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I don't know how to define in the logic of the first order the following statement: "The set of natural numbers $N$ is closed with respect to the sum operation between them".

For this purpose it’s possible to use the operator + in the formulas, in addition to existential and universal quantifiers and logical operators.

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Are you thinking of a first-order theory of natural numbers such as PA, or of sets such as ZF? For a first-order theory of natural numbers you could write $\forall a\forall b\exists c(c=a+b)$, whereas for a first-order theory of sets you could write $\forall a\in\Bbb N\forall b\in\Bbb N(a+b\in\Bbb N)$ (or even $\forall a\in\Bbb N\forall b\in\Bbb N\exists c\in\Bbb N(c=a+b)$).

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  • $\begingroup$ Another possibility I could think of is if you're interested in the first-order theory of $\mathbb{Q}$ or of $\mathbb{R}$ with a predicate symbol $\mathrm{nat}$ and the statement would be $\forall a, b, (\mathrm{nat}(a) \wedge \mathrm{nat}(b)) \rightarrow \mathrm{nat}(a+b)$. $\endgroup$ – Daniel Schepler Apr 12 at 21:48

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