Indecomposability
Encyclopedia
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 {0,1} is constant.
It follows from the indecomposability principle that any property of real numbers that is decided (each real number either has or does not have that property) is in fact trivial
(either all the real numbers have that property, or else none of them do). 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 (every set is either a member of the class, or else not a member of the class) is either empty or the entire universe.
Partition of a set
In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...
into two nonempty pieces. This principle was established by Brouwer
Luitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan Brouwer FRS , usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis.-Biography:Early in his career,...
in 1928 using intuitionistic principles
Constructivist analysis
In mathematics, constructive analysis is mathematical analysis done according to the principles of constructive mathematics.This contrasts with classical analysis, which simply means analysis done according to the principles of classical mathematics.Generally speaking, constructive analysis can...
, and can also be proven using Church's thesis
Church's thesis (constructive mathematics)
In constructive mathematics, Church's thesis is the mathematical assertion that all total functions are recursive. It gets its name after the informal Church–Turing thesis, which states that every algorithm is in fact a recursive function, but the constructivist version is formal and much...
. The analogous property in classical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
is the fact that any continuous function from the continuum to {0,1} is constant.
It follows from the indecomposability principle that any property of real numbers that is decided (each real number either has or does not have that property) is in fact trivial
Trivial (mathematics)
In mathematics, the adjective trivial is frequently used for objects that have a very simple structure...
(either all the real numbers have that property, or else none of them do). 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
Constructive set theory
Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. That is, it uses the usual first-order language of classical set theory, and although of course the logic is constructive, there is no explicit use of constructive types...
, it is consistent to assume the universe of all sets is indecomposable — so that any class for which membership is decided (every set is either a member of the class, or else not a member of the class) is either empty or the entire universe.