Definition
Intuitionism
Intuitionism, founded by L. E. J. Brouwer, is a philosophy of mathematics that regards mathematical objects as arising through mental construction and mathematical assertions as requiring constructive justification.
Establishing
requires constructing a witness and proving ; establishing requires establishing one of the two alternatives. Consequently, the tertium non datur is not accepted universally: a construction of either or is required. Intuitionistic logic formalises this constructive reasoning.
Example
Continuum hypothesis and excluded middle
The continuum hypothesis (
) states that no set has a cardinality strictly between those of the natural numbers and the real numbers . Let ZFC denote Zermelo–Fraenkel set theory with the axiom of choice. Assuming ZFC is consistent,
is independent of its axioms: Here
means “does not prove”. Yet classical excluded middle gives This proves the disjunction without proving either alternative. Under the intuitionistic reading, justifying the disjunction requires a construction establishing one alternative; excluded middle alone supplies neither.
Independence is relative to ZFC. It does not establish that
can never be settled by other principles, or that intuitionism proves its negation.