Indecomposability
In constructive mathematics, indecomposability or indivisibility is the principle that the continuum cannot be partitioned into two nonempty pieces. This principle was established by Brouwer in 1928 using intuitionistic principles, and can also be proven using Church's thesis. The analogous property in classical analysis is the fact that any continuous function from the continuum to is constant.
It follows from the indecomposability principle that any property of real numbers that is decided is in fact trivial. Conversely, if a property of real numbers is not trivial, then the property is not decided for all real numbers. This contradicts the law of the excluded middle, according to which every property of the real numbers is decided; so, since there are many nontrivial properties, there are many nontrivial partitions of the continuum.
In CZF, it is consistent to assume the universe of all sets is indecomposable—so that any class for which membership is decided is either empty or the entire universe.
