Foundations

Set existence is governed by explicit axioms and schemas.

Upper-bounded nonempty real sets have least upper bounds.

Countability includes finite sets in this convention.

An equivalence relation partitions a set into classes.

Semantic consequence and arithmetic completeness are distinct.

Repeated division computes a greatest common divisor.

Positive integers greater than one have unique prime factors.

Congruence identifies integers differing by a multiple of n.

Coprime moduli specify one combined residue.

A group has associative composition, identity and inverses.

Homomorphisms preserve the group operation.

A ring combines addition with distributive multiplication.

A field allows division by every nonzero element.

Nonconstant complex polynomials factor into linear terms.

A metric supplies a distance satisfying three axioms.

A topology specifies which subsets are open.

Compactness reduces every open cover to a finite one.

Continuity preserves openness through inverse images.

Triangle-angle relations depend on the geometry.

Vector-space laws extend beyond coordinate arrows.

A basis spans without linear redundancy.

Kernel and image dimensions sum to domain dimension.

An eigenvector changes only by a scalar factor.

Self-adjoint operators have orthonormal eigenbases.

Any real rectangular matrix has orthogonal factors.

Real convergence can be checked by distances between late terms.

A series converges through its partial sums.

The remainder quantifies a finite polynomial approximation.

One cutoff controls the error at every domain point.

Integration starts from measurable simple functions.