= Foundations of mathematics
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Foundations of mathematics asks what our objects mean, what we assume about them, and why an argument establishes its conclusion. <Set theory> gives a common language for collections and <functions>; <mathematical logic> makes the structure of statements and deductions precise. A <mathematical definition> fixes a meaning, an <axiom> supplies an assumption, and a <mathematical proof> explains what follows. Separating these roles makes results reusable: once the <group> axioms are enough for a conclusion, that conclusion applies to every <group>, regardless of how its members are represented.
The assumptions can change the answer to a familiar calculation. Over the <real numbers>, $x^2=x$ factors as $x(x-1)=0$, so $x$ is zero or one. The same reasoning holds in any <integral domain>, because a product can be zero only when a factor is zero. In the <ring> of <integers> modulo six, however, $3^2\equiv3$ and $4^2\equiv4\pmod6$: there are four solutions, represented by $0,1,3,4$. The extra solutions are <idempotents>, and the failed step involves <zero divisors>. Writing the domain is part of explaining the answer.
This precision also reveals surprising possibilities. A two-dimensional grid of <natural numbers> is a <countable set>, while all infinite <binary sequences> form an <uncountable set>. The <Cantor pairing function> supplies a complete enumeration of the first; <Cantor's diagonal argument> defeats every proposed enumeration of the second. Both arguments are short, yet distinguish two sizes of infinity. The subject's appeal is this ability to turn a vague question into one whose assumptions, construction, and limits can all be inspected.
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