5.2Leng3tow010911

# 5.2Leng3tow010911 - 1 5.2 From length 3 towers to length n...

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1 5.2. From length 3 towers to length n towers. In this section, we obtain a variant of Lemma 5.1.7 (Lemma 5.2.12) involving length n towers rather than length 3 towers of infinite sets. However, we only assert that the sets in the length n tower have at least r elements, for any r 1. Thus we pay a real cost for lengthening the towers. Because the sets in the tower are finite and not infinite, certain indiscernibility properties of the first set in the tower must now be stated explicitly as additional conditions. See Lemma 5.2.12, iii), viii). These indiscernibility properties can of course be obtained from the usual infinite Ramsey theorem by taking a subset of the infinite A N from Lemma 5.1.7 - but then we would only have a tower of length 3. We will apply Lemma 5.1.7 with f arising from term assignments. Thus Lemma 5.2.12 uses g and not f. Recall the definition of the language L (Definition 5.1.8). In order to avoid having to write too many parentheses in terms and formulas of L, we use the following two standard precedence tables. +,- ¬ , , DEFINITION 5.2.1. Let t be a term of L. We write #(t) for the maximum of: the subscripts of variables in t, and the number of occurrences of the symbols 01+-• ()v 1 v 2 ,...log We count log as a single symbol. Note that for all n 0, {t: #(t) n} is finite. DEFINITION 5.2.2. Let ϕ be a quantifier free formula in L. We write #( ϕ ) for the maximum of: the subscripts of

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2 variables in ϕ , and the number of occurrences of the symbols 01+-• ()=< ¬ ∧∨→↔ v 1 v 2 ,...,v r log in ϕ . Note that for all n 0, { ϕ : #( ϕ ) n} is finite. DEFINITION 5.2.3. For all r 1, let β (r) be the number of terms t in L with #(t) r. We fix a doubly indexed sequence t[i,r] of terms in L, which is defined if and only if r 1 and 1 i β (r). For each r 1, the sequence t[i,r], 1 i β (r), enumerates the terms t with #(t) r, without repetition. DEFINITION 5.2.4. For all r 1, let γ (r) be the number of quantifier free formulas ϕ in L with #( ϕ ) r. We fix a doubly indexed sequence ϕ [i,r] of quantifier free formulas in L, which is defined if and only if r 1 and 1 i γ (r). For each r 1, the sequence ϕ [i,r], 1 i γ (r), enumerates the quantifier free formulas ϕ with #( ϕ ) r, without repetition. We adhere to the convention of displaying all free variables (and possibly additional variables). Thus t(v 1 ,...,v n ) and ϕ (v 1 ,...,v m ) respectively indicate that all variables in the term t are among the first n variables v 1 ,...,v n , and all variables in the quantifier free formula ϕ are among the first m variables v 1 ,...,v m . Note that all terms t[i,r] have variables among v
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5.2Leng3tow010911 - 1 5.2 From length 3 towers to length n...

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