On Transfinite Sequences of Projective Sets with an Application to ∑^1_2 Equivalence Relations

Abstract

This chapter focuses on transfinite sequences of projective sets with an application to Σ^1_(∼2n) equivalence relations. From the full axiom of determinacy (AD), the chapter demonstrates that the class of Σ^1_(∼2n) sets is closed under well ordered unions. This result is then applied to compute the maximum length of transfinite 1–1 sequences of ^1_(∼2n) sets, from AD again. As a further application, the chapter demonstrates that in the real world, every Σ^1_2 equivalence relation has either ≤_א1k, or 2_0^א equivalence classes. In addition, the cardinality of the maximum length of a transfinite sequence of ^1_(∼2n) sets without repetitions is computed in the chapter.

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