# Trivial vs degenerate cases

(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?

• 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. Jul 18, 2017 at 2:58
• Things are trivial when they are "like $1$" and degenerate when they are "like $0$". Jul 18, 2017 at 23:51
• From another point of view, an object is trivial if you specifically exclude it from your proof, and degenerate if you specifically include it. Jan 5, 2021 at 7:37

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.

1. Things are trivial when they are "like 1" and degenerate when they are "like 0".

2. From another point of view, an object is trivial if you specifically exclude it from your proof, and degenerate if you specifically include it.

... as a point of departure: triviality and degeneracy thus appear as reciprocals or inverses of each other in some sense, or as exemplars of category-theoretic duality. But triviality seems to not be the only reciprocal of degeneracy: a degenerate polygon, for example, has too little of something that other polygons have, whereas then we can imagine something having too much of the relevant content and so speak of obgenerate cases as well. Now, in line with the idea of numerical associates of triviality and degeneracy, let us say:

• Apply the concept of triviality to sets of solutions to problems in general, or category-theoretically, take something like Problem and go to empty, trivial, and degenerate problems (objects in the category) with, in turn, empty, trivial, and degenerate solutions. Differentiate between (A) quantitative triviality and (B) qualitative triviality: (A) occurs when a solution-set is too large, i.e. has absolutely infinitely many solutions in it (with or without a comparable filter; see the example of ZFC powerset forcing below), and (B) occurs when a solution is absolutesimally close to a minimum non-empty solution. An absolutesimal will be a surreal number such as $$\frac{1}{\textbf{No}}$$ for the proper class $$\textbf{No}$$ of surreals (we might also work with $$\frac{1}{ORD}$$, etc.), i.e. an infinitesimal that is absolutely infinitely small. In this way, we can hold to the reciprocity of the two types of triviality, in advance of analyzing the problem of characterizing degeneracy more generally too.

So, a tangent: just to illustrate what is "at stake" with respect to quantitative triviality, go to ZFC and observe that the natural powerset can be forced to equal a proper class' worth of alephs. So far, then, ZFC offers a quantitatively trivial number of solutions to the powerset question. However, in ZFC, we can rule out e.g. $$\aleph_{\omega} = 2^{\aleph_0}$$, or if we force $$2^{\aleph_0} = 2^{\aleph_1}$$, then we can also rule out $$\aleph_{\omega_1} = 2^{\aleph_0}$$, etc. Also, there is a theory other than ZFC where the natural powerset can be inflated to proper-class scale (see Matthews[21], e.g. pg. 34), so that such an inflation is blocked in ZFC lends some qualitative nontriviality to the theory, here.

Insofar as degeneracy resembles triviality, then, but is also opposed by obgeneracy, and an obgenerate case has "too much" of something that a degenerate case has "too little" of, we can assimilate the two conditions to the numerical associates of the two types of triviality indicated (we make no claim that these are the only types of triviality, although they are perhaps the most often considered): again, a degenerate case is "like 0" in a way that an obgenerate case is like unrestricted infinity.K

Then:

• Empty cases $$\rightarrow$$ 0
• Degenerate cases $$\rightarrow$$ 0, $$\frac{1}{\textbf{No}}$$
• Trivial Cases $$\rightarrow$$ $$\frac{1}{\textbf{No}}$$, 1
• Nontrivial cases $$\rightarrow$$ 1, 1+
• Obgenerate cases $$\rightarrow$$ 1+, $$\textbf{No}$$

KPhilosophically, we might say that with respect to the so-called antinomies of pure reason, each thesis represents a degenerate solution, each antithesis an obgenerate one, in line with Kant's talk of various solutions as either too small or too large for "the" understanding.