intuitionistic analysis In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics. This contrasts with ''classical analysis'', which (in this context) simply means analysis done according to the (more comm ...

and in

computable analysis In mathematics and computer science, computable analysis is the study of mathematical analysis from the perspective of computability theory. It is concerned with the parts of real analysis and functional analysis that can be carried out in a compu ...

, indecomposability or indivisibility (german: Unzerlegbarkeit, from the adjective ''unzerlegbar'') is the principle that the continuum cannot be partitioned into two nonempty pieces. This principle was established by

Brouwer Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'. Brouwer * Adriaen Brouwer (1605–1638), Flemish painter * Alexander Brouwer (b. 1989), Dutch beach volleyball player * Andries Br ...

in 1928 English translation of §1 see p.490–492 of: using intuitionistic principles, and can also be proven using Church's thesis. The analogous property in classical

analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...

is the fact that every continuous function from the continuum to 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 Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...

(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 In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...

, 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 constructive set theory (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.

References

* * *{{cite book , first=Michael , last=Rathjen , chapter-url=http://www.maths.leeds.ac.uk/~rathjen/tklracend.pdf , chapter=Metamathematical Properties of Intuitionistic Set Theories with Choice Principles , title=New Computational Paradigms , editor1-last=Cooper , editor2-last=Löwe , editor3-last=Sorbi , publisher=

Springer Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinationa ...

, location=New York , isbn=9781441922632 , year=2010 Constructivism (mathematics)

See also

References