basic constructions:
strong axioms
A large cardinal is a cardinal number that is larger than can be proven to exist in the ambient set theory, usually ZF or ZFC. Large cardinals arrange themselves naturally into a more or less linear order of size and consistency strength, and provide a convenient yardstick to measure the consistency strength of various other assertions that are unprovable from ZFC.
Set theorists often adopt the existence of certain large cardinals as axioms in the foundation of mathematics.
Here is a diagram showing the relation between these:
A general axiomatic framework for large cardinal axioms is proposed in
Arthur Apter, Carlos Diprisco, James Henle, William Swicker, Filter spaces: towards a unified theory of large cardinal and embedding axioms, Annals of Pure and Applied Logic Volume 41, Issue 2, 6 February 1989, Pages 93–106
Arthur Apter, Carlos Diprisco, James Henle, William Swicker, Filter spaces. II. Limit ultraproducts and iterated embeddings, Acta Cient. Venezolana 40 (1989), no. 5-6, 311–318.