Is HOTT, a new attempt at foundation of mathematics, free from incompleteness theorem or is it still suffering?

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Do mathematicians who study Homotopy Type Theory think that it can be completely free from Godel-Rosser theorem?

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The goals of use HTT or variants thereof are to provide a better foundation than ZFC in a variety of respects such as being easier to work with in computer checked proofs. They don't intend to avoid Godel's theorems, and you cannot. If your system has a recursively enumeralable axiom list, is consistent, and can model Peano Arithmetic then Godel's theorems apply. Heck, you can get away with even weaker conditions: You can replace being able to model PA with just being able to model Robinson Arithmetic https://en.wikipedia.org/wiki/Robinson_arithmetic and the results still apply.