Mathematical Rigor: Unicity in Defining Irrational Numbers Sparks Debate

A recent tweet from Paul Raymond-Robichaud has ignited discussion within mathematical circles regarding the formal definition of irrational numbers. Raymond-Robichaud asserted that any definition of a mathematical object must adhere to the principle of "unicity," meaning an object cannot be defined by a non-unique property. He specifically criticized the use of "multiple series representations" for irrational numbers as a valid definitional approach, stating, "This is formally invalid. A core requirement for a mathematical definition is unicity. An object cannot be defined by a non-unique property. Since irrationals have multiple series representations, this fails.

Raymond-Robichaud then posed the question, "So how are they defined correctly? First, by construction. (1/3)." This highlights a fundamental aspect of modern mathematics: the rigorous construction of number systems. Irrational numbers are a subset of real numbers that cannot be expressed as a simple fraction (p/q, where p and q are integers and q≠0). Their decimal expansions are non-terminating and non-repeating.

The core of Raymond-Robichaud's argument rests on the idea that while various infinite series can converge to the same irrational number (e.g., different series can represent π or √2), using these series as the definition itself is problematic because the choice of series is not unique. A definition, in a formal mathematical sense, should uniquely identify the object being defined.

Instead, mathematicians typically define irrational numbers through constructive methods. One prominent approach involves Dedekind cuts, where an irrational number is defined by partitioning the set of rational numbers into two non-empty sets. Another method uses Cauchy sequences of rational numbers that converge to the irrational number. These constructions provide a unique and unambiguous way to establish the existence and properties of irrational numbers, forming the bedrock of real analysis. This rigorous approach ensures that the definition is independent of any particular representation, such as a specific infinite series.