Set theorists use a technique called forcing to obtain independence results and to construct models of set theory for other purposes.

This is also wrong, at least as a formal statement, since it presuming some underlying model of set theory.

These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant.

However, there are models of set theory in which the ordering principle holds while the order-extension principle does not.

Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory.

However, there are various valuable applications of "real" cardinal numbers in various models of set theory.

Certain properties are absolute to all transitive models of set theory, including the following (see Jech (2003 sec.

There are certain large cardinals that cannot exist in the constructible universe (L) of any model of set theory.

Nevertheless, the constructible universe contains all the ordinal numbers that the original model of set theory contains.

So these sets are equinumerous in any model of set theory which includes them.