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Math Help - Questions in mathematical logic - compactness theorem

  1. #1
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    Questions in mathematical logic - compactness theorem


    L1 = <S> is a lexicon (alphabet).
    T1 is a set of sentences (theory) in L1 that have true value in the structure (model) M = [{1,2,3}, <].
    We define L2 = L1 ∪ [S, c1, c2, c3, e].
    We define T2 = T1 ∪ {c1<e, c2<e, c3<e}.
    Is T2 consistent? Why?



    L1 = <S, c1, c2, c3> is a lexicon.
    T1 is a set of sentences in L1 that have true value in the structure M = [{1,2,3}, <].
    We define L2 = L1 ∪ [S, c1, c2, c3, e].
    We define T2 = T1 ∪ {c1<e, c2<e, c3<e}.
    Is T2 consistent? Why?



    L1 = <S, c0, c1, c2, c3, c4, …> is a lexicon.
    T1 is a set of sentences in L1 that have true value in the structure M = [ℕ, <, 0, 1, 2, 3, 4…].
    We define L2 = L1 ∪ [e].
    We define T2 = T1 ∪ {Cn<e : n is natural}.
    Is T2 consistent? Why?
    Last edited by math0; May 5th 2014 at 03:02 PM.
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  2. #2
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    Re: Questions in mathematical logic - compactness theorem

    Quote Originally Posted by math0 View Post
    L1 = <S> is a lexicon (alphabet).
    T1 is a set of sentences (theory) in L1 that have true value in the structure (model) M = [{1,2,3}, <].
    We define L2 = L1 ∪ [S, c1, c2, c3, e].
    We define T2 = T1 ∪ {c1<e, c2<e, c3<e}.
    Is T2 consistent? Why?
    Why is L1 written using angular brackets while the definition of L2 uses square brackets? Why do we add S to L1 if it is already there?

    How do we interpret S in T1? I thought that since this is the only non-logical symbol in the alphabet, it is interpreted by the single relation <. However, T2 uses <, which is not in the alphabet, instead of S.

    It seems that T2 is consistent: just interpret c1 = c2 = c3 as 1 and e as 2.
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