Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 3 down vote favorite

This is a really "elementary" question, forgive the pun.

What is the difference between an elementary submodel and an elementary substructure (in first-order Logic)?

Sincere thanks for help.

share|improve this question
2  
Model is a structure which satisfies a particular theory. Without the context of a theory this is just a substructure. When adding the context of a theory this is the same thing because both structure satisfy the same sentences [read: axioms]. – Asaf Karagila Apr 24 at 14:24
2  
None. Both terms denote the same objects. It is simply that sometimes we talk of models and sometimes we talk of structures. Typically, "model" is reserved for "model of a theory $T$". If no theory is being discussed, "elementary submodel of $\mathcal M$" means "elementary substructure that is a model of the theory of $\mathcal M$", but of course, any elementary substructure of $\mathcal M$ is a model of this theory. So the two uses are equivalent. It is more a matter of personal taste than anything else. – Andres Caicedo Apr 24 at 14:24
1  
@Asaf: What if the sentences are not all equations (universal sentences)? E.g. I can imagine that a subring of a field is a substructure (in the language of rings), but the submodels are only the subfields... $\quad$ well.. if they are elemantary, then of course, this differentiation doesn't play.. – Berci Apr 24 at 15:05
1  
Same thing. In set theory submodel seems to be more common than elsewhere. – André Nicolas Apr 24 at 15:10
1  
@Berci: I don't understand your comment. – Asaf Karagila Apr 24 at 16:52

Know someone who can answer? Share a link to this question via email, Google+, Twitter, or Facebook.

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.