Q:

Mathematical logic: a canonical model for Lpi

Here is an excerpt from some lecture notes about mathematical logic:
Assume that T is a theory in Lpi (where Lpi is a variant of a predicate language obtained by adding the symbol = to its vocabulary). A canonical model for T cannot be constructed by using the set of terms of Lpi as a domain and considering an assignment in which every term denotes itself. For given two terms t and t', it may happen that t = t' ∈ T while t = t' is not satisfied (how this can be possible??). To prove that, for every formula A, the chosen assignment satisfies A if and only if A ∈ T, it must be the case that t and t' have the same denotation if and only if t = t' ∈ T. This is obtained if the domain is a set whose elements are equivalence classes of terms of Lpi, that is, sets obtained by grouping the terms of Lpi on the basis of the following condition: t and t' belong to the same set if and only if t = t' ∈ T (I don't understand very well what is meant here by "equivalence classes"). So we respect the intended meaning of =, because we rule out that distinct terms can denote distinct objects when their identity is provable in T.
I've put my questions in bold. Your help would be greatly appreciated.
Thanks a lot
Fish

A:

If $1+1$ and $2$ are terms then they are not the same (the first has three characters and the second only one and none of the characters are common). On the other hand it may be possible to prove $1+1=2$ in the theory.
If you want to interpret the statement $1+1=2$ in your model as meaning the thing that represents $1+1$ is the same as the thing that represents $2$, then representing the terms by themselves doesn't work.
If for all terms $t=t$ can be proved and whenever $t=t'$ can be proved so can $t'=t$ and whenever $t=t'$ and $t'=t''$ can each be proved so can $t=t''$, then $t=t'$ is provable is an equivalence relation on the terms (https://en.wikipedia.org/wiki/Equivalence_relation). If instead of representing a term $t$ by itself you represent it by its equivalence class then in the above case you can indeed interpret $1+1=2$ in the model as the the thing that represents $1+1$ is the same as the thing that represents $2$, because the equivalence classes of $1+1$ and $2$ are the same set.  

