First of all. Very interesting and beneficial discussion for me. Nice to see one more person join us.
I've taken my time to read your explanations clic-clac, and I understand your point about closure and functions now. I blame my previous inability to understand what you said by being unfamiliar with the "<E>"-style notation, and my insecurities about what closure under a domain actually means. Your "translation" into another approach was very usefull also, thanks!
My side note was not that important.
emakarov: We are not supposed to prove L÷wenheim-Skolem via the completeness theorem for first order logic, which is also a part of this course.
Your model theoretic description seems like it might be really beneficial for me, but I have little clue how to read the notation you use. Remember I'm not a math student, so I'm not familiar with that notation. I believe I understand the first two steps about closure intuitively now, and understand somewhat how to describe it in syntax.
To clarify one thing. What kinds of objects are in a union of interpretations?
Also, now if we have done the first two steps correctly, the last step seems to flow pretty effortlessly from our construction? So how would one go about to prove this, would it require a lengthy proof. Being a beginner I don't have a feeling for how much one have to write or make explicit etc...