Reddit reviews Set Theory An Introduction To Independence Proofs (Studies in Logic and the Foundations of Mathematics (Volume 102))

I would learn some mathematical logic (Enderton, Leary or Chiswell and Hodges are the usual suggestions. I'd avoid Enderton's, personally), and then come back to Kunen or Hrabeck and Jech (pdf). If you're feeling particularly ambitious, Jech's other book (pdf) is the reference for working set theorists, so may want to take glances at that while reading Kunen or H&J.

> It is really hard to find something that actually is - AFAIK (I'm not very advanced in math so I could be wrong) nobody has actually found a statement that we know fits the example.

I'm not sure what you mean by "the example" here, but this doesn't sound right. Godel's own proof constructed such a statement that was independent of Russel & Whitehead's theory. The R&W stuff has fallen out of favor these days, but Godel's construction has been shown to work everywhere relevant.

The new hotness is a set theory call Zermelo-Frankel Set Theory (colloquially, ZF). And we now know, for example, that several really important hypotheses are independent of ZF (and PA):

• The Continuum Hypothesis cannot be proved in ZF
  
• The Axiom of Choice cannot be proved in ZF
  
  Scott Aaronson has a very interesting survey on the state of P vs. NP that covers relevant ground:
  
  
• http://www.scottaaronson.com/papers/pnp.pdf
  
  The definitive work on Independence Proofs (afaik):
  
  
• http://www.amazon.com/Introduction-Independence-Studies-Foundations-Mathematics/dp/0444868399
  

If you need an introductory text into Set Theory and Logic, you should try Kunen's Set Theory: An Introduction to Independence Proofs or Jech's Set Theory.

Then I would recommend reading Aczel's paper on Non-well-founded sets (1988).

For some historical context, I would urge you to read the amazing graphic novel Logicomix.

All of these books can be found online.