nLab
large cardinal

Idea

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.

List of large cardinal conditions

• axiom of infinity – a large cardinal axiom relative to finitist theories, but not relative to ZF
• inaccessible cardinal – the smallest sort of large cardinal in ZF, equivalent to the existence of a Grothendieck universe.
• measurable cardinal – the boundary between “small” large cardinals and “large” large cardinals
• elementary embedding – a tool used in the study of large large cardinals
• supercompact cardinal
• extendible cardinal
• C(n)-extendible cardinal
• Vopěnka's principle – a large cardinal axiom with important implications for the behavior of locally presentable categories and accessible categories.

Here is a diagram showing the relation between these:

References

• Wikipedia has a list of large cardinal properties.