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structure, interpretation
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"Structures" or "models" that is the question.

I am no logician, but in every encounter I have had
with logic, this concept was named a "model".
Witness the fact that "model" theory is an established branch of
mathematical logic.

So my question is: where is your "structure" terminology
coming from. Is this a personal preference, or
is there a pattern of widespread usage to back up
your choice of words?

In regards to your structures entry, here is a
question that has been bothering me for a good long while:

Are empty models/structures allowed?

Of course, all foralls are true in an empty model,
and all "there exists" false.

Usually the standard texts say "no empty models allowed"
and I always wondered: "how come?"

The closest I've been able to come to an explanation,
is that people seem to want to be able to deduce

(For all x) (Px) |- (Exists x)(Px)

and to do this, you need to forbid empty models.

This always struck me as a question of convention,
an arbitrary decision someone made long ago.

Am I missing something? What is so abhorrent about
empty models that they must be forbidden?

This is standart nomenclature. In some other books or references, the term "model" is also used for structures. The term "model theory" comes from the notion of satisfaction : when we write the symbol A|= phi, we read "A models phi" or "A is a model of phi".

Usually, when the term "model" is used for a "structure", the author means a "structure" that "models" a bunch of formulas. I hope this is clear.

I think it's just a matter of taste up to some point. It is preferable that models be not empty for fome very technical reasons. The reason why models should be non-empty comes from symtactical concerns, because you would like the completeness theorem to be true.

So how would you render concepts/sentences like

"countable model of set theory"

"Up to isomorphism, the real numbers are the
unique model of the theory of complete, ordered

Would your choice of usage be to replace "model"
with "structure" in the above?

note that you have used "model of a theory" in your question. A "countable model of set theory" is a countable structure which is a model of the axioms of set theory.

Sometimes I've seen "model" used for "structure", especially when we are talking about structures that are models of a theory.

Just to give you a quick example, any graph is a structure for the signature of set theory, but to get a model of set theory, you must have more than a mere graph.

Thank you. Your point has finally sunk in.

Language -> structure
Theory -> model

Again, thanks for your efforts.

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