What are the real numbers, really? - by Joel David Hamkins

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2024-05-13 02:30:04

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.

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