[bees_buzz]

> he never mentioned Principia Mathematica again.

???

This paper contains a nice summary of Gödel's interactions with Russell. https://www.cambridge.org/core/services/aop-cambridge-core/c...

Gödel's future work on constructible hierarchies and the dialectica interpretation were directly inspired by Principia Mathematica. I agree that characterizing Gödel's view on Principia as a "step backwards" is overly simplistic, but he certainly mentions and takes inspiration form the work throughout his career.

> as he was mainly criticizing the syntax

If you mean to say that Gödel's criticisms of Principia were just about syntax - that is also incorrect. Gödel had substantial complaints about the truthfulness of the axiom of reducibility that (IIRC) Russell himself thought posed a valid attack on the system in Principia.