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3.15observ030409 - 1 3.15 Some Observations Recall the...

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1 3.15. Some Observations. Recall the Template and Extended Template introduced at the beginning of this Chapter before section 3.1. We now justify the claims TEMP 1,2, and ETEMP, which were also presented there. TEMP 1. Every one of the 6561 assertions in the Template is either provable or refutable in SMAH+. There exist 12 assertions in the Template, provably equivalent in RCA 0 , such that the remaining 6549 assertions are each provable or refutable in RCA 0 . Furthermore, these 12 are provably equivalent to the 1-consistency of SMAH over ACA’ (Theorem 5.9.11). To see how the Annotated Table of section 3.14 justifies Temp 1, recall how it was constructed. The ordered pairs of clauses in the Annotated Table comprise a list of representatives from each equivalence class of the ordered pairs of clauses under the equivalence relation used in section 3.1. The entries that correspond to the assertions in the Template are the entries in the Annotated Table with INF or ¬ INF. The 12 Exotic Cases (see Definition 3.1.2) correspond to the single entry in 28 under ACBC, INF. Every entry in the Annotated Table, with the sole exception of this single entry for INF, was justified in sections 3.3 – 3.13. All of the arguments in sections 3.3 – 3.13 were conducted within RCA 0 .
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