I think the most interesting way (at least for me) of learning about the Gödel’s Incompleteness Theorems is by reading Douglas Hofstadter´s book "Gödel, Escher, Bach: An Eternal Golden Braid".

Both content and form are carefully chosen, in such a way that you are introduced to very difficult concepts in a, let´s say, artistical way. And of course it is completely intentional. The structure of
the book is, as it has been said, a counterpoint between Dialogues and Chapters.

The author makes a explicit analogue of Bach´s music in his book. And, I think,
it shows the universality of the concepts he wants to show.

As a fugue, he first shows the main themes, and masterly proceeds to makes variations of it. He presents the concept of recursivity and isomorphism from an artistic point of view. He presents what he calls a "strange loop" in terms of music and painting. As he defines them it is a phenomenon that occurs whenever, by moving upwards (or downwards) through the levels of some hierarchical system, we unexpectedly find ourselves rightback where we started. You can find this in Bach´s music and in Escher paintings.

He also explains that implicit in the concept of Strange Loops is the concept of infinity, and the conflict between the finite and the infinite in Bach´s and Escher´s works.

He introduces the main theme as the so-called Epimenides paradox, or liar paradox. Epimenides was a Cretan who made one immortal statement: "All Cretans are liars." What can be said about that statement? It is true or false?

Godel´s idea was to use mathematical reasoning in exploring mathematical reasoning itself, and came up with the conclusion that "All consistent axiomatic formulations of number theory include undecidable propositions.

Hofstadter masterly shows the importance of strange loops in Gödel´s proof of his theorem. Once a formal system ask about its consistency and completeness, it cannot reach a conclusion. The equivalent
of Epimenides paradox in mathematics it´s "This statement of number theory does not have any proof". Whereas the Epimenides statement creates a paradox since it is neither true nor false, the Gödel sentence is unprovable (inside its own formalization) but true.

As Hofstadter concludes, Gödel’s Theorem shows that there are fundamental limitations to consistent formal systems with self-images. In particular, it cannot proves its own consistency.

It has been used to prove that we cannot compute the human mind, because we would be incomplete. There is the argument against this of humans not being consistent, so we could be inconsistent Turing machines, and therefore computable. Read more about the computability of the mind and the Gödel´s
theorems here.

I must say that it is not an easy book to read. You have to pay a lot of attention to it. But it is worth reading.

As a mathematical representation of one of Bach´s compositions, here you have the "Crab canon", from his Musical Offering, with which Hofstadter starts his book.

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