Why mathematicians treat 'infinity' as an actual destination
In short: In standard arithmetic, infinity is a direction—a process of growing without end. But in the extended real number system, it becomes a specific point on the map. By treating these potential infinities as actual values, mathematicians unlock powerful tools for calculus and measure theory that would otherwise remain unreachable.
The standard real number system is bounded by its own limitations; it cannot contain a value larger than every other real number. To solve this, the extended real number system (denoted as $\overline{\mathbb{R}}$) adjoins two specific elements: $+\infty$ and $-\infty$. This transformation allows mathematicians to treat infinitely increasing sequences or infinitely decreasing series as 'actual' infinities rather than just ongoing processes.
In practical application, this isn't just a philosophical quirk; it is a computational necessity. In calculus and mathematical analysis, using these actual limits significantly expands possible computations. For instance, the sequence of natural numbers has no upper bound in standard reals, but in the extended system, $+\infty$ serves as its least upper bound and limit. This framework also enables a formal definition for 'limits at infinity,' allowing functions like $1/x^2$ to be evaluated consistently as they approach zero or grow infinitely large.
The utility of $\overline{\mathbb{R}}$ is most profound in measure theory and integration. It allows for the description of sets with infinite measure and integrals that result in infinite values, such as $\int_{1}^{\infty} dx/x$. The monotone convergence theorem and the dominated convergence theorem lose their meaning if functions are prohibited from reaching infinite values. Furthermore, $\overline{\mathbb{R}}$ possesses the property of compactness, meaning every subset has a supremum and an infimum, which simplifies complex topological definitions into more manageable ones.
It is worth noting that there is a distinct 'projectively extended real line' where $+\infty$ and $-\infty$ are not distinguished. In that specific model, there is only one actual infinity for both increasing and decreasing sequences. However, the standard extended real number system remains the primary tool for most analysis because it preserves the ordering of numbers while providing a home for the infinite.
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- Summary by Intellect · source: Extended real number line
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