Infinity

Infinity is an unbounded quantity or extent, with no finite limit. In mathematics, it can describe processes that grow without bound or completed infinite collections.

Limit (mathematics): A value that a function or sequence approaches as its input or index approaches a specified point. Limits describe how finite values can approach infinity without ever reaching it.
Infinite series: A sum with infinitely many terms, whose value is defined through the limit of its partial sums. Series turn infinitely many additions into a finite value when their partial sums converge.
Finite set: A set whose elements can be counted by a natural number. Finite sets provide the direct opposite of infinite collections.
Zeno's paradoxes: A set of ancient Greek arguments claiming that motion or plurality leads to contradiction. Zeno's puzzles made infinite division central to debates about motion and quantity.
Unboundedness: The property of a set or function whose values exceed every finite bound. Unboundedness captures one central way infinity appears in analysis.
Convergence: The tendency of a sequence or series toward a finite limit. Convergence explains how an infinite process can yield a bounded result.
Countable set: A set whose elements can be matched one-to-one with a subset of the natural numbers. Countability identifies the smallest infinite size shared by the integers and rational numbers.
Euclid's theorem on primes: The theorem that there are infinitely many prime numbers. Euclid's proof is an early rigorous demonstration that a collection can be infinite.
Cardinality: A measure of the number of elements in a set, including infinite sets. Cardinality distinguishes infinite collections by their sizes.
Divergence: The failure of a sequence or series to approach a finite limit. Divergence marks processes whose values may grow without bound or fail to settle.

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