Good book for Gödel's incompleteness theorems

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This discussion centers on the recommendation of textbooks covering Gödel's incompleteness theorems, specifically highlighting "Gödel's Theorem Simplified" by Harry Gensler as a valuable resource. Participants clarify that Gödel's theorem demonstrates the existence of self-referential statements within formal systems, particularly through the encoding of statements into Gödel Numbers, which leads to inherent incompleteness. The conversation also addresses misconceptions about the depth of the theorem, emphasizing its significance beyond mere self-referential paradoxes.

PREREQUISITES
  • Understanding of formal logic and symbolic systems
  • Familiarity with Gödel Numbers and their encoding
  • Knowledge of the Principia Mathematica framework
  • Basic concepts of mathematical completeness and consistency
NEXT STEPS
  • Read "Gödel's Theorem Simplified" by Harry Gensler
  • Explore the implications of self-referential statements in formal logic
  • Investigate the historical context of Gödel's work within mathematical logic
  • Study the relationship between completeness and consistency in formal systems
USEFUL FOR

Students of mathematics, philosophers interested in logic, and educators seeking to deepen their understanding of Gödel's incompleteness theorems.

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Can anyone recommend a good textbook that would include Gödel's incompleteness theorems?

Also I have some basic questions:

From the stuff I read on the web it seems that Gödel's incompleteness theorem, basically just created a statement which is unprovable by its nature of being self-referential. as explained http://www.scienceforums.net/topic/29955-godels-theorem-for-dummies/"

Godel found a way of encoding a statement to the effect of "This statement is unprovable" into the symbolic logic system defined in Principia Mathematica (PM). The notable aspect of the statement is that it is self-referential, which Godel managed to accomplish by encoding statements in PM into "Godel Numbers." Thus the actual statement in PM refers to its own Godel Number.

To boil it down into a nutshell, I'd say it means that any system which is expressive enough to be consistent and complete is also expressive enough to contain self-referential statements which doom it to incompleteness.

Is that really all there is to it? In other words, do all of the unprovable statements take the form of a self-referential paradox? In which case, why do people think it is so deep? If that is the case, then it is trivial, because obviously if you construct a self-referential statement designed to paradoxical, then you won't be able to prove it, but so what?
 
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