Suppose I have two elements, $a$ and $b$. Suppose now that I want to say (using some appropriate first order language endowed with the membership symbol):
- there is a set $B = \{a, b\}$,
- the element $a$ is a member of the set $B$.
What does a very simple model of that look like? Does the structure need to contain three items: $a$, $b$, and $B$, plus the membership relation $\{(a, B), (b, B)\}$?
Or is there really just two items in the structure, namely $a$ and $b$, and then $B$ is somehow constructed using just logical operators or something? (If so, how is it constructed?)
Perhaps I just need a simple example of how I would use an appropriate first order language to write the sentences (1) and (2) above and how I would construct a model of them.
Objects that exist need to be an actual object in your model, so $a$ and $b$ and $B$ need to correspond to objects in your model.
However, you did not have rules stating that $a$, $b$, and $B$ are distinct, so they might as well be the same object.
Also, you did not have rules stating that other objects don't exist, so your model might have extra elements.