Logical independence

In mathematical logic, a statement S is independent of a theory T if it is impossible to prove S from T and it is impossible to prove not S from T. Many interesting statements in set theory are independent of ZF. It is possible for the statement "S is independent from T" to be itself independent from T. This reflects the fact that statements about proofs of mathematical statements when represented in mathematics become themselves mathematical statements.

Theorems relevant to independence

Kurt Gödel proved the completeness theorem and the incompleteness theorem

The completeness theorem states (Assuming ZFC) A theory T is consistent iff T has a model.

The incompleteness theorem states (Assuming ZF) In any consistent formalization of mathematics that is sufficiently strong to define the concept of natural numbers, one can construct a statement that can be neither proved nor disproved within that system.

Independence results in set theory

The following statements in set theory are known to be independent of ZF:

• The axiom of choice
• The continuum hypothesis and The generalised continuum hypothesis
• The Souslin conjecture
• The Kurepa hypothesis

The following statements (none of which have been proved false) cannot be proved to be independent (but may be so):

• The Axiom of determinacy
• The existence of Strongly inaccessible cardinals
• The existence of Large cardinals