Set theory

Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.

The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.

Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

Θ (set theory)

In set theory, Θ (pronounced like the letter theta) is the least nonzero ordinal α such that there is no surjection from the reals onto α.

If the axiom of choice (AC) holds (or even if the reals can be wellordered) then Θ is simply , the cardinal successor of the cardinality of the continuum. However, Θ is often studied in contexts where the axiom of choice fails, such as models of the axiom of determinacy.

Θ is also the supremum of the lengths of all prewellorderings of the reals.

Proof of existence

It may not be obvious that it can be proven, without using AC, that there even exists a nonzero ordinal onto which there is no surjection from the reals (if there is such an ordinal, then there must be a least one because the ordinals are wellordered). However, suppose there were no such ordinal. Then to every ordinal α we could associate the set of all prewellorderings of the reals having length α. This would give an injection from the class of all ordinals into the set of all sets of orderings on the reals (which can to be seen to be a set via repeated application of the powerset axiom). Now the axiom of replacement shows that the class of all ordinals is in fact a set. But that is impossible, by the Burali-Forti paradox.

Set theory (music)

Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson (1960) in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte (1973), drawing on the work in twelve-tone theory of Milton Babbitt. The concepts of set theory are very general and can be applied to tonal and atonal styles in any equally tempered tuning system, and to some extent more generally than that.

One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered or unordered, and can be related by musical operations such as transposition, inversion, and complementation. The methods of musical set theory are sometimes applied to the analysis of rhythm as well.

Mathematical set theory versus musical set theory

Although musical set theory is often thought to involve the application of mathematical set theory to music, there are numerous differences between the methods and terminology of the two. For example, musicians use the terms transposition and inversion where mathematicians would use translation and reflection. Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences (though mathematics does speak of ordered sets, and although these can be seen to include the musical kind in some sense, they are far more involved).