The description of a rāga as a “non-constructible set in music” is essentially useless, because if you follow the source that is referred to on Wikipedia, and its citations for the phrase “non-constructible set”, you arrive at things like these (1, 2, 3, 4) from which, after unpacking a fair bit, it turns out to just mean roughly “something hard to define”.
However, the source that Wikipedia quotes for the “non-constructible set” actually has a great description of what its author intended. The source is the article by Wim van der Meer titled “Audience Response and Expressive Pitch Inflections in a Live Recording of Legendary Singer Kesar Bai Kerkar”, in a book Expressiveness in Music Performance. This is what it says:
Raga is sometimes described as a set of melodic rules to which the musician must adhere, but it is now generally accepted that this is too narrow a viewpoint. Perhaps a raga is best described as a non-constructible set along the same lines as language is seen as a non-constructible set by Kortlandt (1983, 2003) and Van Driem (2004, 2007).
(These citations are the 4 articles I linked above.) But it goes to say something more helpful:
In fact, in India, raga is often seen as an entity in the sense of a living being, a creature, an organism. And what the artist does is to bring it to life in front of the audience. Undoubtedly, expressivity in Indian music has much to do with this very conception of raga. When a musician manages to touch the core of a raga, and to portray its salient characteristics, this surely elicits a strong response from the audience, possibly with specific interjections such as “yeh [raag] hai” (this is the raga). is is not an analytical or intellectual process; the listener does not sit comparing the notes being performed with a formal definition of the characteristics of a particular raga. Instead, the experience is immediate and intuitive in the Bergsonian sense (Bergson 1889), in that the audience “knows” the raga without any intervention of reason. Many members of the audience may know the raga without being able to identify it by name; they are familiar with its melodic patterns or gestalt, in much the same manner that we may know or recognize a person but not their name. The comparison to people is drawn also for the immediate nature of the cognition of ragas, in the sense that when we come across a person in the street we recognize them without resorting to a table of physiognomies.
That's fair enough, at least as a sociological description of how listeners experience rāgas (even if it's not useful to other musicians as a technical definition), and explains what the author meant by “non-constructible set”.
The rest of the Wikipedia section (permanent link) has more useful things to say about rāgas too.
[…] Jones translated it as "mode" of European music tradition, but Willard corrected him […] a Raga is both mode and tune. […] Stern refined this explanation to "the raga is more fixed than mode, less fixed than the melody, […] richer both than a given mode or a given melody; it is mode with added multiple specialities"
[…] the concept has no direct Western translation […] Raga is a fusion of technical and ideational ideas found in music, and may be roughly described as a musical entity that includes note intonation, relative duration and order, in a manner similar to how words flexibly form phrases to create an atmosphere of expression. […]
[…] A raga has a given set of notes, on a scale, ordered in melodies with musical motifs. […] a rasa (mood, atmosphere, essence, inner feeling) that is unique to each raga. […] For most artists, their basic perfected repertoire has some forty to fifty ragas. […]
[…] The goal of a raga and its artist is to create rasa (essence, feeling, atmosphere) with music […]
[…] defined Raga as a "tonal framework for composition and improvisation." […]