A Refinement of the Ramsey Hierarchy via Indescribability

Abstract

We study large cardinal properties associated with Ramseyness in which homogeneous sets are demanded to satisfy various transfinite degrees of indescribability. Sharpe and Welch [25], and independently Bagaria [1], extended the notion of -indescribability where to that of -indescribability where . By iterating Feng’s Ramsey operator [12] on the various -indescribability ideals, we obtain new large cardinal hierarchies and corresponding nonlinear increasing hierarchies of normal ideals. We provide a complete account of the containment relationships between the resulting ideals and show that the corresponding large cardinal properties yield a strict linear refinement of Feng’s original Ramsey hierarchy. We isolate Ramsey properties which provide strictly increasing hierarchies between Feng’s -Ramsey and -Ramsey cardinals for all odd and for all . We also show that, given any ordinals the increasing chains of ideals obtained by iterating the Ramsey operator on the -indescribability ideal and the -indescribability ideal respectively, are eventually equal; moreover, we identify the least degree of Ramseyness at which this equality occurs. As an application of our results we show that one can characterize our new large cardinal notions and the corresponding ideals in terms of generic elementary embeddings; as a special case this yields generic embedding characterizations of -indescribability and Ramseyness.

Ideal operators and higher indescribability

“…We can show yet another form of proper containment of ideals when O = I. An analogous result for the operator R was claimed by the first author in [Cod20], see our Question 10.6 below.…”

“…In this section, we analyze the interplay between our generalized operators, and ideals of higher indescribability. Such an analysis for the Ramsey operator R and the ideals Π 1 ξ (κ) for ξ < κ has been performed by the first author in his [Cod20]. Let us start this section by citing a classic result of Baumgartner, that we will extend afterwards.…”

“…The next result that is claimed in [Cod20] is its [Cod20, Theorem 4.2], which suffers the same kind of problem as does its [Cod20, Theorem 4.1] (and it is now seen to be wrong for it includes the base case when α = 0, which is [Cod20, Theorem 4.1]). However, if its statement is modified according to the modification of [Cod20,Theorem 4.1] that we provided in Theorem 10.3 above, it is not clear as to whether Question 10.4. Assume that κ is a cardinal, that γ < κ is an ordinal, that S ∈ R γ+1 ([κ] <κ ) + , and that T = {α < κ | ∀ξ < α + S ∩ α ∈ R γ (Π 1 ξ (α)) + }.…”

“…The proof that is provided however is slightly flawed, and in fact only yields a somewhat weaker result, namely a weak form of the analogue of Theorem 3.8 for the Ramsey operator R rather than the ineffability operator I, in the special case when γ = 0 (see below). We first want to provide a counterexample for the above statement that is claimed in [Cod20], and then follow it with a corrected version of that theorem. We then shortly discuss the consequences that this has on other results of [Cod20].…”

“…We first want to provide a counterexample for the above statement that is claimed in [Cod20], and then follow it with a corrected version of that theorem. We then shortly discuss the consequences that this has on other results of [Cod20]. For the very start, we need an auxiliary result.…”