Is this a demonstration or a definition?

Question

Some people says that this can be demonstrated $0!=1$, but other say that this is a definition. Which one is correct?

Let's given $n\in\mathbb{N}$:

$$(n+1)!=(n+1)\cdot n!$$ $$(0+1)!=(0+1)\cdot 0!$$ $$1!=1\cdot 0!$$ $$1=0!$$

Answer 1

Depending on the equations suitable to define $n!$, there are multiple possible answers:

If $n! := n (n-1)!, 0!=1$ is used, it's a plain definition.
If $1! = 1$ is used as the starting point, it can be proved that $0!:=1$ is a consistent extension. (this is what you did)
If $n! := \prod_{i\in \mathbb N, i \le n} i$, it follows from the definition of the empty product, $\prod_{k\in\emptyset} f(k) := 1$.
If $n! := \Gamma(n+1), n\in\mathbb N$, it follows from the definition because $\Gamma(1) = 1$.
If $n! := |S_n| = |\{f: A\to A \text{ bijective}\}|$ with $|A| = n$, there is only one set with $0$ elements, $\emptyset$ and only one function $f: \emptyset \to\emptyset$

Answer 2

There are multiple equivalent ways of defining the factorial function. Since the exact choice of definition doesn't matter in the long run, people don't generally make a big deal about which definition they're choosing. In some contexts, this would be a definition, whereas in others it would be a proof from the definition.

Answer 3

In demonstrating that $0! = 1$, you assumed that $1! = 1$.

If you define $n!$ by

1. $1! = 1$

2. $n! = n(n - 1)!$ for $n \in \mathbb{N}$

with $\mathbb{N}$ defined as the set of positive integers, then you can demonstrate that $0! = 1$ by substituting $1$ for $n$.

If, as some mathematicians do, you define $\mathbb{N}$ to be the set of non-negative integers, it would make sense to define $n!$ by

1. $0! = 1$

2. $(n + 1)! = (n + 1)n!$ for $n \in \mathbb{N}$, with $n \geq 1$

Answer 4

Another way of showing why this should be true: \begin{align} 5!&=5\cdot4\cdot3\cdot2\cdot1\cdot(1)\\ 4!&=4\cdot3\cdot2\cdot1\cdot(1)\\ 3!&=3\cdot2\cdot1\cdot(1)\\ 2!&=2\cdot1\cdot(1)\\ 1!&=1\cdot(1)\\ 0!&=(1) \end{align} The first five lines are easy to verify, and the last line follows by analogy.