Elementary equivalence using countable models

Logic Cloud

1. The problem statement, all variables and given/known data
Prove that [tex](\mathbb{R}, <)[/tex] and [tex](\mathbb{R} \backslash \{0\}, <)[/tex] are elementary equivalent using the fact that there exist countable models [tex](A, <_0)[/tex] and [tex](B, <_1)[/tex] which are elementary equivalent with [tex](\mathbb{R}, <)[/tex] and [tex](\mathbb{R} \backslash \{0\}, <)[/tex] respectively.


2. Relevant equations
n/a


3. The attempt at a solution
Once you prove that the two countable models are elementary equivalent, the desired result follows immediately. So suppose the countable models are not elementary equivalent. Then there exists a sentence s such that s is satisfied by A and ~s is satisfied by B. I don't know how to proceed from here. I'll probaly have to use the uncountable models in some way as well but I don't see how. Any help will be much appreciated.
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution