Incompleteness theorem validity on assertions with no referential cycles Popularized expositions of the Gödel Incompleteness theorem typically show the theorem to rely on a self-referential assertion that cannot be proved from the existing axioms. 
Self-referential assertions are known to be problematic as they can lead to verbal paradoxes as Russell's paradox
Hence this leads me to question: Consider the subset of mathematical assertions that have referential cycles (i.e. their assertions have references that can lead to the original assertion), not sure if this subset has a customary name in the field, but let's call it the set of self-cyclic assertions $\bf{SC}$. Now consider the complement of this set ($\bf{SC^\complement}$), which might include assertions that have references that can be mapped to some directed acyclic graph.
Questions in my mind:
1) is $\bf{SC^\complement}$ a well-defined set? one in which one or more assertions can either belong to it or not in a provable way
2) Does the Incompleteness theorem still apply to assertions in $\bf{SC^\complement}$?
Addendum
It seems that the notion of reference of assertions is ill-defined as long as many assertions with the same amount of information are formally different assertions.
Consider the assertion (think of $P$ as a Gödel index)


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*$P$: Assertion $P$ is false
And consider now these two assertions (again, think of $M$ and $N$ as a Gödel indices):


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*$M$: Assertion $N$ is false 

*$N$: Assertion $M$ is false
Ideally, one should be able to define an equivalence class for assertions identity such that one can "quotient" the two assertions $M$ and $N$ into being identical to $P$. Tentatively I would hope that such equivalence $E(M) = E(N) = E(P)$ would make $\bf{SC} \Big/ E$ (and its complement) well-defined sets
 A: The obvious problem with this idea is, how exactly do you define SC? The whole point of Godel's theorem is that we can hide self-reference inside purely arithmetic statements - we may think of the Godel sentence as saying "This sentence is unprovable" (or similar), but the sentence is really "just" a statement about numbers and arithmetic.


*

*In particular, there isn't actually any "referential structure" in the sentence itself - that's a component of our analysis of the sentence.


The MRDP theorem pushes this to a further level: the theorem (or rather, the proof of the theorem + a little bit of auxiliary argument) shows that for every appropriate theory $T$ there is a Diophantine equation $\mathcal{E}_T$ such that $\mathcal{E}_T$ has no solutions but $T$ cannot prove that $\mathcal{E}_T$ has no solutions. Sentences of the form $$\forall x_1,...,x_n(f(x_1,...,x_n)\not=0)$$ are about as concrete as it gets, yet by the above observation wind up yielding incompleteness.
Put another way, there is no clear culprit for incompleteness. That isn't to say that there aren't reasonably-large decidable subtheories of a given theory (e.g. addition without multiplication is decidable), but this basically kills the hope of restricting incompleteness to the "unnatural" sentences.
