What is a real number?
Let us consider the real continuum. The classical discovery of irrational numbers reveals gaps in the rational number line: the place where √2 would be, if it were rational, is a hole in the rational line. Thus, the rational numbers are seen to be incomplete. One seeks to complete them, to fill these holes, forming the real number line ℝ.
Please enjoy this free extended excerpt from Lectures on the Philosophy of Mathematics, published with MIT Press 2021, an introduction to the philosophy of mathematics with an approach often grounded in mathematics and motivated organically by mathematical inquiry and practice. This book was used as the basis of my lecture series on the philosophy of mathematics at Oxford University. Lectures on the Philosophy of Mathematics, MIT Press 2021
Dedekind cuts
Dedekind (1901, I.3) observed how every real number cuts the line in two and found in that idea a principle expressing the essence of continuity:
If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions. —Dedekind, 1901
For Dedekind, the real numbers are what we now call Dedekind complete: every cut is filled. In the rational line, some cuts, determined by a rational number, are already filled; but other cuts correspond to holes in the rational line, not yet filled. For any such unfilled cut, Dedekind proposes that we may imagine or “create” an irrational number in thought precisely to fill it. In this way, we shall realize the real number line as the Dedekind-completion of the rational number line.
And if we knew for certain that space was discontinuous there would be nothing to prevent us, in case we so desired, from filling up its gaps, in thought, and thus making it continuous; this filling up would consist in a creation of new point-individuals and would have to be effected in accordance with the above principle. —Dedekind, 1901
Theft and honest toil
Russell explains how one may undertake this creation process explicitly, building the real numbers as a mathematical structure that fulfills Dedekind's completeness property. In a truly elegant construction, he forms the Dedekind-completion of the rational line from the set of all Dedekind cuts themselves, viewing each cut as constituting a single new point. A Dedekind cut in the rational line is a bounded nonempty initial segment of the rationals with no largest element. The no-largest-element requirement ensures that rational numbers are represented uniquely, since otherwise we could place the rational limit point on either side, forming two distinct cuts where only one is wanted.
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