Abstract
In this chapter we shall investigate combinatorial properties of sets of partitions of ω, where we try to combine as many topics or voices (to use a musical term) as possible. As explained in Chapter 11, partitions of ω are to some extent the dual form of subsets of ω. Thus, we shall use the term “dual” to denote the partition forms of Mathias forcing, of Ramsey ultrafilters, of cardinal characteristics, et cetera. Firstly, we shall investigate combinatorial properties of a dual form of unrestricted Mathias forcing (which was introduced in Chapter 24). In particular, by using the Partition Ramsey Theorem 11.4, which is a dual form of Ramsey’s Theorem 2.1 (and which was the main result of Chapter 11), we shall prove that dual Mathias forcing has pure decision. Secondly, we shall dualise the shattering number \(\mathfrak{h}\) (introduced in Chapter 8 and further investigated in Chapter 9), and show how it can be increased by iterating dual Mathias forcing (cf. Proposition 24.12). Finally, we shall dualise the notion of Ramsey ultrafilters (introduced and investigated in Chapter 10), and show—using the methods developed in Part II and the previous chapter—that the existence of these dual Ramsey ultrafilters is consistent with ZFC+CH as well as with ZFC+¬CH.
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Halbeisen, L.J. (2012). Combinatorial Properties of Sets of Partitions. In: Combinatorial Set Theory. Springer Monographs in Mathematics. Springer, London. https://doi.org/10.1007/978-1-4471-2173-2_26
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DOI: https://doi.org/10.1007/978-1-4471-2173-2_26
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