Σ 1-wellorders without collapsing

Abstract

Given an uncountable cardinal κ that satisfies κ <κ = κ, we provide a forcing that is <κ-closed, has size 2 κ and is κ + -cc (and thus in particular preserves all cofinalities), to introduce a Σ 1 -definable wellorder (with parameters) of H(κ + ). This improves (and also simplifies the proof of) the main result of Holy and Lücke (Fundam Math 226 (3):221-236, 2014), where such a wellorder is introduced by a forcing which potentially collapses cardinals, and where the additional requirement that 2 κ be regular is needed. As an application, we use this to infer that Σ 1 -definable wellorderings (using parameters) of H(κ + ) can be introduced for many different cardinals κ simultaneously, while preserving a lot of ground model structure, improving results of Friedman and Holy (Fundam Math 215 (2):133-166, 2011) and Friedman and Lücke (Ann Pure Appl Logic, accepted). Moreover the results of this paper answer Holy and Lücke (2014, Questions 5.2 and 5.3).

Simplest possible locally definable well-orders

“…3 While the forcing construction in our paper is based on the construction in [11], one could instead base it on the construction that was later provided in [10]. This would eliminate the assumption that 2 κ be regular and would yield a forcing that preserves all cofinalities (rather than just those less than or equal to 2 κ ) in our below results.…”

“…10 will make sure that the complexity of those definitions will in fact be ∆ 1 -definable. Note that eachS α is a fat stationary subset of κ.…”

“…9 We will show later that this case never occurs (see Corollary 5.12). 10 The idea behind this construction is that the set a p collects information about the interpretations of names in A p that is already decided by the condition p. This will allow us to use the almost disjoint coding part of the forcing (see Clause (iv), (b)) to add a subset of κ that in the end codes p∈G a p and thus also p∈G A p whenever G is Given conditions p and q in P γ , we define q ≤ Pγ p to hold if s p = s q (β p + 1), t p = t q (β p + 1), d p,α = d q,α (β p + 1) for every α ≤ β p , a p ⊆ a q , A p = A q γ p and c p,x = c q,x (β q + 1) for every x ∈ a p . Proposition 5.3.…”