Definition of integers in higher-order logic There's the classical of natural numbers in higher-order logic (see the introduction of this page for example).
Is there something similar for integers (elements of $\mathbb{Z}$) ? I didn't find this on internet.
And what about rational and real numbers ?
 A: Yes. To be precise, say that a second-order-logic sentence $\psi$ characterizes a structure $\mathcal{A}$ iff the structures in which $\psi$ is true are exactly the structures isomorphic to $\mathcal{A}$, and say that a structure $\mathcal{A}$ is second-order characterizable iff such a $\psi$ exists. Here are some examples:

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*$\mathbb{R}$ (with addition and multiplication) is the unique-up-to-isomorphism Dedekind-complete orderable field.


*$\mathbb{Q}$ (with addition and multiplication) is the unique-up-to-isomorphism infinite divisible orderable abelian group with no nontrivial proper divisible subgroup.


*$\mathbb{Q}$  (with ordering alone) is the unique-up-to-isomorphism countable dense linear order without endpoints (this is due to Cantor).


*$\mathbb{Z}$ (with addition and multiplication) is the unique-up-to-isomorphism orderable ring with no proper subring.


*$\mathbb{Z}$ (with ordering alone) is the unique-up-to-isomorphism nonempty linear order without endpoints in which all intervals between elements are finite.
Each of these descriptions is straightforwardly convertible into a second-order characterizing sentence.
Interestingly, every naturally-occurring mathematical structure seems to be second-order characterizable; this theme is discussed in Vaananen's paper Second-order logic or set theory?, which I recommend. Moreover, it is undecidable from the usual axioms of set theory whether every countable structure is second-order characterizable (see the discussion here).
