Works of Kurt Gödel I'd like to know how to get started with Gödel's work and theorems. I have a decent knowledge of tensors, Einstein field equations. When it comes to logic and set theory, I'm a beginner. Can anyone suggest some introductory texts that can prepare me for Gödel's theories?
 A: Gödel did epoch-making work in a number of fields:


*

*In pure logic, he was the first to prove the completeness of a system of the predicate calculus. 

*In what we might call the proof theory of formal systems, he proved the incompleteness [different sense!] of any formal system strong enough to encode a certain amount of arithmetic. (This required developing some of the theory of computable functions.)

*In set theory, he proved the consistency of the axiom of choice and of the continuum hypothesis with other, more basic, axioms of the set theory.

*He wrote a number of important philosophical essays, on the significance of these and other results for foundational questions about the nature of mathematics and our knowledge of it.


And that's not the end of it. E.g. you happen to mention Einstein's field equations. Gödel found a new solution which (in as sense) allows time travel. 
So what do you need to appreciate Gödel's work in various areas? Leaving aside his contribution on General Relativity, to get the full broad picture you'll need to develop a solid grounding in core mathematical logic, and in the philosophy of mathematics too. That's quite an undertaking!
But one aspect of his work -- what people often in fact have in mind when they talk about "Gödel's Theorem(s)" -- is actually both surprisingly accessible and particularly intriguing. That's (2), the incompleteness theorem(s). There are a number of nice introductions to this. @lhf mentions Torkel Franzen's very readable short book. And if you want quite a bit more detail, though still in a form that doesn't presuppose too much background, there is e.g. my Introduction to Gödel's Theorems (CUP, 2nd edn. 2013).

Gödel's own formal papers, by the way, tend to be terse, masterpieces of compression. He makes few concessions to his reader. So it is indeed best to approach his work via the plentiful secondary literature.
A: Try the book Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. 
