Unformatted text preview: σ is smaller at the ﬁrst position they diﬀer. Compare this to the lexicographic ordering < L extended to all of 2 <ω , which is deﬁned by σ < L τ if and only if σ ⊂ τ or σ is less than τ at the ﬁrst position they diﬀer. For example 1 < LL 01 but 01 < L 1. It is easy to check that < LL well orders 2 <ω . However, < L is not a well order. For example, the set { n 1  n ∈ N } ⊆ 2 <ω has no < Lleast element. The deﬁnition of a ﬁnished tableau τ from a set of premises Σ requires that Tα must appear on every noncontradictory path through τ for every α in Σ. Technically you also check the CST has this property. The check is simple. Every noncontradictory path in τ extends some noncontradictory path in τ m , the tableau we had at stage m , and at stage m we added Tα m to the end of each such path when we form τ m +1 . 1...
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 Spring '08
 DORAIS
 Math, Logic, Order theory, Total order, Paul Shafer, [email protected]

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