Simplest possible locally definable well-orders

Abstract

Abstract. We study the provable consequences of the existence of a well-order of H(κ + ) definable by a Σ 1 -formula over the structure H(κ + ), ∈ in the case where κ is an uncountable regular cardinal. This is accomplished by constructing partial orders that force the existence of such well-orders while preserving many structural features of the ground model. We will use these constructions to show that the existence of a well-order of H(ω 2 ) that is definable over H(ω 2 ), ∈ by a Σ 1 -formula with parameter ω 1 is consistent with a failure of the GCH at ω 1 . Moreover, we will show that one can achieve this situation also in the presence of a measurable cardinal. In contrast, results of Woodin imply that the existence of such a well-order is incompatible with the existence of infinitely many Woodin cardinals with a measurable cardinal above them all.

Forcing lightface definable well-orders without the GCH

“…Given an uncountable cardinal κ that satisfies κ <κ = κ and such that λ <λ < κ for every λ < κ, under additional anti-large cardinal hypotheses 2 it is possible to use the methods developed by the second and third authors in [9] and introduce a Σ 1 -definable well-order of H(κ + ) that only uses κ as parameter (and is thus Σ 3 -definable over H(κ + ) without parameters) by <κ-closed forcing that has the κ + -cc and has size 2 κ . In particular, this shows that it is consistent to have 2 κ large in the presence of a lightface definable well-order of H(κ + ).…”

“…Although we will never explicitly make use of this coding forcing, it will be woven into our main forcing construction in Section 5 and some of our later proofs will be variations of the arguments given in this section. Moreover the forcing itself might prove to be interesting (in fact it has already been made use of in [8] and [9]). We will call the coding we want to introduce club coding (relative to a stationary set).…”

Σ 1-wellorders without collapsing

“…This can for example be achieved by almost disjoint coding at κ (see [6,Sections 2 and 4]). 5 The core of our construction is to define the forcing used at the third stage.…”

“…Note however that as defined in[6, Section 2], the almost disjoint coding forcing at κ is not uniformly definable in parameter κ, as it depends on a prior choice of an enumeration of <κ κ. However uniformity can easily be achieved by first choosing such an enumeration generically.…”

“…However uniformity can easily be achieved by first choosing such an enumeration generically. The resulting coding forcing will then be as desired, the required dense set of conditions being those that decide for such an enumeration 6. We emphasize here that this last statement is about names and not their evaluations.…”