I have been working through an assortment of problems in logic and model theory to solidify my own understanding of the subject before the upcoming school year and I am unsure on some of the details of a particular problem. The exercise in question is number 1.3.2 from Chang & Keisler. Here is a copy of the problem:
A model is isomorphically embedded in if there is a model ℭ and an isomorphism such that :≃ℭ and ℭ⊂. The subset symbol in this context refers to a submodel
Let $\mathfrak{A}\overset{\sim}{\subset} \mathfrak{B}$ mean that $\mathfrak{A}$ is isomorphically embedded in $\mathfrak{B}$. Show that the relation $\overset{\sim}{\subset}$ is reflexive, transitive but not antisymmetric. Let $N$ be the set of all natural number $\{0,1,2,...\}$. Decide if the following are true or false
- $\langle N,\leq, +,0\rangle \overset{\sim}{\subset}\langle N,\leq, *,1\rangle$
- $\langle N,\leq, *,1\rangle \overset{\sim}{\subset}\langle N,\leq, +,0\rangle$
- $\langle N-\{0\},\leq, *,1\rangle \overset{\sim}{\subset}\langle N,\leq, +,0\rangle$
- $\langle N-\{0\}, *,1\rangle \overset{\sim}{\subset}\langle N, +,0\rangle$
- $\langle N-\{0\}, *\rangle \overset{\sim}{\subset}\langle N, +\rangle$
So I have shown the relation to be reflexive and transitive but have failed to show its not antisymmetric. I understand that since it is $\textit{not}$ antisymmetric that it is possible to have $A\overset{\sim}{\subset}B$ and $B\overset{\sim}{\subset}A$ while $A\neq B$ but I do not know which examples of the ones given fufill this. Also when I attempted to prove 1 true or false I tried to use the map of taking addition to multiplication by sending the number $k \rightarrow 2^k$ but I am not sure if this is the correct way to proceed or how to reverse the process for multiplication to addition. I am also not sure how to show that no such map exists if that is the case.
Thanks! Any help is greatly appreciated.