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Elementary equivalence using countable models

  1. Aug 19, 2012 #1
    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
     
  2. jcsd
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