I try to solve the problem 1.3.14 in Chang and Keisler's Model theory:
For each $n\in\omega$, find a model $\mathfrak{A}_n$ for $\mathcal{L}$ a language with only a finite number of symbols, which has exactly $n$ undefinable elements.
It is easy to solve when $n\neq 1$ (just take the language which only has a equality symbol and $n$-element set as a domain.) However, as this textbook remarked, the case of $n=1$ is extremely hard for me. All of my attempts are failed and I don't know how to solve the problem.
So my question is: there is a model which has only one undefinable element over a language with only a finite number of symbols? Thanks for any help.
This question is not duplicate with that question because this question only consider the model for language with finitely many symbols. In this question, I regard the equality symbol as logical symbols.
This is apparently very difficult. There is an article by Harvey Friedman that gives an example
https://u.osu.edu/friedman.8/files/2014/01/UniqueUndElt010913-2b6widm.pdf
In general, the double-starred problems in CK are closer to research problems than exercises...