Please help me with confusions of following terms: 1.Incompleteness 2.undefinable 3.non-standard model
If a structure has non-standard counterpart under a certain set of axioms, does that mean such structure is not definable in a certain language? (N,<) in first order logic would be an example. Does this also mean the axiom of (N,<) is not complete? Since we could create a sentence discribing something not in N,< but also prove it’s in N,<
Also, why non-standard model don’t exist in second order logic? An example would be R,
Thank you.
Any first order theory that has infinite models has many different models, due to the Lowenheim-Skolem theorem. This does not necessarily imply incompleteness. For instance, the theory of all first order sentences satisfied by the structure $(\mathbb Z, \le)$ is complete, but it has many models (they just all have the same first order properties). In fact the models are exactly $(L\times \mathbb Z, \preceq),$ where $L$ is any linearly ordered set and $\preceq$ is the dictionary order. But, for any incomplete theory $T$, if we let $\phi$ be some sentence that is neither provable nor refutable from the theory, the completeness theorem says there are models of $T+\{\phi\}$ and also models of $T+\{\lnot \phi\}.$
I would not say that the structure is not definable cause that term is used in a different way. However, it does mean that no set of first order sentences determine the structure up to isomorphism.
A second order theory is not necessarily categorical (i.e. does not necessarily have a unique model). However, due to the additional expressivity of second order semantics, much more control is available over the models: there is no Lowenheim-Skolem theorem in second order logic. For instance, we cannot express the fact that a structure is countable (or uncountable) in first order logic, but we can in second order.
You are probably under the impression that a second order theory has to be categorical because second order arithmetic famously is. The reason is cause you can write a sentence saying "the domain is $\mathbb N$" in the second order language (more precisely: "every element of the domain is in any set that contains zero and is closed under successor") and then can prove that sentence holds by induction (which is trivial if a bit disorienting).
(Note also these categoricity results require that we are only considering full models of second order logic, where the predicate variables actually must range over all predicates for the domain. Considering the more general class of models for second order logic (Henkin models) makes the semantics effectively first order.)
I’ve avoided the term “non-standard model” because the term is predicated on there being an “intended model”, which is not always the case.