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I am considering formal system of elementary number theory as given in Kleene's "Introduction to Metamathematics". If you want me to specify axioms please let me know. In his book "Mathematical logic" he writes that "Although only polynomials can be expressed in this formal system by means of terms, the representing predicates $F(x_1, ..., x_n, y)$ of a vastly greater class of functions can be expressed." He gives an example that "Let $F(x,y)$ express $x!=y$. Take the proposition $(x+1)! = (x+1)x!$, which is an informal theorem about factorial function. This can be paraphased in terms of predicate $F(x,y)$ e.g. thus : $$\exists u \exists v (F(x+1,u) \land F(x,v) \land u=v\cdot (x+1))."$$

My question is how one expresses $x! = y$ as $F(x,y)$? Also, for other functions like $x^{n}=u$ how does one express them using formulas of the formal system?

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These are primitive recursive functions, so can be implemented in arithmetic (if not already in the language) using the beta function trick.

The beta function is an arithmetical function such that if $(a_0,a_1,\ldots, a_n)$ is a sequence, then there are natural numbers $a,b$ such that $\beta(a,b,i)=a_i$ for all $0\le i \le n.$

For $x!=y,$ we say "there is a sequence $a_0,a_1,\ldots, a_x$ such that $a_0=1,$ $a_x=y$ and for all $i < x,$ $a_{i+1} = (i+1)\cdot a_i$". In terms of the beta function, this is $$ \exists a,b(\beta(a,b,0)=1\land \beta(a,b,x)=y\land \forall (i< x)(\beta(a,b,i+1)=(i+1)\cdot\beta(a,b,i)).$$

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