A few days ago, I heard a biologist mention that one implication from Gödel's Incompleteness Theorems is that an unlimited number of general statements can account for a given set of observations. Therefore, confirmation of observational consequences of a given general statement does not only confirm that statement, but also all the others and thus nothing is learned about reality. Initially, I doubted the validity of the claim, but after a while, I could see that this could be a reasonable implication. I'm wondering if somebody could confirm why this claim is true, or explain why it is false.
I'd offer this as a comment, but for some reason I can't comment until I have 50 reputation, and right now I only have 41.
There might be a relevant implication of the Incompletness Theorems that I'm forgetting, but this sounds to me more like a version of the problem of induction than incompleteness. Spurious references to incompleteness are pretty common; for examples, see the Wikipedia and the Stanford Encyclopedia entries on the Incompleteness Theorems.
This is just another way of putting Popper's objection to verification in science in general.
Statistically, and in other objective ways, only the experimental failure of a statement can really mean something, in the sense of clearly causing other statements to be more or less likely to be true, and even then only to a very limited degree. Since failure to fail can be taken as a sort of meta-statement, verification has an effect, but is an almost meaninglessly weak force, operating very indirectly upon our warranted expectations. (Since it has significant psychological force, that indicates that verification acts primarily upon our unwarranted expectations.)
It is not really a consequence Goedel, but of the overall structure of model theory and mathematical models in general. Whether or not a scientific theory is completable in principle, it is never complete in practice, and what applies to its truth is more of a matter of informal statistics than formal logic.
If you wanted to put a basis under this idea from formal logic, it would not be Goedel, but something like the Lowenheim-Skolem theorem, that every axiomatization with any infinite model has more than infinitely many different models -- at least one of every cardinality. And any individual fact from one of those models can be 'observed' and added and still leave more than infinitely many options. But that would still be overkill.
@jobermark mentioned Popper, but I think neglected to explicitly mention his major contribution, falsifiablilty, relevant to your question (though I think he hinted at it). What's not relevant is that observations may confirm one or more theories. Conversely, it's the sine qua non of a (scientific) theory that it must be falsifiable. Then any observation contrary to a theory's predictions disproves it.
You're kind of trying to say that observations confirm theories, and in fact may confirm many different theories. But no number of observations confirm any theory; they're merely consistent with one (or more) theories. And there's no problem about the consistency of a set of observations with several different theories. Popper's point is that as you make more and more observations, you'll (hopefully) begin to come across inconsistent observations that eventually falsify all-but-one of your theories. And even then, the remaining theory hasn't been absolutely confirmed, per se. Theories aren't confirmable, they're falsifiable.