Let σpo := (<) be the usual signature for strict partial orders. A linear order (A, <) is called dense if for any a, b ∈ A with a < b, there is c ∈ A such that a < c < b. a. Write down a σpo -theory DLO axiomatizing the class of all dense linear order without endpoints (i.e. without a least / largest elements). b. Verify that (ℚ, <) is a countable (if it is finite or admits a bijection to ℕ) model of DLO. You may use that ℚ is countable without proof. Note: It is a theorem that DLO has only one countable model up to isomorphism, i.e. all countable models are isomorphic to each other.
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Let σpo := (<) be the usual signature for strict partial orders. A linear order (A, <) is called dense if for any a, b ∈ A with a < b, there is c ∈ A such that a < c < b.
a. Write down a σpo -theory DLO axiomatizing the class of all dense linear order without endpoints (i.e. without a least / largest elements).
b. Verify that (ℚ, <) is a countable (if it is finite or admits a bijection to ℕ) model of DLO. You may use that ℚ is countable without proof. Note: It is a theorem that DLO has only one countable model up to isomorphism, i.e. all countable models are isomorphic to each other.
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