Where does set collide?

A collection of distinct objects considered as a single whole, its members being anything definite and clearly told apart from one another. Cantor's 1874 Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen began the study of such collections by proving that the real numbers cannot be listed the way the counting numbers can, and his 1895 Beiträge zur Begründung der transfiniten Mengenlehre gave the working definition of a set (Menge) as any gathering of definite, distinct objects of thought into a whole. The referent is an abstract collection whose only structure is which objects belong to it, not a group of pitches heard in music and not a readiness fixed in a person's mind.[1][2]

Unit

One collection of distinct members, identified entirely by which objects belong to it

Recovery means

Not applicable

Measured by

Specified by listing its members or by a defining property, and compared by size through one-to-one matching (cardinality)

A readiness to approach a task in a fixed, habitual way, carried over from what worked before even when it no longer fits. Luchins's 1942 Mechanization in problem solving: The effect of Einstellung showed this with water-jar problems: people who had learned one solution method kept applying it to later problems that had a simpler answer, and many never noticed the shorter route. The referent is a mental predisposition inside a person, read from how it steers behavior, and it is not an abstract collection of objects in mathematics and not a group of pitches in music.[3]

Unit

One person's fixed readiness to respond to a class of tasks in the same learned way

Recovery means

The fixation eases when a person is prompted to reconsider the task, but it tends to persist without such a cue

Measured by

Operationalised with tasks like the water-jar problems, scoring whether a person keeps using a learned method when a simpler one is available

An unordered collection of pitch classes, the distinct note names of the chromatic scale treated apart from octave and repetition. Babbitt's 1961 Set Structure as a Compositional Determinant carried the mathematical idea of a set into the analysis of twelve-tone and atonal music, and Forte's 1973 The Structure of Atonal Music built it into the pitch-class set and the catalogue of set classes now used to compare chords and motifs. The referent is a group of pitch classes studied for its interval content, not the abstract mathematical collection it takes its name from and not a readiness fixed in a listener's mind.[4][5]

Unit

One unordered group of pitch classes, drawn from the twelve of the chromatic scale

Recovery means

Not applicable

Measured by

Described by its normal form, prime form, and interval-class vector, and grouped into set classes for comparison