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I added some categorical POV on structure in model theory (which is being touched upon in another thread).
I tried to make the relation between structure in model theory and model come out more clearly:
at model I tried to make sure that all pointers to “structure” point to structure in model theory.
at structure in model theory I made this sentence appear:
In model theory, given a language $L$, a structure for $L$ is the same as a model of $L$ as a theory with an empty set of axioms. Convervsely, a model of a theory is a structure of its underlying language that satisfies the axioms demanded by that theory.
Hope that’s about right. Please feel free to improve on this.
I’ll look more closely later, but your boxed statement is certainly correct.
I am wondering if the supposed disambiguation at structure that superceded the plain redirect to stuff, structure, property is really justified.
Do model theorists say “structure” in the sense of formalizing the notion of “structure” as such? It seems to me to be far from that. Don’t they actually rather say “$L$-structure”? Much like, say t-structure in triangulated category theory or similar? This being a particular kind of structure that can be put on something?
it seems to me that we should reorganize the entry structure such as to give the pointer to stuff, structure, property as a definition right at the beginning, and then maybe list “$L$-structure in model theory” just as an example.
But maybe I am wrong. Let me know.
An $L$-structure in model theory isn’t something that you put on a set or collection of sets, it’s the set(s) together with that stuff you put on it. Am I misunderstanding your question?
Exactly, that’s precisely what I mean. Thanks for confirming. It seems obvious, but it does contradict the current organization of the entry structure!
That’s my point: I think we should re-organize this entry. I just thought I’d double check if you’d all agree. Zoran might diagree…
Okay, I have changed the entry structure accordingly. But let’s discuss this in the corresponding thread.
Wait, I thought I was disagreeing with you!
Following disucssion in another thread I have edited at structure in model theory a bit:
gave it an actual Idea-section and renamed the previous Idea-section to “Definition”
added references
Make that structure in model theory.
Thanks, fixed.
I have also added a subsection that points to the entries of elementary classes etc.
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