①

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?

②

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?

③

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?