# If it is trivial that 𝘟, is it trivial that it is trivial that 𝘟?

Differentiate between empty, trivial, and nontrivial solutions to problems. From a category-theoretic point of view (or maybe just mathematics/logic historically), one has that empty solutions are specifically correlated with functions mapping to 0, while trivial solutions are specifically correlated with functions mapping to 1 (or "identity"). But further differentiate quantitative from qualitative triviality:

1. A set of solutions is maximally quantitatively trivial if the set is absolutely infinite.
2. A set of solutions is maximally qualitatively trivial if there is no filter on types of solutions in the set.

These conditions are not perfectly distinct; we'll try to show why below. Now, temporarily suspending (1) and (2), consider a generic statement:

1. It is trivial that it is trivial that X.

I don't know that (3) is always true, i.e. I don't know that:

1. If it is trivial that X, it is trivial that it is trivial that X.

For example, take powerset questions in ZFC. On the one hand, there is a quantitatively trivial set of forcing solutions to these questions in that one can force 2κ to equal an absolutely infinite number of things (or, as they put it, there is a proper class of κ+a that 2κ can be forced to equal). However, there are a few qualitative restrictions: cf(2κ) > cf(κ), so for example 20 ≠ ℵω or ℵω+ω, etc. Moreover, in ZFC, the natural powerset can't be inflated to the size of a proper class. Now, there are a proper class of transfinite cardinals that are cofinal with the zeroth aleph, so ZFC does filter out a proper class of forcing solutions to 20 = κ, and generally there are class-many solutions to other such equations that are filtered out for reasons of cofinality.

So, in one sense, if one "works in" a specific model of ZFC where 20 is set to ℵ2, say, it seems possibly trivial (in the model) that 20 = ℵ2. However, it doesn't seem trivial outside the model that the natural powerset can be forced to be so many things. So it doesn't seem as though triviality must always iterate over itself.

But then is triviality relative? Or, we might differentiate between:

1. It is quantitatively trivial that it is quantitatively trivial that X.
2. It is qualitatively trivial that it is qualitatively trivial that X.
3. It is quantitatively trivial that it is qualitatively trivial that X.
4. It is qualitatively trivial that it is quantitatively trivial that X.

And so on and on. I would be tempted to think, per the "too simple to be simple" theme in category theory, that triviality does end up being relative, although that seems to conflict with the idea that a solution-set is trivial when it is absolutely infinite (c.f. the understanding of the explosion argument as one based on avoiding "trivialism," when every proposition is true without distinction).

Notably, Hawkins does contain the phrase "it is trivial that it is trivial" in relation to reflection on Fregeanism, and there is a MathSE post where someone says that some fact about intersections is trivially trivial. The Wikipedia article on mathematical triviality testifies to the relativity of the concept. However, so far as I so far know, distinctions like "quantitative vs. qualitative triviality" do not seem to often, if ever, be explicitly made in mathematics (a Google search's first-page results linked to essays in other disciplines than mathematics, except with respect to something called a "Lipschitz flow-box").

Kantian postscript: I suppose the question is at least similar to asking about the following:

1. If it is analytically true that X, is it analytically true that it is analytically true that X?
2. If it is synthetically true that X, is it analytically true that it is synthetically true that X?
• "Milk and cookies kept you awake?" :-) You come up with the most interesting stuff. Jun 13 at 17:52
• @ScottRowe most of my questions are about parts of a large argument I'm trying to formalize. Sometimes I wonder if a step of the argument is acceptable at all, sometimes I don't know how to frame a step to sound reasonable, sometimes I wish I didn't seem to add more and more steps every month or so. I tried writing it down in full last week, and it came out to about 50 lines, but most of those require a lot more to be interpreted usefully in the first place, so I don't know if it'll ever be resolved. Jun 13 at 18:09
• And I mean, the target conclusion is a bizarre attempt to justify some obscure commentary by Martin Buber, of all people :P Jun 13 at 18:12
• Old mathematician joke: Professor writes a proof on the board. He says "this step is trivial". Then he thought hard for 20 minutes and said "yes, it is really trivial". Jun 13 at 19:15
• If by "relative" you mean context-dependent then it is trivial (pardon the pun) that triviality is context-dependent. The use of 0 is trivial now, it was not at all trivial to create a conceptual framework that made it trivial, as history indicates, the same with negative and imaginary numbers. It is said that mathematics develops by turning theorems into definitions, what used to be a hard result (Pythagorean theorem in Euclidean geometry) becomes trivial in a new setting (inner product spaces). But this is different from asking if triviality obeys the analog of KK rule in a fixed setting. Jun 13 at 19:18