Interesting question. As far as I know, the answer would be "no".

Here's as far as I've seen anyone go with it: Define the natural numbers, some include zero, some don't. However, the Naturals are defined axiomatically. Meaning, they are assumed to be things that exist that don't need to be justified. We would define them based on something like the

Peano axioms. We would denote the set of natural numbers as \(\displaystyle \mathbb{N}\), and think of it as \(\displaystyle \mathbb{N} = \{ 0,1,2,3,\dots \}\).

We would then define the integers as \(\displaystyle \mathbb{N} \cup (- \mathbb{N})\), where I'm using \(\displaystyle -\mathbb{N}\) to denote the negatives of all the natural numbers. We then denote the set of integers by \(\displaystyle \mathbb{Z}\). Where the "Z" comes from the German word for number, "zahl".

But honestly, after you do that, we usually just invoke them by saying something like, "Let \(\displaystyle n\) be an integer." or "Let \(\displaystyle n \in \mathbb{Z}\)." And no one questions it. The integers are thought of as the building blocks, the atoms of number systems. We build the other number sets from them via operations, but the integers themselves just...are.