I was going to write this in a comment, but I think it's better if I gave an answer.
I'm going to begin by pointing out things about the two statements in the question.
The first statement does have the intended meaning. I think it can still be made more precise.
The second statement is circular, or ambiguous at best. There is no way of saying what the largest natural number is until it has been decided definitely whether it exists or not.
Finally, although the domain has been described in the worded statement, for the logical statement to make sense, it must include the correct domain.
Since T. Gunn's answer very nicely covers the question, I will instead give an example of a correct statement which is useful in the study of modern algebra, how number (and other) sets are constructed, the relations between them, and the rigorous framework used to describe these ideas:
$$\forall n \in \mathbb N,n + 1 > n \land n + 1 \in \mathbb N$$
Though, to be precise this is a logically equivalent statement, not an identical one. And I have assumed, the addition operator and ordering relation ">" have already been defined.
This statement makes the intended meaning, and its demonstration, more explicit. It describes a function which guarantees the required property will be satisfied.