# Non-absoluteness of Hjorth's Cardinal Characterization

@inproceedings{Lucke2021NonabsolutenessOH, title={Non-absoluteness of Hjorth's Cardinal Characterization}, author={Philipp Lucke and Ioannis A. Souldatos}, year={2021} }

In [5], Hjorth proved that for every countable ordinal α, there exists a complete Lω1,ω-sentence φα that has models of all cardinalities less than or equal to אα, but no models of cardinality אα+1. Unfortunately, his solution does not yield a single Lω1,ω-sentence φα, but a set of Lω1,ω-sentences, one of which is guaranteed to work. It was conjectured in [9] that it is independent of the axioms of ZFC which of these sentences has the desired property. In the present paper, we prove that this…

## References

SHOWING 1-10 OF 13 REFERENCES

The Bounded Proper Forcing Axiom

- Computer Science, MathematicsJ. Symb. Log.
- 1995

The bounded proper forcing axiom BPFA is equivalent to the statement that two nonisomorphic models of size @1 cannot be made isomorphic by a proper forcing notion, and the consistency strength of the bounded properforcing axiom is exactly the existence of a §1-re∞ecting cardinal.

Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence

- Mathematics, Computer ScienceNotre Dame J. Formal Log.
- 2014

The set of cardinals that are characterized by a Scott sentence is closed under successors, countable unions and countable products, and it is proved that if $\aleph_\alpha$ is characterized byA Scott sentence, at least one of $\ aleph_alpha$ and $ is homogeneously characterizable.

Directed sets and cofinal types

- Mathematics
- 1985

We show that 1, S. @1, X x xl and [l]<@ are the only cofinal types of directed sets of size Sl, but that there exist many cofinal types of directed sets of size continuum. A partially ordered set D…

Bounded forcing axioms as principles of generic absoluteness

- Computer Science, MathematicsArch. Math. Log.
- 2000

It is shown that Bounded Forcing Axioms imply a strong form of generic absoluteness for projective sentences, namely, if a $\Sigma^1_3$ sentence with parameters is forceable, then it is true.

Knight's Model, its automorphism Group, and Characterizing the uncountable Cardinals

- Mathematics, Computer ScienceJ. Math. Log.
- 2002

We show that every ℵα (α<ω1) can be characterized by the Scott sentence of some countable model; moreover there is a countable structure whose Scott sentence characterizes ℵ1 but whose automorphism…

DISJOINT AMALGAMATION IN LOCALLY FINITE AEC

- Computer Science, MathematicsThe Journal of Symbolic Logic
- 2017

The first examples of a class of models of a complete sentence in ${L_{{\omega _1},\omega }}$ where the spectrum of cardinals in which amalgamation holds is other that none or all are provided.

The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal

- Computer Science
- 1999

The second edition of a well-established monograph on the identification of a canonical model in which the Continuum Hypothesis is false is updated to take into account some of the developments in the decade since the first edition appeared.

The nonstationary ideal in the Pmax extension

- J. Symbolic Logic,
- 2007