It sounds like you might be looking for Peano’s axioms for arithmetic (which essentially formalize addition as being repeated “add 1” and multiplication as being repeated addition) or perhaps explicit constructions of various number systems (like those described here).
The drawback of these definitions is that they don’t properly situate these numbers systems as “core” examples of rings. For example, one way to define the integers is to first define a ring and then define the integers to be the “smallest” or “simplest” ring (formally: the initial object in the category of rings). From this, you can deduce that all integers can be formed by repeatedly summing 1s or −1s (else you could make a smaller ring by getting rid of the elements that aren’t sums of 1s and −1s) and that multiplication is repeated addition (because a⋅b=a⋅(1+⋯+1)=a+⋯+a where there are b terms in these sums).
(It’s worth noting that it’s not the case in all rings that multiplication, addition, and “plus 1” are related in these ways. E.g. it would be rough to argue that if A and B are matrices then the product AB corresponds to summing A with itself B times. So I think it’s a reasonable perspective that multiplication and addition are independent “in general” but the simplicity of the integers forces them to be intertwined).
Some other notes:
Defining −a to be the additive inverse of a is the same as defining it as the solution to a+x=0. No matter which approach you take, you need to prove the same theorems to show that the notion makes sense (e.g. you need to prove that −a+−b=−(a+b)).
Similarly, taking Q to be the field of fractions of Z is equivalent to adding insisting that all equations ax=b have a solution, and the set of theorems you need to prove to make sure this is reasonable are the same.
In general, note that giving a definition doesn’t mean that there’s actually any object that actually satisfies that definition. E.g. I can perfectly well define α to be an integer such that α⋅0=3 but I would still need to prove that there exists such an integer α. No matter how you define the integers, rational numbers, etc., you need to prove that there exists a set and some operations that satisfy that definition. Proving this typically requires giving a construction along the lines of what you seem to be looking for. So these definitions aren’t really meant to be a substitute for constructing models of the number systems.
It sounds like you might be looking for Peano’s axioms for arithmetic (which essentially formalize addition as being repeated “add 1” and multiplication as being repeated addition) or perhaps explicit constructions of various number systems (like those described here).
The drawback of these definitions is that they don’t properly situate these numbers systems as “core” examples of rings. For example, one way to define the integers is to first define a ring and then define the integers to be the “smallest” or “simplest” ring (formally: the initial object in the category of rings). From this, you can deduce that all integers can be formed by repeatedly summing 1s or −1s (else you could make a smaller ring by getting rid of the elements that aren’t sums of 1s and −1s) and that multiplication is repeated addition (because a⋅b=a⋅(1+⋯+1)=a+⋯+a where there are b terms in these sums).
(It’s worth noting that it’s not the case in all rings that multiplication, addition, and “plus 1” are related in these ways. E.g. it would be rough to argue that if A and B are matrices then the product AB corresponds to summing A with itself B times. So I think it’s a reasonable perspective that multiplication and addition are independent “in general” but the simplicity of the integers forces them to be intertwined).
Some other notes:
Defining −a to be the additive inverse of a is the same as defining it as the solution to a+x=0. No matter which approach you take, you need to prove the same theorems to show that the notion makes sense (e.g. you need to prove that −a+−b=−(a+b)).
Similarly, taking Q to be the field of fractions of Z is equivalent to adding insisting that all equations ax=b have a solution, and the set of theorems you need to prove to make sure this is reasonable are the same.
In general, note that giving a definition doesn’t mean that there’s actually any object that actually satisfies that definition. E.g. I can perfectly well define α to be an integer such that α⋅0=3 but I would still need to prove that there exists such an integer α. No matter how you define the integers, rational numbers, etc., you need to prove that there exists a set and some operations that satisfy that definition. Proving this typically requires giving a construction along the lines of what you seem to be looking for. So these definitions aren’t really meant to be a substitute for constructing models of the number systems.