Example 3.1.8.
Suppose we claim that there is no smallest number. We can translate this into symbols as
\begin{equation*}
\neg \exists x \forall y (x \le y)
\end{equation*}
(literally, “it is not true that there is a number \(x\) such that for all numbers \(y\text{,}\) \(x\) is less than or equal to \(y\)”).
However, we know how negation interacts with quantifiers: we can pass a negation over a quantifier by switching the quantifier type (between universal and existential). So the statement above should be logically equivalent to
\begin{equation*}
\forall x \exists y (y \lt x)\text{.}
\end{equation*}
Notice that \(y \lt x\) is the negation of \(x \le y\text{.}\) This literally says, “for every number \(x\) there is a number \(y\) which is smaller than \(x\text{.}\)” We see that this is another way to make our original claim.