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.



  • 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.


  • 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.


  • 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.

Related Posts

Just because 2 things are dual, doesn't mean they're just opposites

Boolean Algebra, Arithmetic POV

discontinuous linear functions

Continuous vs Bounded

Minimal Surfaces

November 2, 2023

NTK reparametrization

Kate from Vancouver, please email me

ChatGPT Session: Emotions, Etymology, Hyperfiniteness

Some ChatGPT Sessions