I do not get the intuition behind the structure (I think it's called "interpretation" in other texts) of a first order language. Here I use the following definition, given in Shoenfield's Introduction to Mathematical Logic:
I do get the definitions, but I am at loss to why such definitions were invented in the first place, and some things are unclear to me. Specifically:
I do not that the paragraph "We want to define a formula $A$ [called "well-closed formulas", or "wfs," in some texts"] to be valid... this leads us to the followong definitions." Could you provide a specific example that explains the intuition behind the definitions?
It's written "for each individual $a$ of $\alpha$, we choose a new constant, called the name of $a$." Where does the new constant come from? Are we assuming descriptive set theory and choose an elment not in the $0$-ary function symbol and adding it to the language being considered?
If $A$ is $p$ with $p$ a $0$-ary predicate symbol, then $\alpha(A)= T$ iff $\alpha(\emptyset)$ belongs to the predicate. However, we haven't defined what $\alpha(\emptyset)$ is,unless we are making another assumption that there exist an individual of $|\alpha|$ that has $\emptyset$ as its name. Am I mistaken?