How does the video prove that there are infinitely many natural numbers?

Conditions

• The proof method is proof by contradiction (PBC).
• The set under discussion is $\mathbb{N}$, the natural numbers.
• The arithmetic fact used is that adding 1 to a natural number yields a larger natural number.

Reasoning, step by step

1. Assume the opposite of the target statement: that there is a finite amount of $\mathbb{N}$.
2. From the finiteness assumption, infer that there must be a last natural number, and name it $x$.
3. Record that $x \in \mathbb{N}$.
4. Construct $x+1$ and assert that $x+1 \in \mathbb{N}$.
5. Compare sizes and note that $x+1 > x$.
6. Identify the contradiction: $x$ was assumed to be the last natural number, but $x+1$ is a larger natural number.
7. Conclude that the assumption was false, so there is an infinite amount of $\mathbb{N}$.

Example

The handwritten proof shows: "Pbc.", "Assume that there is a finite amount of $\mathbb{N}$.", "let x be the last $\mathbb{N}$.", "$x \in \mathbb{N}$", "$x+1 \in \mathbb{N}$", "$x+1 > x$", and finally "=> There is an infinite amount of $\mathbb{N}$."

Common misconceptions

• Thinking that intuitive obviousness is sufficient for a mathematical proof.
• Believing that the proof requires advanced axioms like Peano axioms, whereas the video uses informal arithmetic facts.

Watch the explanation

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