Good book for Gödel's incompleteness theorems

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In summary, Gödel's incompleteness theorems state that any system that is expressive enough to be consistent and complete will also contain self-referential statements that are unprovable. This was demonstrated by Godel through his encoding of statements into "Godel Numbers" in Principia Mathematica. This means that all unprovable statements in such a system will take the form of a self-referential paradox. While this may seem trivial, it is considered deep because it has profound implications for the foundations of mathematics and logic. "Godels theorem simplified" by Harry Gensler is a recommended book for further understanding.
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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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FAQ: Good book for Gödel's incompleteness theorems

What is the significance of Gödel's incompleteness theorems?

Gödel's incompleteness theorems revolutionized the field of mathematics by proving that any formal system complex enough to encompass basic arithmetic will have statements that cannot be proven or disproven within that system. This means that no single system can ever be complete and consistent at the same time, challenging the notion of a perfectly logical and complete mathematical universe.

What is the best book for understanding Gödel's incompleteness theorems?

The best book for understanding Gödel's incompleteness theorems is "Gödel, Escher, Bach: An Eternal Golden Braid" by Douglas Hofstadter. This book not only explains the theorems in a clear and accessible manner, but also delves into their philosophical implications and connects them to other fields such as music and art.

How does Gödel's incompleteness theorems relate to the concept of truth in mathematics?

Gödel's incompleteness theorems demonstrate that there are statements in mathematics that cannot be proven true or false within a given system. This challenges the idea that mathematics is a completely objective and certain field, as it shows that there will always be statements that are undecidable or unknowable.

Can Gödel's incompleteness theorems be applied to other fields besides mathematics?

Although Gödel's theorems were originally formulated in the context of mathematics, their implications can be extended to other fields such as philosophy, computer science, and artificial intelligence. The idea that no system can be both complete and consistent has profound implications for our understanding of knowledge and truth in general.

Are there any criticisms or counterarguments to Gödel's incompleteness theorems?

There have been various attempts to refute or circumvent Gödel's theorems, but none have been widely accepted by the mathematical community. Some argue that the theorems only apply to formal systems and not to real-world applications of mathematics, while others argue that the theorems are not as revolutionary as they are often portrayed. However, the majority of mathematicians and philosophers accept Gödel's incompleteness theorems as fundamental and groundbreaking discoveries.

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