Axiomatic System Problem: Checking Models and Isomorphism
Consider an infinite set of undefined elements S and the undefined relation R that satisfies the following axioms:
Axiom 1. If a, b S and aRb, then a does not equal b.
Axiom 2. If a, b, c S, aRb, and bRc, then aRc.
(i) Show that an interpretation with S as the set of integers and aRb as "a is less than b" is a model for the system.
(ii) Would S as as the set of integers and aRb interpreted as "a is greater than b" also be a model?
(iii) Are the models in parts (i) and (ii) isomorphic?
(iv) Would S as the set of real numbers and aRb interpreted as "a is less than b" be another model?
(v) Is the model in Part(iv) isomorphic to the model in Part (i)?
Part (i) and (ii) are models for the system because the axiom statements are correct.
Part (iii) The models are isomorphic because they are the same except the notation changes.
Part (iv) This interpretation is also a model.
Part (v) The model is not isomorphic because Part i's S is set of integers while Part iv's S was set of real numbers.
I was wondering if someone could check and make sure everything is okay.