In intuitionistic analysis and in computable analysis, indecomposability or indivisibility (, from the adjective ''unzerlegbar'') is the principle that the continuum cannot be partitioned into two nonempty pieces. This principle was established by Brouwer 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 (38 ...

is the fact that every

continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

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 (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 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 , access-date=2008-05-14 , archive-date=2011-05-19 , archive-url=https://web.archive.org/web/20110519065906/http://www.maths.leeds.ac.uk/~rathjen/tklracend.pdf , url-status=dead Constructivism (mathematics)

See also

References