Godel assumed $\omega$-consistency for his proof of the first incompleteness theorem to run. But, isn't it the case that if we assume that a theory $T$ is effective, complete and $\omega$-consistent, then this would entail that $ T \vdash \operatorname {Con}(T)$? On the other hand, if we assume that $T$ is effective, complete, and proves its own consistency; then would this necessarily entail that $T$ is $\omega$-consistent? If NO, then demanding $ T\vdash \operatorname{ Con}(T)$ would be a weaker assumption that does the job! So, it can prove incompleteness even for some theories that are not $\omega$-consistent. By then, the question would be why Godel chose $\omega$-consistency, while he could have chosen a weaker obvious assumption. If YES, then $\omega$-consistency would be a simpler statement. So, which case is true?
Based on comments by @spaceisdarkgreen it would be more reasonable to ask about replacing the $\omega$-consistency assumption by the condition of $T \not \vdash \neg \operatorname{Con}(T)$, i.e., the theory not proving its own inconsistency.
So, Godel's incompleteness would be presented as:"every effective (represents all computable functions) theory that doesn't prove its own inconsistency, is incomplete".
This will cover all effective $\omega$-consistent theories, and as well would add other theories on the table! For example, an arithmetically sound but $\omega$-inconsistent theory.
The whole setting of this question is about assumptions for the original Godel proof to run. And, so it is not about Rosser's improvements. So, here, I'm arguing that $T \not \vdash \neg \operatorname {Con}(T)$ might constitute a better alternative to $\omega$-consistency, for that particular purpose.
