If some infinitary sentence provides a counterexample to
Vaught's Conjecture, then there is an infinitary sentence which also
provides a counterexample but has no model of cardinality bigger than
ℵ₁.
@article{1172787544,
author = {Hjorth, Greg},
title = {A Note on Counterexamples to the Vaught Conjecture},
journal = {Notre Dame J. Formal Logic},
volume = {48},
number = {1},
year = {2007},
pages = { 49-51},
language = {en},
url = {http://dml.mathdoc.fr/item/1172787544}
}
Hjorth, Greg. A Note on Counterexamples to the Vaught Conjecture. Notre Dame J. Formal Logic, Tome 48 (2007) no. 1, pp. 49-51. http://gdmltest.u-ga.fr/item/1172787544/