TITLE: What is the symbolic form of "there does not exist a largest natural number "
QUESTION [16 upvotes]: Other students in office hour said this is the correct form
$(\forall x)(\exists y)(y>x)$
{ 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 ?

REPLY [5 votes]: 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.