i was going through the original statement of ɷ-consistency as stated in godel's 1st incompleteness proof, but i saw this pararaph on wikipedia here - https://en.wikipedia.org/wiki/%CE%A9-consistent_theory#Definition ( its the 2nd para ).
A $\bf T$ that interprets arithmetic is ω-inconsistent if, for some property P of natural numbers (defined by a formula in the language of T), T proves P(0), P(1), P(2), and so on (that is, for every standard natural number n, T proves that P(n) holds), but T also proves that there is some natural number n such that P(n) fails.[2] This may not generate a contradiction within T because T may not be able to prove for any specific value of n that P(n) fails, only that there is such an n. In particular, such n is necessarily a nonstandard integer in any model for T (Quine has thus called such theories "numerically insegregative").
" In particular, such n is necessarily a nonstandard integer in any model for T (Quine has thus called such theories "numerically insegregative"). " going by this statement, this should mean that no godel statements exists in the system, since no such integer exist in our system for which prove godel incompleteness theorem thus the system is complete..
and if not, then what exactly does this means ?
$ $ $ $
edit - after Mauro's answer -
this makes the whole addition of ɷ-consistency to a system and concluding non-standard models - a bit pretentious, just for the sake of proving incompleteness.
In other words, is godel's 1st theorem just another way of saying " any theory , satisfying godel hypotheses, must have a model with an axiom added explicitly - which isn't syntactically constructible ( without using self-reference - even then we don't actually know its godel number ) in a negated version, to make other models purposefully seem syntactically incomplete ( which they shouldn't be cuz the godel sentence is nowhere actually to be find )" ?
even rosser's sentence on a second look can be seen as a clever trick to incorporate the essence of ɷ-consistency in the self-referential sentence itself, to remove restriction of ɷ-consistency ....
even in tony dean's comment in mauro's answer, that extra axiom seems counter-intuitive and for the sake of excluding ɷ-consistency restriction...
why we are so much obscessed with incompleteness ?
Long comment
The phenomenon of $\omega$-inconsistency con be described with this example (due to Henkin): for a theory $T$ that interprets arithmetic, if $N$ is a model of the theory there is no contradiction between assuming the validity of $0 \in N, 1 \in N, \ldots$ and the assumption that there is some $x \in N$ such that $x \ne 0, x \ne 1, \ldots$.
This means that, using theory $T$, we cannot prove that $N$ contains just $0, 1, \ldots$ and no other nameless objects besides this.
These "rogue" objects are called non-standard numbers.
This is the meaning of "numerically insegregative": theory $T$ is unable to "segregate" all and only those elements of its models that are "usual" numbers.
See W.V.O. Quine, On ω-inconsistency (JSL, 1953) and C.-Y. Cheng, On referentiality and its conditions (NDJFL, 1974).
But this does not contradicts Gödel First Incompleteness Theorem.
You can see the Wiki's examples regarding the theory $\mathsf {PA} + ¬ \text {Con}(\mathsf{PA})$ [where $\mathsf {PA}$ is first-order arithmetic]: this is a case of consistent but $\omega$-inconsistent theory.