I was wondering what is the set of formulas of first-order logic that is satisfiable only iff the size of the domain is 3?
I was also wondering how we can use the above formulas to find another set of formulas that is satisfactory iff the size of the domain is infinite.
I'm pretty confused about how I can cook up some formulas and link one question to another, any help will be appreciated
I have found another answer here Does there exist a formula of first-order logic that is satisfiable only on structures with infinite domains? but I'm not sure how to find a SET of formulas for infinite and size 3 domain
As Mauro mentioned in his comment, a formula that is satisfiable iff the size of the domain is at least 3 is, $$\exists x_1,\exists x_2, \exists x_3(x_1\neq x_2 \wedge x_1\neq x_3 \wedge x_2\neq x_3).$$
If you want those to be the only 3 elements then you also need the universal statement Mauro mentioned.
Similar to the existential formula, you can write a formula that is satisfiable iff the size of the domain is at least 4.
You can do this for every $n$ and have a set of formulas that is satisfiable iff the domain is infinite (note that you don't want the universal formula previously mentioned to be a part of this set).