Gödel's Theorems and Zermelo's Axioms

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The book delves into the foundations of mathematics, starting with essential concepts of mathematical logic, including axioms and formal proofs. It meticulously presents Gödel's completeness and incompleteness theorems, featuring a comprehensive proof of the second incompleteness theorem, which asserts the impossibility of proving arithmetic's consistency within its own axioms. Additionally, it introduces modern axiomatic set theory based on Zermelo's axioms and explores Gödel's constructible universe, emphasizing various models of theories like Peano and Presburger arithmetic.