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What were the benefits of arithmetizing syntax for Gödel? What did the arithmetization of syntax allow for Gödel that was otherwise not possible?

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Arithmetization of syntax allows Gödel to show that statements about number theory are also statements in number theory. This allows him to construct self-referential statements about number theory in a simple way. This in turn allows him to show that Self-reference is inevitable, and that it is impossible to avoid self-reference when trying to construct complete formal axiomatic systems rich enough to describe arithmetic (And thus refuting Russell's logicist project).

There are many resource, but a good informal one which accessible to mathematicians and non-mathematicians alike is Douglas Hofstadter's "I am a Strange Loop" Chapter 10 - Gödel's Quintessential Strange Loop.

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Self reference. When a formula can be encoded by a number and also take numbers as arguments, it can also take its own number as an argument, allowing to formulate something like "this sentence is not provable".

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