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A set S is finite if there exists a bijection between S and {1, 2, ..., n} for some natural number n, or S is empty. Equivalently, S is finite if and only if there is no injection from S into any proper subset of S. This purely set-theoretic definition of finiteness works without relying on prior notion of 'natural number'.
You know how to describe set membership, how injections and surjections work, and how counting principles apply to finite collections. But notice a circularity lurking in everyday reasoning: when we say a set is finite because it has "a certain number of elements," we are presupposing the natural numbers. In axiomatic set theory, the natural numbers must themselves be *constructed* from sets. So we need a definition of finiteness that does not already assume we have the natural numbers in hand — one that is purely structural.
The standard definition says: S is finite if there exists a bijection between S and the initial segment {1, 2, ..., n} for some n, or S is empty. This definition uses the natural numbers, but it uses them only as a measuring stick — once we construct ℕ inside set theory (which can be done), this definition is available. The bijection requirement makes counting precise: S is finite if its elements can be listed without repetition or omission in a list of length n. Your knowledge of injections (no repetition) and surjections (no omission) makes the bijection condition exact.
There is a second, more internal characterization that avoids reference to specific numbers: S is finite if and only if there is no injection from S into any proper subset of S. This captures the defining property of finite sets — you cannot pair all their elements with a strict sub-collection. Contrast with infinite sets: the even numbers inject into the natural numbers (via n ↦ n), and the natural numbers include the even numbers as a proper subset. For finite sets, any attempt to map all elements into fewer elements must produce a collision — this is the pigeonhole principle, which is provable from the injection characterization using mathematical induction. The two definitions are equivalent (given induction), and together they reveal that finiteness is a structural property of how a set relates to its own subsets.
Why does this matter beyond pedantry? In axiomatic set theory (ZFC), the definition of finiteness underlies the entire theory of cardinality: the distinction between finite and infinite, the definition of countably infinite sets, and the hierarchy of infinite cardinal numbers all depend on getting finiteness right. Moreover, in models of set theory without the axiom of choice, the two definitions can come apart — Dedekind-finiteness (no injection into a proper subset) and Tarski-finiteness (bijects with some {1,...,n}) are equivalent only given choice. This subtle divergence, invisible in everyday mathematics, is the kind of phenomenon that a rigorous definition makes visible. The definition of finite sets is the foundation from which all of infinite combinatorics and cardinal arithmetic grows.