Non-absoluteness of Hjorth's Cardinal Characterization

TL;DR

This paper proves that Hjorth's characterization of certain infinite models is independent of ZFC axioms, showing the property varies across different set-theoretic assumptions like BPFA and CH.

Contribution

It establishes the independence of Hjorth's cardinal characterization from ZFC by introducing a diagonalization principle and analyzing its behavior under different set-theoretic axioms.

Findings

Hjorth's solution differs in models of BPFA versus CH.

The diagonalization principle can be forced over CH models.

Large cardinals are not necessary for the independence result.

Abstract

In [5], Hjorth proved that for every countable ordinal $\alpha$, there exists a complete $\mathcal{L}_{\omega_1,\omega}$-sentence $\phi_\alpha$ that has models of all cardinalities less than or equal to $\aleph_\alpha$, but no models of cardinality $\aleph_{\alpha+1}$. Unfortunately, his solution does not yield a single $\mathcal{L}_{\omega_1,\omega}$-sentence $\phi_\alpha$, but a set of $\mathcal{L}_{\omega_1,\omega}$-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 conjecture is true. More specifically, we isolate a diagonalization principle for functions from $\omega_1$ to $\omega_1$ which is a consequence of the Bounded Proper Forcing Axiom (BPFA) and then we use this principle to prove that Hjorth's solution to characterizing $\aleph_2$ in models of BPFA is different than in models of CH. In addition, we show that large cardinals are not needed to obtain this independence result by proving that our diagonalization principle can be forced over models of CH.