# Thread: Questions in mathematical logic - compactness theorem

1. ## 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?

2. ## Re: Questions in mathematical logic - compactness theorem

Originally Posted by math0
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.