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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 1-consistency. 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 . These are defined inductively as follows. DEFINITION 4.1.1. The strongly 0-Mahlo cardinals are the strongly inaccessible cardinals (uncountable regular strong limit cardinals). The strongly n+1-Mahlo 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 < , every strongly m-Mahlo cardinal is a strongly n-Mahlo cardinal. There is a closely related notion: n-Mahlo cardinal. DEFINITION 4.1.2. The 0Mahlo cardinals are the weakly inaccessible cardinals (uncountable regular limit cardinals). The n+1-Mahlo cardinals are the infinite cardinals all of whose closed unbounded subsets contain an n-Mahlo cardinal. Again, for all n < m < , every m-Mahlo cardinal is an n- Mahlo cardinal. NOTE: Sometimes (strongly) n-Mahlo cardinals are called (strongly) Mahlo cardinals of order n. Also, sometimes what we call n-Mahlo cardinals are called weakly n-Mahlo cardinals. 2 The well known relationship between n-Mahlo cardinals and strongly n-Mahlo cardinals is given as follows. THEOREM 4.1.1. The following is provable in ZFC. Let n < . A cardinal is strongly n-Mahlo if and only if it is n-Mahlo and strongly inaccessible. Under the GCH, a cardinal is strongly n-Mahlo if and only if it is n-Mahlo. Proof: For the first claim, note that it is obvious for n = 0. Assume that every strongly inaccessible n-Mahlo cardinal is strongly n-Mahlo. Let be a strongly inaccessible n+1- Mahlo cardinal. Let A be closed and unbounded. Since is strongly inaccessible, the set B consisting of the strong limit cardinals in A is closed and unbounded. Let B be an n-Mahlo cardinal. As previously remarked, is an inaccessible cardinal. Since is a strong limit cardinal, is a strongly inaccessible cardinal. By the induction hypothesis, is a strongly n-Mahlo cardinal. We have thus shown that every closed unbounded A contains a strongly n-Mahlo element. Hence is strongly n+1-Mahlo. For the final claim, assume the GCH. By an obvious induction, every strongly n-Mahlo cardinal is an n-Mahlo cardinal. For the converse, let be an n-Mahlo cardinal. As previously remarked, is a weakly inaccessible cardinal. Hence is a strongly inaccessible cardinal (by GCH). By the first claim, is a strongly n-Mahlo cardinal. QED We now develop the essential combinatorics of strongly Mahlo cardinals of finite order used in this Chapter....
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- Fall '08