Even Natural Numbers are Infinite

Theorem

The set of even natural numbers is infinite.


Proof

Let $E$ denote the set of even natural numbers.

Aiming for a contradiction, suppose $E$ is finite.

Then there exists $n \in \N$ such that $E$ has $n$ elements.

Let $m$ be the greatest element of $E$.

But then $m + 2$ is an even natural number.

But $m + 2 > m$, and $m$ is the greatest element of $E$.

Therefore $m + 2$ is an even natural number that is not an element of $E$.

So $E$ does not contain all the even natural numbers.

From that contradiction it follows by Proof by Contradiction that $E$ is not finite.

$\blacksquare$


Sources

  • 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 1$ What is infinity?
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