Why 0 to the power of 0 is 1

A set/category theoretic interpretation. Here’s a much more thorough elaboration. I’m proud to say that it may be the most comprehensive video explanation of this fact.

Encoding 0 and 1 as sets

The standard encoding of the natural numbers as sets gives:

• 0 is the empty set $\{\}$.
• 1 is a set that contains 0. In set notation, it’s $\{0\}$. Unfolding the definition further, it’s a set that contains the empty set, $\{\{\}\}$.

Counting functions with exponents

Target^Source can be interpreted as the set of all functions from Source to Target. This is seen in the powerset notation $2^X$, where we can identify subsets of $X$ with maps into a 2 element set, where 1 element indicates “yes this is in the subset” and the other indicates no.

Examples

Let

• Source := $\{1,2\}$
• Target := $\{3\}$

There is exactly 1 function between these sets, which sends both 1 and 2 to 3. The exponential formula predicts $1^2=1$ possible functions, which checks out.

Let

• Source := $\{\}$
• Target := $\{1\}$

There is exactly 1 function between them, the empty set.

The empty set can be interpreted as a function because functions are subsets of the Cartesian product Source x Target. An empty set meets this criteria.

Let

• Source := $\{1\}$
• Target := $\{\}$

There are no functions between them because all elements of Source must be mapped to something, but there’s nothing to map to. The exponential formula gives $0^1=0$ total functions.

Finally, let

• Source := $\{\}$
• Target := $\{\}$

There’s nothing in Target to map to, but there’s also nothing that needs mapping from since Source is empty too. So the empty set is a valid function between them.

Therefore, the set of all functions between empty sets (aka 0) is a set that contains the empty set. Which is 1, by the standard encoding given above.

A nice confluence.