6  Axioms and Definitions

The last three chapters used the words “definition” and “well-defined” freely without ever pinning down what they mean. That was deliberate — it’s much easier to explain what a definition is after you’ve seen several, the way it’s easier to explain what a sentence is after you’ve read some. Now that sets, relations, and functions are on the table, it’s worth stepping back and being precise about the scaffolding they were built with.

6.1 Definitions don’t assert, they abbreviate

A definition introduces a new term as shorthand for something already expressible in terms you have. “A number is even if it equals \(2k\) for some integer \(k\)” doesn’t discover a new fact about the world — it’s a naming convention, and the only thing that can go wrong with it is ill-formedness (does “even” pick out a well-defined set of numbers, with no ambiguity about which numbers are in it?), not falsehood. This is worth being explicit about because it’s easy to treat a new definition as though it were a claim requiring justification the way a theorem does. A definition requires only that it be coherent — self-consistent and unambiguous — not that it be true.

That said, definitions are not arbitrary. Mathematicians choose which concepts to name based on which combinations of properties turn out to recur, connect to other ideas, or make an important theorem statable cleanly. “A number divisible by 2” got a name; “a number whose digits sum to a multiple of 7” didn’t, despite being an equally legitimate definition — because the second property essentially never reappears anywhere useful. Naming is a judgment call about which patterns are worth having a handle for, made after the fact by an entire mathematical community, not before.

6.2 Axioms don’t get proven — they get assumed

An axiom is a starting statement, accepted without proof, from which other statements are derived. Unlike a definition, an axiom does assert something — but the assertion isn’t backed by a proof from more primitive facts, because there’s nowhere more primitive left to appeal to. Every formal system needs at least a few: Logic and Proof already leaned on this without naming it — a proof is a chain of inference starting somewhere, and axioms are that somewhere.

This is exactly where the word “true” gets subtle for the rest of this site. Whether the parallel postulate holds isn’t a fact about the physical universe waiting to be discovered — Euclidean geometry (assuming it) and hyperbolic geometry (assuming its negation) are both fully consistent formal systems, and asking which one is “really true” outside the context of a specific application (which geometry describes spacetime, say) is a category error. Modern mathematics generally treats axioms as chosen: you pick a set of axioms, and every downstream theorem is true relative to that choice — “\(P\)” in mathematics almost always means “\(P\), given these axioms,” with the given part left implicit because it rarely changes within a single piece of work.

A theorem is then any statement derivable from the axioms via valid inference (the rules from Logic and Proof). The whole edifice — axioms plus everything derivable from them — is what “a mathematical theory” formally means, distinct from the looser, more common sense of “theory” this site uses elsewhere to mean a field of study like linear algebra.

6.3 Why these particular axioms?

This is the question this site returns to more than any other, starting with Groups and Vector Spaces: given a structure, why these axioms and not some other list? Two forces usually decide it. First, necessity — drop an axiom and some construction you actually need stops working (drop the existence of inverses from a group and you can no longer “undo” a combination). Second, fruitfulness — a minimal, well-chosen axiom list turns out to have surprisingly rich and useful consequences, which is discovered gradually as a community works with a structure, not decided in advance. Axioms that seem natural in hindsight were often arrived at only after decades of specific examples were already in circulation — the historical graph, again, running behind the conceptual one.

6.4 A formalist century, and its limit

The push to make “derivable from stated axioms” the entire standard for mathematical truth — rather than an ideal mathematicians approximated informally — is largely a late-19th/early-20th-century project, closely tied to the response to Russell’s paradox in Sets. David Hilbert’s program (from around 1920) aimed higher still: not just axiomatizing each theory, but proving, using only elementary methods, that mathematics as a whole was consistent and that every true statement was provable. Kurt Gödel’s incompleteness theorems (1931) showed this second, larger goal is unreachable in principle — any consistent formal system powerful enough to describe basic arithmetic contains true statements it cannot prove. This is a genuinely deep result and this paragraph does not do it justice; it’s mentioned here mainly so “just add more axioms until everything is settled” doesn’t seem like an available option later in this site, and flagged, like the rest of this chapter’s history, as simplified.

6.5 Where this leads

With logic, sets, functions, and now the axiomatic method itself named, Number Systems puts all of it to work: constructing \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\), and \(\mathbb{C}\) from nothing but \(\mathbb{N}\), and showing what each construction buys you.