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Keywords:

  • Reflection principles;
  • William Tait;
  • large cardinals;
  • Silver indiscernibles;
  • MSC(2010);
  • 03E55

Abstract

In 5, Tait identifies a set of reflection principles called equation image-reflection principles which Peter Koellner has shown to be consistent relative to the existence of κ(ω), the first ω-Erdős cardinal 1. Tait also defines a set of reflection principles called equation image-reflection principles; however, Koellner has shown that these are inconsistent when m > 2, but identifies restricted versions of them which he proves consistent relative to κ(ω) 2. In this paper, we introduce a new large-cardinal property, the α-reflective cardinals. Their definition is motivated by Tait's remarks on parameters of third or higher order in reflection principles. We prove that if κ is ℶκ + α + 1-supercompact and 0 < α < κ then κ is α-reflective. Furthermore we show that α-reflective cardinals relativize to L, and that if κ(ω) exists then the set of cardinals λ < κ(ω), such that λ is α-reflective for all α such that 0 < α < λ, is a stationary subset of κ(ω). We show that an ω-reflective cardinal satisfies some restricted versions of equation image-reflection, as well as all the reflection properties proved consistent in 2.