Why is this set determined to be empty? From Eccles' Introduction to Mathematical Reasoning, problem 7.1 asks you to determine the set for:

$${\{n \in \mathbb{Z}^+ \mid \forall m \in \mathbb{Z}^+, m \leq n \}}$$

The answer provided in the back of the book is $\emptyset$. Why is $\{1\}$ not an answer? It satisfies $\mathbb{Z}^+$ and $m \leq n$, does it not?
Is $1\geq m$ for every positive integer $m$? In words, the set is the set of all positive integers greater than or equal to all positive integers.
Suppose that the statement below holds:
\begin{align}
&amp;\exists x(x\in\Bbb Z^+\land\forall y(y\in\Bbb Z^+\to y\le x))\\
&amp;\alpha\in\Bbb Z^+\land\forall y(y\in\Bbb Z^+\to y\le \alpha)&amp;&amp;\text{so there is an $\alpha$ satisfing the statement}\\
&amp;\forall y(y\in\Bbb Z^+\to y\le \alpha) &amp;&amp; \text{pick $y\in\Bbb Z^+$ such that $\alpha &lt;y$}\\
&amp;y\le\alpha\land\alpha&lt;y\\
&amp;y&lt;y &amp;&amp; \text{but '&lt;' relation isn't reflexive, i.e. $\forall x\neg(a&lt;a)$}\\
&amp;\bot
\end{align}
It shows that our initial assumption was wrong and there is no such element exists.