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In axomiatic set theory one can define functions $f : C \to D$ between any two classes $C, D$ (even if one or both of them are proper classes).

Let $C$ be a proper class and $D$ be a set. Consider all constant functions $f : C \to D$. Is it legit to say that these functions form a set?

There is a $1$-$1$-correspondence between these function and the elements $d \in D$ which suggests that these functions form a set. On the other hand, these functions can formally can be regarded as product classes $C \times \{d\}$ with $d \in D$, thus they cannot be elements of a set.

Is there any variant of the axioms of set theory which allows to form a set of the above constant functions?

Moreover, I think it is okay to form the class $\mathcal C(C,D)$ of constant functions. Then we have a bijection between this class and the set $D$. It seems weird to have a bijection between a proper class and a set.

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    $\begingroup$ There is a one-to-one correspondence of the constant functions $D\to C$ with the elements of $C$, not with the elements of $D$. Is that a typo? $\endgroup$ Dec 3, 2023 at 16:40
  • $\begingroup$ (My guess is OP intended to say "Consider all constant functions $f: C \to D$", which would fix both problems in addition to the fact that $f$ is supposed to be "to a set" and fits with the usual "alphabetical order" convention) $\endgroup$ Dec 3, 2023 at 17:15
  • $\begingroup$ @spaceisdarkgreen There was a typo which I corrected. Izaak van Dongen's guess was correct. $\endgroup$ Dec 3, 2023 at 23:28
  • $\begingroup$ Ther are only set-many such functions, but the domain of each is a proper class, thus so is each such function — too big to be a member of anything. $\endgroup$
    – BrianO
    Dec 7, 2023 at 22:42

1 Answer 1

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The main issue here is that the "collection" of all constant functions $C\to D$ (when $C$ is a proper class and $D$ is a set) is not even a class. Let me describe how two common theories (fail to) deal with this "collection".

In ZFC, where every object is a set, proper classes and class functions are not actually first-class objects. Instead, they are meta-theoretic objects. If I write down a property $P$, I can conceive of the class of all objects satisfying $P$, but this class doesn't exist as an object itself (unless I can prove that there is a set whose elements are exactly the objects satisfying $P$). Now given two classes (defined by properties $P$ and $Q$, I can describe a way of assignment a unique object satisfying $Q$ to each object satisfying $P$, but this assignment (called a class function) does not exist as an object itself (unless $\{x\mid P(x)\}$ and $\{x\mid Q(x)\}$ are sets). Thus "the collection of (constant) class functions $C\to D$" is not even a class, since its elements are not first-class objects. At the meta-theoretic level, we can write down a particular class function and prove it's constant. But we don't have a way of quantifying over classes in ZFC, so "all constant class functions" doesn't even make sense. And given a particular class function, it may not even be clear whether it is constant - for example, I could write down a class function whose behavior depends on whether the Continuum Hypothesis is true, in such a way that we cannot prove whether or not it is a constant function.

In NBG, both sets and proper classes exist as first-class objects, so if $X$ and $Y$ are classes, I can treat a class function $f\colon X\to Y$ as a first-class object. But if either $X$ or $Y$ is a proper class, $f$ will be a proper class, and hence $f$ cannot be an element of a class. So again, there is no object (set or class) whose elements are class functions $X\to Y$.

In both theories, the question "is the collection of all constant class functions $C\to D$ a set?" does not even make sense, because there is no such collection.

On the other hand, assuming $D$ is a set, we do have a way of talking about "the constant class function with value $d$" for each $d\in D$.

For example, in ZFC, we can use a parametric definition: the constant function $F_d$ is defined by $y = F_d(x) \iff y=d$. We can prove that for all $d\in D$, this definition gives a class function $C\to D$, and if $d\neq d'$, $F_d\neq F_{d'}$. Moreover, for each class function $G\colon C\to D$, we can prove (as a theorem schema) that if $G$ is constant, then there exists $d\in D$ such that $G = F_d$. Taken together, all this allows us to think of the "collection" of constant class functions as being in a kind of meta-theoretic bijection with the elements of $D$. We can use the elements of $D$ as definable "codes" for constant class functions, and the collection of codes is a set ($D$ itself). This will be sufficient in practice to treat the "collection" of constant class functions $C\to D$ as if it were a set.

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  • $\begingroup$ Thank you for this profound answer. I understand that in the standard axiomatic frameworks the "collection" of constant class functions $C→D$ is not a class, let alone a set. Neverheless there is a "stale taste" that a solution requires mental acrobatics to cover something which seems so obvious. But "seeming obvious" is of course not a reasonable concept. $\endgroup$ Dec 3, 2023 at 23:50
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    $\begingroup$ @KritikerderElche There is still one "out", but it's not widely used these days. If you invent a theory with three kinds of object - sets, arbitrary classes of sets, and arbitrary collections of classes - your desired collection exists as the final kind only. You're welcome, of course, to have infinitely many layers like this. And that's the point of stratification. $\endgroup$
    – J.G.
    Dec 3, 2023 at 23:55
  • $\begingroup$ @J.G. Thank you, interesting point of view! $\endgroup$ Dec 4, 2023 at 9:02
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    $\begingroup$ Another common approach, along the lines of @J.G.'s comment, and which is widely used these days: Assume we have a whole heirarchy of universes. Agree to work in some universe $U$, so that every time we would say "set", we say "$U$-small set", i.e., "element of $U$". Now among the subsets of $U$, some are $U$-small, but others are not - the non-$U$-small subsets are the analogue of proper classes. But these are still sets, and in fact they are $U'$-small for some larger universe $U'$. $\endgroup$ Dec 4, 2023 at 14:27
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    $\begingroup$ This trick of allowing ourselves to move to larger universes is a nice bookkeeping device that avoids having to introduce new names for objects at each level ("set", "class", "hyperclass", "conglomerate", whatever...) @KritikerderElche See also here and here $\endgroup$ Dec 4, 2023 at 14:31

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