Are all empty maps the same?

Question

In set theory, map $f:X\rightarrow Y$ is interpreted to be a subset of the product $X\times Y$ satisfying some properties. If $X=\varnothing$ then $f \subseteq \varnothing\times Y = \varnothing$ and all empty maps are the same regardless of whether they have different codomains $Y$.

However, it is said that it matters what the codomain of a map is. If $f:X\rightarrow Y$ and $f':X\rightarrow Y'$ and the two codomains are different, then $f\ne f'$.

So what gives? If $f:\varnothing\rightarrow Y$ and $f':\varnothing\rightarrow Y'$ are maps, are the two maps equal or not? Does the answer depend on the choice of foundations you use?

Edit: I think at this point, I'm just looking for a citation that defines functions in terms of set theory keeping domains and codomains in mind.

Answer 1

There are two incompatible yet closely related definitions of what a function is. One definition states that a function is a triple $(X,Y,F)$ such that $F\subseteq{X\times{Y}}$ and $\forall{x\in{X}},\exists{y\in{Y}},\{\{x\},\{x,y\}\}\in{F}$ and $\forall{x\in{X}},\forall{y,y'\in{Y}},\left(\{\{x\},\{x.y\}\},\{\{x\},\{x,y\}\}\in{F}\implies{y=y'}\right)$. This definition is more closely aligned with the category-theoretic notion of a morphism in the category of sets. Under this definition, $(X,Y,F)$ and $(X,Y',F)$ are different functions if $Y\neq{Y'}$. As such, under this definition, every empty function is different.

Another definition, more commonly used in most other contexts, is that $F$ itself is the function. Under this definition, $F$ is invariant under a change from $Y$ to $Y'$. This implies that "the" codomain of $F$ is not unique, in this definition: it can be any superset of $F[X]$. As such, under this definition, the empty function, or empty map, is unique, and is just equal to the empty set itself. For this reason, it is common to hear about mathematicians talking about "the" empty function.

Some authors prefer to keep the notion of "function" and "map" as separate and distinct, while still allowing for both definitions to be used. In that case, $F$ is taken to be a function, and $(X,Y,F)$ is taken to be a map, or a morphism, in category-theoretic terms. Having both definitions available but distinct is probably the most useful way to go about this, but the issue is that many other authors use the words "function" and "map" interchangeably.

Answer 2

You could say the same thing about the inclusion $i:\Bbb Z\hookrightarrow\Bbb Q$ compared to the inclusion $i':\Bbb Z\hookrightarrow\Bbb R$ (where I'm assuming $\Bbb Z\subset\Bbb Q\subset\Bbb R$). As sets, they're both given by $\{(x,x) \mid x\in\Bbb Z\}$. Does this make them equal? If you take these to be equal, then you're correct: by your reasoning, all empty maps are the same.

However, if you somehow insist that the above maps should be different, then this should likely also mean that empty maps differ if their codomains differ. One way, I suppose, you could have this is by encoding a map $f:X\to Y$ as the pair $((X,Y),\{(x,f(x)) \mid x\in X\})$ or something.

Answer 3

A map is typically defined as a triple of $\langle D, C, F \rangle$ where $F \in C^D$. Since C differs between the empty maps then it is different.

Answer 4

With regard to received views, you are comparing apples and oranges. Implicit to "set theory," is the assertion, "Mathematics is extensional." Everything is a set. Functions are to be admissible only in terms of set representations. So, the axiom of extensionality determines "sameness."

Along similar lines, it makes no sense to speak of partial functions in set theory.

If you want a paradigm differentiating functions on the basis of domains and codomains, pick up a copy of Lawvere and Rosebrugh. The notion of set described using category theory is very different. One thing to notice is their criticism of a narrative found in Russell's "Principles of Mathematics" and repeated in Potter's "Set Theory and Its Philosophy." Russell credits Peano as correctly distinguishing between membership and containment. Implicit to this attribution is a criticism of Dedekind. Apparently, Dedekind initially used containment in earlier work. And, according to Potter, he later changed the notation he used.

Potter refers to this as a clear sign of mereology. Since mereology has enjoyed a recent resurgence, Hamkins and a colleague have investigated treating containment as a mereological relation. It would transform set theory into a decidable theory.

However, category theory treats inclusions as primitive. One can formulate a notion of "part" using inclusions. And this is precisely what you will find in Lawvere and Rosebrugh. But it is not the simple-minded idea of containment treated as parts.

Whether or not the mathematical notion of a set as a collection taken as an object ought to coincide with the philosophical emphasis on comprehensions has not been clearly decided. Bolzano explicitly distinguishes the two. Lawvere and McCarthy cite an untranslated work by Cantor which would indicate a similar view. And the translations of Cantor which are available definitely indicate the influence of Bolzano upon his work.

The real shame in all of this is that the intellectual giants run around teaching incompatible views to students who are trying to learn difficult material expressed with difficult symbolic language. What happened on the FOM mailing list with Awodey and Bauer several years ago is typical of how people segregate themselves into groups of like-minded individuals instead of actually addressing controversy.

Students should not be having to sort out ism-mongering when they are simply trying to learn mathematics.