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(Throughout, triviality is taken to be in the sense of the first meaning given by the answer by 5xum to the question What defines "triviality"?)

There is, of course, no strict definition of trivial cases in mathematics, and as far as I know none for degenerate cases either. However, people seem to have some sort of intuitive sense for cases of triviality, e.g. the set of the identity element, the factors of a prime number, the indiscrete topology etc. Examples of degenerate cases include the circle of radius zero, or perhaps a counting system with radix 0, I would call these degenerate.

The case of the empty set seems to me degenerate, but is often cited as an example of a trivial case. Not that the two have to be mutually exclusive, but what then is the relationship between the two? Triviality occurs in cases which can be in a sense "taken for granted" or are explicitly stated in the definition of the objects in question, and degeneracy occurs in cases which are "useless" to what we are studying, and I could fairly easily convince myself that every example of degeneracy is also an example of triviality (I can't think of any examples where I would call something degenerate but not concede that it was also pretty trivial).

Is this a fair conclusion? Is calling a case degenerate stronger than calling it trivial in a meaningful (if not explicitly definable) way?

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  • $\begingroup$ My intuitive feeling is that trivial cases obey some general pattern being studied, but in ways that are immediately obvious from the definitions. While degenerate cases are "corner" cases where the usual theory breaks down and special-case arguments must be employed to study them. $\endgroup$ – Tob Ernack Jul 18 '17 at 2:58
  • $\begingroup$ Things are trivial when they are "like $1$" and degenerate when they are "like $0$". $\endgroup$ – Oscar Cunningham Jul 18 '17 at 23:51
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"trivial Similar to clearly. A concept is trivial if it holds by definition, is immediate corollary to a known statement, or is a simple special case of a more general concept." see: en.wikipedia.org/wiki/List_of_mathematical_jargon

en.wikipedia.org/wiki/Degeneracy_(mathematics) " In mathematics, a degenerate case is a limiting case in which an element of a class of objects is qualitatively different from the rest of the class and hence belongs to another, usually simpler, class. Degeneracy is the condition of being a degenerate case."

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Both words trivial and degenerate, like all other words in mathematics, have precise meanings.

Trivial means "has a most simple structure." Both trivial proofs and trivial objects (e.g. the group of one element) have most simple structures possible. The group of one element is trivial because there's only one interpretation you can give it, making it the most simple possible because that's the most extreme kind of simplicity you can have. Meanwhile, the group of two elements is not trivial (although it is still simple) because it can have multiple interpretations: it can be bits under bitwise xor, it can be the multiplicative group of $\mathbb{Z}/4\mathbb{Z}$, or it can be the group of $\pm 1$ under multiplication. And none of these three cases, although isomorphic, can be interpreted in the same way.

A degenerate object X of a general class of object (henceforth called A) is an object that meets the following two conditions:

  1. X is either a special case or limiting case of the objects in A.
  2. There is another general class of objects B such that $X \in B$ and $B$ is more simple than $A$.

Let's take the class of triangles. This is a general class of objects A. Let's inspect the case of a line segment with a distinguished point. This is the degenerate object X. Does it meet the two condtions?

  1. Is it a special case or limiting case of the objects in A? Yes! It is the limiting case of a triangle when one of the angles is 180 degrees.

  2. Is it a member of another general class of objects B that is simpler than A? Yes! Let B be the set of line segments with a distinguished point. The result that X is an element of B is trivial (because it's the result "a line segment with a distinguished point is a line segment with a distinguished point," which is a tautology, which is a most simple type of result) and B is a simpler set than A because it lacks area, chirality, and other structures triangles have.

Because it meets both of these criteria, therefore the line segment with a distinguished point is a degenerate case of a triangle.

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