If a function $f$ is understood as its graph, i.e. a set of pairs $\langle x,y\rangle$ where $x$ is input and $y$ is output, then the empty set $\emptyset$ is a valid function, and for any set $A$, we have the empty function in $\emptyset\to A$. This makes $\emptyset$ the initial object in the category of sets. By set extensionality, there is only this empty function and this also fits with cardinalities, where we like $|A^{\emptyset}|=|A|^{|\emptyset|}=1$. In logic, we usually take that form absurdum follows everything, ex falso quodlibet, and in the propositions interpretation for types, if $\bf 0$ denotes the empty type (or bottom type), we want a function ${\bf 0}\to \tau$ for all types $\tau$ too.
I've been told in some programs this would be done by defining alla
definition exfalso (a : Type) (x : 0) : a := "case x of -emptyspace-"
Now I try to construct the terms for function types involving the empty type over the standard theories you'd find in books, but I can't quite reproduce the good properties, see below. I wonder
Which type theories actually let you construct a term of a function type from the empty type, and then, is it unique?
My considerations (leading to more question) are this...
Take simply types lambda calculus. We have $(\lambda (x:\tau).x):\tau\to\tau$ and so $(\lambda (x:{\bf 0}).x):{\bf 0}\to{\bf 0}$. In any model of the type theory, this will be the empty function and so I'm okay with that. But how do I construct, from the typing rules of the theory, any type of ${\bf 0}\to{\bf \tau}$, where $\tau$ isn't ${\bf 0}$? And then, how is there only one? Depending on how the theory lets you define functions, I could imagine ${\bf 0}\to\mathrm{Nat}$ to have zero or an infinite number of terms (constant functions which can never take any input). I guess the cardinal arithmetic then only works out once we pass to the set theoretic model, where all those constant functions are, extensionally, the empty function. Is that right? In a type theory with booleans $\bf 2$, how many functions $\bf 0\to\bf 2$ are there? If $a,b:\bf 2$, are $\lambda x.a$ and $\lambda x.b$ generally two function of this type?