What is the significance for contemporary philosophy of mathematics of the fact that the Programme is unachievable?

Style: Chicago/TurabianNumber of source/references: 7Order instructions:A paper in the topic of Philosophy of Mathematics, it is a 3rd year BA paper in Philosophy.
Disclaimer: The topic is Philosophy of Mathematics, but the emphasis should lie more on philosophy than technical maths. Graphs are not discouraged and mathematical theorems can be analyzed, but it is preferred to have a philosophical foundation regarding the topics discussed.
Within the additional materials I can upload other possible essay titles to choose from, but the one I have written in the topic regarding Hilbert’s Programme is the one I would prefer.
Regarding the essay question: the fact that the programme is unachievable links to Gödel’s 2nd incompleteness theorem. The essay does not have to necessarily go through Gödel’s theorems per se, but focus more on the origin and the goal(s) of Hilbert’s Programme as a proposed solution to the foundational crisis of mathematics, while the remaining parts of the essay attempts at answering what implications Gödel’s disproving of Hilbert’s programme gave to the topic of contemporary philosophy of mathematics.
If the writer would like to know anything further, or the readings that are attached to this specific topic, please get in touch.
There is no required amount of sources, but between 4-10 sources seems like a good amount, not all of these sources would have to be cited directly.