I recently open a can of worms for myself by inquiring if there is a difference between a number as a natural or real, and got a fair answer, in doing so I came by an interesting idea about viewing the numbers slightly differently in different sets, such as 'why does -1 have no square root if left in the natural numbers?' My issue with this question is that I do not really understand the idea of a universe.
Let's take a simple example, let's take an element $x$ in a set $A$ and an element of a subset $B$, What I fail to understand is, do we define the universe simply as something we 'discuss', or do we simply pretend that all elements of $A$ that aren't in $B$ simply do not exist? Doing so makes it like we create two different realities, one where they exist, one where they do not, and how can $x$ be the same object in this case?
How can an object exist in different 'universes' if we define them in a way that they sort of exist independently with their own sense of what exists and what does not? Is this too much of me trying to view them as real objects in real 'universes'? And how can we define properties of an object to define it, for example as given before $-1$ has a square root in one universe and doesn't have it in another, is this part of our 'universe' definition, I.E. a relation an object can have with a particular universe.
I understand that in the strictest sense they aren't the same objects, but more interestingly how it works, say I split $N$ into a smaller subset.
If you want to know if $n \in \mathbb{N}$ is the same as $n \in \mathbb{R}$ just look at how these two sets are defined.
The short and useful answer is: Yes, they are equal.
The long and mostly useless (unless you are doing set theory) answer is: no, since $\mathbb{N}$ is built like $\bigl\{ \emptyset, \{\emptyset, \{ \emptyset, \{\emptyset\} \},... \bigr\}$ (Von Neumann definition) and $\mathbb{R}$ is defined via the Dedekind cuts and its elements are different from that of $\mathbb{N}$. However, you can find something identical to $\mathbb{N}$ inside $\mathbb{R}$, so usually you work like if $\mathbb{N}$ is a proper subset of $\mathbb{R}$, leaving this formalisms to set theorists.
Usually, when you talk about natural and real number, you don't talk about the formal sets definitions, so you can assume that you are talking about natural numbers inside real numbers.
Regarding the concept of 'universe', we need a definition. Intuitively, the univers is composed by the elements you are working with, so if you look at $B$ as a subset of $A$, $A$ could be the universe. A formal definition could be taken using the classes. A universe for a property (proposition) $\varphi(x)$ could be defined as a class of the elements on which you want to 'test' the property.
For instance, if your property is $\varphi(x)=$ "$x \in B$", you can consider as class $A$, which, in this case, is also a set.
An example where you use a proper class (a class which is not a set) could be $\varphi(x)=$ "the cardinality of $x$ is finite" and you want to consider as universe the groups.
When you are talking about the square root of $-1$, you want to define it. The definition could be $\forall \ x\in A \ \sqrt{x}=\{y \in A : y^2=x\}$, so, it depends on $A$. If you take $A=\mathbb{R}$ and $x=-1$ you have that this set is empty, while if you take $A=\mathbb{C}$ you will have $\sqrt{x}=\{i, -i\}$. It's all about definitions.
I hope I have given the answer you were looking for, the question was a bit vague.