Why does constructing $x+1$ create a contradiction in the first proof?

Conditions

• The proof assumes there is a finite amount of $\mathbb{N}$.
• $x$ is defined as the last natural number under this assumption.
• The arithmetic property used is that if $x \in \mathbb{N}$, then $x+1 \in \mathbb{N}$ and $x+1 > x$.

Reasoning, step by step

1. Assume there is a finite amount of $\mathbb{N}$ and let $x$ be the last natural number.
2. Construct the successor $x+1$.
3. Assert that $x+1 \in \mathbb{N}$ because adding one to a natural number yields another natural number.
4. Observe that $x+1 > x$ because adding one increases the value.
5. Identify the contradiction: $x+1$ is a natural number larger than $x$, which violates the assumption that $x$ was the last natural number.
6. Conclude that the initial assumption of finiteness is false.

Example

The speaker explains that if you take a natural number and add one, you get a bigger natural number. The written lines show "$x+1 \in \mathbb{N}$" and "$x+1 > x$", which contradict the earlier statement "let x be the last $\mathbb{N}$."

Common misconceptions

• Thinking that $x+1$ might not be a natural number.
• Believing that the contradiction arises from the definition of $x$ rather than the properties of natural numbers.

Watch the explanation

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