What is the symbolic form of "there does not exist a largest natural number " Other students in office hour said this is the correct form
$(\forall x)(\exists y)(y&gt;x)$
{ for all x natural number, there exists y such that y is greater than x }

But
"there does not exist a largest natural number "
$\neg(\exists x)(x\text{ largest natural number})$

Am I even close ?
The formula $\forall x \exists y (y &gt; x)$ reads "for any $x$ there is a $y$ which is larger than $x$" which says that any $x$ is not largest and therefore no $x$ is largest. You can also use the formula $\forall \equiv \neg\exists\neg$ to get

$$
\begin{align*}
\forall x \exists y (y &gt; x) &amp;\equiv \neg\exists x \neg \exists y (y &gt; x) \\
&amp;\equiv \neg \exists x \forall y \neg(y &gt; x) \\
&amp;\equiv \neg \exists x \forall y (y \le x)
\end{align*}
$$

which says that "there is no $x$ such that every $y$ is less than or equal to $x$". This better fits with the "there does not exist a largest natural number" phrasing.
Since you are interested in formalisms, I would suggest to rephrase the statement to be more formal. In Mathematical logic, one usually uses quantors (similarly, the negation operator) and parentheses in the following way:
$$
\forall x (\; \text{logical statement} \;)
$$
So, nesting this, your statement becomes
$$
\forall x \in \mathbb{N}\left(\,\exists y \in \mathbb{N}\left(\,y&gt;x\right) \right)
$$
and the other statement, the one you were asking for, becomes
$$
\neg (\,\exists M \in \mathbb{N}
\,(\,\forall l \in \mathbb{N} \,(\,M \geq l))
$$
where I replaced your variable-symbol $x$ by $M$ and used $l$ as the variable-symbol for your natural number.