arXiv subjects

Ana Njegomir

Publications and source records attributed to Ana Njegomir.

3 recordsLinked to original sources

arXiv2018

Characterizing large cardinals through Neeman's pure side condition forcing

We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $ω_2$ in forcing extensions by the pure side condition forcing introduced by Neeman. The combinatorial properties that we make use of are natural principles, and in particular for inaccessible cardinals, these principles are equivalent to their corresponding large cardinal properties. Our characterizations make use of the concepts of internal large cardinals introduced in this paper, and of the classical concept of generic elementary embeddings.

math.LO↗

arXiv2017

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.

math.LO↗

arXiv2017

Small Embedding Characterizations for Large Cardinals

We show that many large cardinal notions can be characterized in terms of the existence of certain elementary embeddings between transitive set-sized structures, that map their critical point to the large cardinal in question. In particular, we provide such embedding characterizations also for several large cardinal notions for which no embedding characterizations have been known so far, namely for subtle, for ineffable, and for $λ$-ineffable cardinals. As an application, which we will study in detail in a subsequent paper, we present the basic idea of our concept of internal large cardinals. We provide the definition of certain kinds of internally subtle, internally $λ$-ineffable and internally supercompact cardinals, and show that these correspond to generalized tree properties, that were investigated by Weiß in his [16] and [17], and by Viale and Weiß in [15]. In particular, this yields new proofs of Weiß's results from [16] and [17], eliminating problems contained in the original proofs.

math.LO↗