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Unformatted text preview: 1 CHAPTER 4. PROOF OF PRINCIPAL EXOTIC CASE 4.1. Strongly Mahlo Cardinals of Finite Order. 4.2. Proof using Strongly Mahlo Cardinals. 4.3. Some Existential Sentences. 4.4. Proof using 1consistency. 4.1. Strongly Mahlo Cardinals of Finite Order. The large cardinal properties used in this book are the strongly Mahlo cardinals of order n, where n ∈ ω . These are defined inductively as follows. DEFINITION 4.1.1. The strongly 0Mahlo cardinals are the strongly inaccessible cardinals (uncountable regular strong limit cardinals). The strongly n+1Mahlo cardinals are the infinite cardinals all of whose closed unbounded subsets contain a strongly n Mahlo cardinal. It is easy to prove by induction on n that for all n < m < ω , every strongly mMahlo cardinal is a strongly nMahlo cardinal. There is a closely related notion: nMahlo cardinal. DEFINITION 4.1.2. The 0–Mahlo cardinals are the weakly inaccessible cardinals (uncountable regular limit cardinals). The n+1Mahlo cardinals are the infinite cardinals all of whose closed unbounded subsets contain an nMahlo cardinal. Again, for all n < m < ω , every mMahlo cardinal is an n Mahlo cardinal. NOTE: Sometimes (strongly) nMahlo cardinals are called (strongly) Mahlo cardinals of order ≤ n. Also, sometimes what we call nMahlo cardinals are called weakly nMahlo cardinals. 2 The well known relationship between nMahlo cardinals and strongly nMahlo cardinals is given as follows. THEOREM 4.1.1. The following is provable in ZFC. Let n < ω . A cardinal is strongly nMahlo if and only if it is nMahlo and strongly inaccessible. Under the GCH, a cardinal is strongly nMahlo if and only if it is nMahlo. Proof: For the first claim, note that it is obvious for n = 0. Assume that every strongly inaccessible nMahlo cardinal is strongly nMahlo. Let κ be a strongly inaccessible n+1 Mahlo cardinal. Let A ⊆ κ be closed and unbounded. Since κ is strongly inaccessible, the set B ⊆ κ consisting of the strong limit cardinals in A is closed and unbounded. Let λ ∈ B be an nMahlo cardinal. As previously remarked, λ is an inaccessible cardinal. Since λ is a strong limit cardinal, λ is a strongly inaccessible cardinal. By the induction hypothesis, λ is a strongly nMahlo cardinal. We have thus shown that every closed unbounded A ⊆ κ contains a strongly nMahlo element. Hence κ is strongly n+1Mahlo. For the final claim, assume the GCH. By an obvious induction, every strongly nMahlo cardinal is an nMahlo cardinal. For the converse, let κ be an nMahlo cardinal. As previously remarked, κ is a weakly inaccessible cardinal. Hence κ is a strongly inaccessible cardinal (by GCH). By the first claim, κ is a strongly nMahlo cardinal. QED We now develop the essential combinatorics of strongly Mahlo cardinals of finite order used in this Chapter....
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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.
 Fall '08
 JOSHUA
 Math

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