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Write a formula $\phi(x)$ in arithmetic language that claims, that $x$ is a factorial for any number. It means that for every valuation $v:X \to \mathbb{N} \ \ (\mathbb{A}, v) \models \phi $ iff $v(x)$ is $n!$ for any $n$. $\mathbb{A}$ is a standard arithmetic model.

I know that I should use Godel's beta function but I don't know how to start. Please help me.

$$\exists t \exists p \exists n \chi[t, p, 0, 1] \wedge \chi[t, p, n, x] \wedge (\forall i 0 < i < n \implies (\exists a \chi [t, p, i-1, a] \wedge \chi[t, p, i, i \cdot a]))$$

Is it sufficient? It seems that it defines a beta function, therefore it defines a sequence, therefore it sequence a function: $f: \mathbb{N} \to \mathbb{N} $ and especially defined funcion is a factorial. Am I right?

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Hint: If we could quantify over functions, a suitable formula would be $$ \exists y \exists f\bigl[f(0)=1 \land f(y)=x \land \forall n < y\; [ f(n+1)=f(n)\cdot(n+1) ] \bigr]$$

Now just use the $\beta$ construction and standard abbreviations to express this in the language of arithmetic ...

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    Do you have a good reference for the actual use of those $\beta$-functions? Aren't they pretty complicated ... Hence the 'standard abbreviations' ... But what are these 'abbreviations' and what do they abbreviate? Again, if you can point me to a good reference I would greatly appreciate it. Thanks! – 2017-01-22
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    @Bram28: The OP knew enough to ask about the $\beta$ function specifically, so I'm assuming he has a text that defines the function and explains how to use it. (He'll have to do _some_ work for himself here). For the rest of us, the [Wikipedia article](https://en.wikipedia.org/wiki/G%C3%B6del%27s_%CE%B2_function) contains roughly the necessary information; or go to any of the various "gentle" developments of Gödel's incompleteness theorem. As for abbreviations, I'm thinking about, for example, how $a$\exists c: S(a+c)=b$. – 2017-01-22
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    Sorry my comment was not meant as a criticism of your answer to the OP but purely for my own personal interest ... Thanks for the link! – 2017-01-22
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    @Bram28: And the $\beta$ function itself isn't terribly complicated: $\beta(a,b,c)=d$ can be expressed as $\exists x\,[x\cdot(b\cdot c+b+1)+d=a] \land d – 2017-01-22
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    And if I understand this correctly: when $\beta(a,b,c) = d$, we use this to say that where $a$ and $b$ are used to encode a sequence of numbers, the c-th entry of that sequence is d, right? – 2017-01-22
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    @HenningMakholm, I edited. – 2017-01-22