Philipp Lücke

CLASS FORCING, THE FORCING THEOREM AND BOOLEAN COMPLETIONS

The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma can fail for class forcing. In addition to these negative results, we show that the forcing theorem is equivalent to the existence of a (certain kind of) Boolean completion, and we introduce a weak combinatorial property (approachability by projections) that implies the forcing theorem to hold. Finally, we show that unlike for set forcing, Boolean completions need not be unique for class forcing.

Large cardinals and definable well-orders, without the GCH

Abstract We show that there is a class-sized partial order P with the property that forcing with P preserves ZFC, supercompact cardinals, inaccessible cardinals and the value of 2 κ for every inaccessible cardinal κ and, if κ is an inaccessible cardinal and A is an arbitrary subset of κ κ , then there is a P -generic extension of the ground model V in which A is definable in 〈 H ( κ + ) V [ G ] , ∈ 〉 by a Σ 1 -formula with parameters. We use this result to construct a class-sized partial order with the above preservation properties that forces the existence of well-orders of H ( κ + ) definable in the structure 〈 H ( κ + ) , ∈ 〉 for every inaccessible cardinal κ. Assuming the GCH, David Aspero and Sy-David Friedman showed in [1] and [2] that there is a class-sized partial order preserving ZFC and various large cardinals and forcing the existence of a well-order of the universe whose restriction to H ( κ + ) is definable in 〈 H ( κ + ) V [ G ] , ∈ 〉 by a parameter-free formula for every uncountable regular cardinal κ. Our second result can be interpreted as a boldface version of this result in the absence of the GCH.

Automorphism Groups of Ultraproducts of Finite Symmetric Groups

It is consistent that there exists a nonprincipal ultrafilter 𝒰 over ℕ such that every automorphism of the corresponding ultraproduct is inner.

Σ1(κ)-DEFINABLE SUBSETS OF H(κ +)

We study Σ 1 ( ω 1 )-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ 1 -formula with parameter ω 1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ 1 ( ω 1 )-definable, the set of all stationary subsets of ω 1 is not Σ 1 ( ω 1 )-definable and the complement of every Σ 1 ( ω 1 )-definable Bernstein subset of

Characterizing large cardinals in terms of layered posets

{}_{}^{{\omega _1}}\omega _1^{}

Forcing lightface definable well-orders without the GCH

is not Σ 1 ( ω 1 )-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ 1 ( ω 1 )-definable well-ordering of H( ω 2 ) and the existence of a Δ 1 ( ω 1 )-definable Bernstein subset of

Measurable cardinals and good Σ1(κ)-wellorderings: Measurable cardinals and good Σ1(κ)-wellorderings

{}_{}^{{\omega _1}}\omega _1^{}

Small embedding characterizations for large cardinals

. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ 1 ( ω 1 )-definable uniformization of the club filter on ω 1 . Moreover, we prove a perfect set theorem for Σ 1 ( ω 1 )-definable subsets of

Choiceless Ramsey Theory of Linear Orders

{}_{}^{{\omega _1}}\omega _1^{}

LARGE CARDINALS AND LIGHTFACE DEFINABLE WELL-ORDERS, WITHOUT THE GCH

, assuming that there is a measurable cardinal and the nonstationary ideal on ω 1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙ max -forcing. Finally, we also prove variants of some of these results for Σ 1 ( κ )-definable subsets of κ κ , in the case where κ itself has certain large cardinal properties.