Foundations

Continuity connects differentiation with integration.

A probability assigns nonnegative, normalised, countably additive weights to events.

Conditioning restricts attention to an event of positive probability.

Expected value weights possible values by their probabilities.

A confidence level describes repeated-sampling coverage of an interval procedure.

A bit has two possible states; a fixed-width unsigned binary word encodes an integer.

Shannon entropy measures uncertainty in a specified probability distribution.

No total algorithm decides halting for every program and input in a general computation model.

Big O bounds eventual growth up to a fixed constant factor.

Combine input uncertainty through a measurement model.

Use dimensions to detect impossible quantity equations.

Lorentz transformations preserve a combined space-time separation.

Quantifiers bind variables over a specified domain.

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.