5.9ZFCL011011 - 1 5.9. ZFC + V = L + {( ∃κ )( κ is...

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Unformatted text preview: 1 5.9. ZFC + V = L + {( ∃κ )( κ is strongly k- Mahlo)} k + TR( Π 1 ,L), and 1-Con(SMAH). We fix a countable model M + and d 1 ,d 2 ,..., as given by Lemma 5.8.37. We will show that M + satisfies, for each k ≥ 1, that “there exists a strongly k-Mahlo cardinal”. In section 4.1, we presented a basic discussion of n-Mahlo cardinals and strongly n-Mahlo cardinals. The formal systems MAH, SMAH, MAH + , and SMAH + , were introduced in section 4.1 just before Theorem 4.1.7. Recall the partition relation given by Lemma 4.1.2. Note that Lemma 4.1.2 states this partition relation with an infinite homogenous set. A closely related partition relation was studied in [Sc74], for both infinite and finite homogenous sets. In [Sc74] it is shown that this closely related partition relation with finite homogenous sets produces strongly Mahlo cardinals of finite order, where the order corresponds to the arity of the partition relation. We give a self contained treatment of the emergence of strongly Mahlo cardinals of finite order from this related partition relation for finite homogenous sets. We have been inspired by [HKS87], which also contains a treatment of essentially the same partition relation, and answers some questions left open in [Sc74]. Our main combinatorial result, in the spirit of [Sc74], is Theorem 5.9.5. This is a theorem of ZFC, and so we use it within M + . We then show that this partition relation for finite homogenous sets holds in M + . As a consequence, M + has strongly Mahlo cardinals of every finite order. DEFINITION 5.9.1. We write S ⊆ On to indicate that S is a set of ordinals. The only proper class considered in this section is On, which is the class of all ordinals. Hence S must be bounded in On. DEFINITION 5.9.2. We write sup(S) for the least ordinal that is at least as large as every element of S. 2 DEFINITION 5.9.3. We write [S] k for the set of all k element subsets of S. We say that f:[S] k → On is regressive if and only if for all A ∈ [S\{0}] k , f(A) < min(A). DEFINITION 5.9.4. We say that E is min homogeneous for f if and only if E ⊆ S and for all A,B ∈ [E] k , if min(A) = min(B) then f(A) = f(B). DEFINITION 5.9.5. We write R(S,k,r) if and only if S ⊆ On, k,r ≥ 1, and for all regressive f:[S] k → On, there exists min homogenous E ∈ [S] r for f. DEFINITION 5.9.6. We say that S ⊆ On is closed if and only if the sup of every nonempty subset of S lies in S. Thus ∅ is closed. Note that every nonempty closed S has sup(S) ∈ S. DEFINITION 5.9.7. Let f:[S] k → On. When we write f( α 1 ,..., α k ), we mean f({ α 1 ,..., α k }), and it is assumed that α 1 < ... < α k . LEMMA 5.9.1. The following is provable in ZFC. Suppose R(S,k,r), where S ⊆ On\ ω . Let n ≥ 1 and f 1 ,...,f n each be regressive functions from [S] k into On. There exists E ∈ [S] r which is min homogenous for f 1 ,...,f n ....
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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5.9ZFCL011011 - 1 5.9. ZFC + V = L + {( ∃κ )( κ is...

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