"The young Austrian mathematician Kurt Gödel tackled the problem and found that Hilbert’s expectations where [sic] fulfilled for first-order logic."
This (i.e., "Hilbert's expectations were fulfilled for first-order logic") is a common misconception, but it is not the case. For instance, ZFC is an example of a first-order theory that does not satisfy Hilbert's expectations mentioned in the article (consistency, completeness and decidability). Notably, if ZFC is consistent, then it is incomplete. Another example of a first-order theory that is essentially incomplete and not decidable is Robinson arithmetic:
https://en.wikipedia.org/wiki/Robinson_arithmetic
The confusion likely stems from the fact that Gödel proved first-order logic to be semantically complete (Gödel's completeness theorem), i.e., all its valid formulae can be derived as theorems. This does not mean that all first-order theories are syntactically complete, i.e, for every statement S, either S or its negation ¬S is in the theory. This kind of incompleteness is what Gödel's incompleteness results are about, and this phenomenon already arises in first-order logic. The common independence results of ZFC are well-known examples of this kind of incompleteness: ZFC, if consistent, does not prove them, and also does not prove their negation. Gödel's first incompleteness theorem shows that every sufficiently powerful formal system exhibits this kind of incompleteness. Also first-order theories exhibit it if they are sufficiently powerful.
With today's terminology and models of computation, it is easy to show this, because first-order logic is sufficient to express a Turing machine as a theory, and if we let S be the (first-order) statement "The machine reaches a halting state", then, if we had a mechanism that always terminates and always correctly decides whether S or its negation ¬S is in the theory, then we would have an algorithm that solves the halting problem, which, however, is easily shown to be only semi-decidable, not decidable. Hence, not all first-order theories are consistent, complete and decidable.