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For any first-order formula $X$ in the first-order language $\langle 0, S, \le\rangle$ (possibly with free variables) does there necessarily exist another open formula $Y$ such that the equivalence $X \equiv Y$ is true on the set of all nonnegative integer numbers?

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    $Y = X\land X?$ – 2017-01-05
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    @bof Depending on how complicated $X$ is, that isn't an **open formula**. – 2017-01-06
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    @NoahSchweber How can $X$ be an open formula and $X\land X$ not an open formula? – 2017-01-06
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    @bof Where was $X$ asserted to be open? (I read "another open formula $Y$" as "another formula $Y$, which is open," - note that $X$ itself was only described as a "first-order formula".) – 2017-01-06
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    @NoahSchweber "**another** open formula" – 2017-01-06
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    @bof See the edit to my comment. The OP should definitely clarify, but I'm pretty sure that's what this is asking. – 2017-01-06
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    @NoahSchweber Seemed like a silly question anyway. $X\land x=x$ is certainly an open formula and is logically equivalent to $X.$ – 2017-01-06
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    @bof Only if $X$ itself is open. – 2017-01-06
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    @NoahSchweber An [open formula](https://en.wikipedia.org/wiki/Open_formula) is a formula that contains at least one free variable. Perhaps you are thinking of a *quantifier-free* formula? – 2017-01-06
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    @bof That's not universally true - indeed, the definition I learned was that open = quantifier free. For a usage of this, see page 99 of [this book](https://books.google.com/books?id=9XxDAAAAQBAJ&pg=PA100&lpg=PA100&dq=%22open+formula%22+mathematical+logic&source=bl&ots=rDpA2H5tZN&sig=iQ-ON4ES0tWiOmgFkh1i_zmcq8k&hl=en&sa=X&ved=0ahUKEwiXxdSio6zRAhUG5yYKHRx3BngQ6AEIYTAP#v=onepage&q=%22open%20formula%22%20mathematical%20logic&f=false). This is also how "open" is used in the study of models of arithmetic ("[open induction](http://mathoverflow.net/questions/23796/a-question-about-open-induction)"). – 2017-01-06
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    @Noah: For what it’s worth, I learned the same definition as **bof**: a formula with at least one free variable. – 2017-01-06
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    @NoahSchweber According to Kleene's *Introduction to Metamathematics*, p. 151, an open formula is a formula with at least one free variable. I'm sorry to hear that some bad person has apparently written a textbook using "open" to mean "quantifier-free". – 2017-01-06
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    @bof Well, the OP should indeed clarify what they mean, but I'll bet money that they mean "quantifier-free." And I think "some bad person" is misleading - the term "open formula" is used this way by a large portion of the logic community (as witnessed by the fact that "open induction" is the standard name for a particular theory of arithmetic). But whatever. – 2017-01-06

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Yes, this is true. The structure $(\mathbb{N}; 0, S, \le)$ admits quantifier elimination, and the proof of this is via induction on the complexity of $X$. For an example of such a proof, see this, which goes through the proof of quantifier elimination for a version of Presburger arithmetic.