What is the symbolic form of "there does not exist a largest natural number "

The formula $\forall x \exists y (y > x)$ 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

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

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.

The statement as you have written it is false. If you read it out loud, $x$ has to be chosen before $y$ and for all $x$ you can find a greater $y$. You want to say that if you choose $y$. You want to choose $y$ first, then say that it is not greater than all $x$.

Yes, you are close.

So far, you have:

$\neg \exists x \,\,(x\text{ largest natural number})$

Since you're looking for the "symbolic form", your next step is to convert "largest natural number" to symbols.

• Note that "largest natural number" is the same thing as "number that is greater than all other natural numbers";

• To make things easier, note that the above is the same thing as "number that is greater than or equal to all natural numbers" (therefore, including itself without a problem!);

• Further rephrase it as "number such that all numbers are less than or equal to it", so we have:

$x$ is a number such that all numbers are less than or equal to $x$

• Observe that this last sentence can be easily translated to symbols as

$$\forall n \,\,\, n \le x$$

Now, just plug that into your original sentence, obtaining:

$\neg \exists x \,\,(\forall n \,\,\, n \le x)$

Note 1: This is not the only correct way to do this. It is possible to express the same fact differently.

Note 2: in the context of this question, it is clear that we are talking about natural numbers. But generally, it would be better to specify this, by writing:

$\neg \exists x \in \mathbb{N} \,\,(\forall n \in \mathbb{N} \,\,\, n \le x)$

Bonus: The students you mentioned are correct too. They chose to rephrase the sentence in a different sentence (but still equivalent). Instead of saying "there is no largest natural number", they are saying "all natural numbers have the property of being smaller than some number", which is the same thing, in the end.

It depends on whether with largest natural number you mean a greatest or a maximal element, though the difference would be important only for a non-total order.

If $(X,<)$ is a partially ordered set, we say that $a\in X$ is

• a greatest element if $\forall x\in X\colon (x< a\lor x=a)$
• a maximal element if $\forall x\in X\neg(a<x)$

(In a totally ordered set, exactly one of $x<a$, $x=a$, $x>a$ must be true, hence therre the notions of greatest and maximal element coincide).

Hence "There is no $a$ such that $a$ is a greatest/maximal element" translates "literally" to either $$ \neg\exists a\in \Bbb N\colon \forall x\in\Bbb N\colon (x<a\lor x=a)$$ or $$ \neg\exists a\in \Bbb N\colon \forall x\in\Bbb N\colon \neg(a<x)$$

If you make use of "$\neg\exists=\forall\neg$" and "$\neg\forall=\exists\neg$", you might equivalently write $$ \forall a\in \Bbb N\colon \exists x\in\Bbb N\colon (x\not <a\land x\ne a)$$ and $$ \forall a\in \Bbb N\colon \exists x\in\Bbb N\colon (a<x),$$ respectively.