Here are some suggestions and references. (This is a draft.)
Introduction to logics and Gödel's incompleteness theorems
• Good reference for learning logics: https://github.com/hongseok-yang/logic25
• Lean tutorial: https://adam.math.hhu.de/#/g/leanprover-community/nng4
We read M. Nakahara’s Geometry, Topology and Physics to learn the mathematical language commonly used in mathematical physics, and A. Yu. Kitaev’s paper Fault-tolerant quantum computation by anyons (Annals of Physics, 303(1):2–30, January 2003) to study the toric code.