  The proofs of Kleene, Chaitin and Boolos for G\"odel's First Incompleteness
Theorem are studied from the perspectives of constructivity and the Rosser
property. A proof of the incompleteness theorem has the Rosser property when
the independence of the true but unprovable sentence can be shown by assuming
only the (simple) consistency of the theory. It is known that G\"odel's own
proof for his incompleteness theorem does not have the Rosser property, and we
show that neither do Kleene's or Boolos' proofs. However, we show that a
variant of Chaitin's proof can have the Rosser property. The proofs of G\"odel,
Rosser and Kleene are constructive in the sense that they explicitly construct,
by algorithmic ways, the independent sentence(s) from the theory. We show that
the proofs of Chaitin and Boolos are not constructive, and they prove only the
mere existence of the independent sentences.
