Since we use the terms "complete" and "consistent" not only in formal logic, but in general discourse, it's tempting to think that Gödel's theorem tells us something about completeness and consistency even when we're talking about the Bible, Objectivism, and so on. However, it does not, for "complete" and "consistent" are not formally defined concepts in such contexts, and mathematical theorems (such as Gödel's theorem) only apply where the formally defined concepts used in the theorem can be applied.
A formal system has a certain associated formal language, and a set of associated formal axioms and rules of inference. "Formal" here means that the language and the axioms are defined precisely, like the syntax of a programming language, in a way that makes it a mechanical matter to decide whether something is a proof in the system. When we say that a formal system is complete, this also has a precise sense: for every sentence A in the language of the system, either A or its negation is a theorem of the system, i.e. can be derived using the axioms and rules of inference. Similarly, consistency has a precise sense: the system is consistent if there is no A such that both A and its negation can be derived.
In the case of the Bible, there is no formally defined language, and there are no formally defined axioms or rules of inference. What does or does follow from the Bible is not a mathematical question, but a question of understanding, interpretation, belief, judgment. Similarly for Objectivism, the Constitution, and so on. There aren't any mathematical theorems about what does or does not follow, in any ordinary sense, from the Bible, the Constitution, etc.
In particular, whether a theory, philosophy, story, legal document and so on is consistent or not is not a mathematical question, but one that turns essentially on questions of understanding, interpretation, belief, judgment. Consistency in the sense of Gödel's theorem, on the other hand, is a purely formal and mathematical matter.
Note that we didn't need Gödel's theorem to arrive at this conclusion. Indeed, we never need Gödel's theorem to conclude that stories, theories, philosophies, and so on, are "incomplete" in a sense analogous to the formal logical concept of incompleteness. The Constitution doesn't say anything about whether or not dancing the polka in Congress is allowed, so it is incomplete. Marxism doesn't imply any answer to the question how many elephants there are in the world, so it's incomplete. And so on. Clearly these are not profound observations.
When people think there is a point in invoking Gödel's theorem to support the view that "the Bible is incomplete or inconsistent", they don't mean such trivial incompleteness. Rather, they seem to have in mind something else, something that a Bible adherent might be expected to object to, for example that the Bible is incomplete in the sense of not adequately explaining every aspect of life. But all such non-trivial informal notions of incompleteness have nothing to do with incompleteness in the sense of Gödel's theorem. Using an informal analogue of incompleteness in the Gödelian sense we only get trivialities that we don't need Gödel to point out: the Bible is incomplete, Marxism is incomplete, Objectivism is incomplete, the morning paper is incomplete, everything is incomplete.
Goedel's proof is formulated in terms of number theory, but IS perfectly general. It applies equally well to living systems.or
The Gödelian structure of the unity of levels of Reality associated with the logic of the included middle implies that it is impossible to construct a complete theory for describing the passage from one level to the other and, a fortiori, for describing the unity of levels of Reality.or
By equating existence and consciousness we can apply pure mathematical results (including Gödel 's incompleteness theorem) to evolution.or again
Non-standard models and Gödel's incompleteness theorem point the way to God's freedom to change both the structure of knowing and the objects known.None of the "applications" or "implications" of Gödel's theorem referred to in such passages are applications or implications in any formal sense. They are suggested extrapolations or analogues of Gödel's theorem, or just vague ideas inspired by Gödel's theorem.