# Elementary equivalence using countable models

1. Aug 19, 2012

### Logic Cloud

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