is by Cauchy sequences. The construction by Cauchy sequences was introduced by Georg Cantor.

Let me explain the construction by Dedekind cuts first. Let $A=\{x\in\mathbb{Q}: x^2<2\}$, i.e. the set of all rational numbers whose square is less than 2, and $B=\{x\in\mathbb{Q}:x^2>2\}$, i.e. the set of all rational numbers whose square is greater than 2. Then there is a gap or a hole between the two sets $A$ and $B$, that cannot be filled by a rational number. If $A$ were to have the least upper bound (also called the supremum) or $B$ were to have the greatest lower bound (also called the infimum), the gap could be filled, and the filler is none other than what we call $\sqrt{2}$. So, how do we know the set $A$ has the least upper bound or $B$ has the greatest lower bound? Actually we don’t, but we want it to exist. So here comes an axiom called the L.U.B. (Least Upper Bound) Property (or the Supremum Property): Every infinite set which is bounded above has the least upper bound. Its dual property is the G.L.B. (Greatest Lower Bound) Property (or the Infimum Property): Every infinite set which is bounded below has the greatest lower bound. This is one of the most basic properties regarding real numbers. The axiom is so fundamental that if we were not to accept this axiom, we would’t have had mathematics as we know it.

A sequence $\{x_n\}$ is called a Cauchy sequence if given a positive number $\epsilon>0$ (no matter how small it is), one can find a natural number $N$ such that for all $m,n\geq N$, the distance between $x_m$ and $x_n$ is smaller than $\epsilon$. Suppose that a sequence $\{x_n\}$ converges to $x$. Then one can easily show that $\{x_n\}$ is a Cauchy sequence using the triangle inequality $$|x_m-x_n|\leq |x_m-x|+|x-x_n|$$ How about the converse? Is a Cauchy sequence necessarily a convergent sequence? Often, we introduce the statement “every Cauchy sequence (of real numbers or of complex numbers) is a convergent sequence” in class as if it were a proven theorem. But the truth is that it is not a theorem because we can’t prove that it is true (without assuming an axiom such as the l.u.b. property), though we want it to be true for a good reason. Georg Cantor introduced the statement that “every Cauchy sequence is a convergent sequence” as an axiom and used it to construct real numbers. For instance, define a sequence $\{x_n\}$ of rational numbers recursively as follows: $$x_1=1,\ x_{n+1}=\frac{x_n}{2}+\frac{1}{x_n}$$ Then $\{x_n\}$ is a Cauchy sequence. By the Cantor’s axiom, it must converge to a number $x$ and it satisfies the equation $x^2=2$, i.e. the solution $\sqrt{2}$ exists.

The rational numbers have density property, meaning between any two rational numbers one can find another rational number (in fact infinitely many rational numbers) in between them, nonetheless  they still have gaps (irrational numbers) and those gaps can be filled by Dedekind cuts or by Cauchy sequences, hence the (real) number system becomes complete or a continuum. The existence of real numbers appears to be inconceivable without having the natural order relation $\leq$. But then another baffling mystery comes in. In set theory, the so-called Well-Ordering Theorem states that “every set can be well-ordered”. A totally ordered set or a chain (a set with an order relation in which any two elements are related by the order) is said to be well-ordered if its every non-empty subset has a least element. Although it is called a theorem, it is actually an axiom. It can be proved by assuming another axiom, for example, Axiom of Choice. According to Well-Ordering Theorem, the set of real numbers can be well-ordered also, though no one has ever found one yet. It’s quite puzzling as to how the set of real number can possibly be well-ordered. By the way, a consequence of Well-Ordering Theorem or Axiom of Choice is Banach-Tarski paradox. What it states is that a solid ball in 3-dimensional space can be decomposed into a finite number of disjoint pieces in such a way that those pieces can be put back together differently via translations and rotations (without changing their shapes) to yield two identical copies of the original ball. Banach-Tarski paradox is pretty counter-intuitive and for that reason it is called a paradox.

As we have seen, the existence of real numbers (irrational numbers) hangs on the axioms like the l.u.b. property or the convergence of Cauchy sequences. So, can we still say real numbers are definitely real? You may say yes if your notion of reality is that of Platonism or of Max Tegmark’s M.U.H. (Mathematical Universe Hypothesis): Platonism in hyperdrive. You may be less convinced, however, if your notion of reality is physical one in usual sense, that can be observed or tested. In quantum field theory, particles are treated as mathematical points and that is believed to be the source of divergence issues arising from quantum field theory. Superstring theory is an attempt to reformulate quantum field theory by replacing point particles by vibrating strings or membranes (in M-theory, M stands for Membranes or Mother meaning the Mother of all string theories). Physicists also realized that to extend the notion of particles from mathematical points in a physically consistent manner, the spacetime itself needs to be quantized (i.e. discrete) as well. Such an attempt to reformulate quantum field theory by quantizing the spacetime goes back to Yukawa Hideki’s elementary domain theory and a notable modern attempt is loop quantum gravity. Someday, we may find out that the spacetime is actually quantum rather than continuum. If and when that happens, the reality of real numbers and continuum could very well be in question.

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