At the time this book was written, Raymond Smullyan was one of the world's leading experts on Godel's Incompleteness Theorems -- some of the deepest mathematical results of the 20th century.
This book is actually a gentle intro to these topics, and the most amazing part of it is that Dr. Smullyan keeps the level suitable for children.
That does NOT mean this book is not suitable for adults. It is extremely entertaining no matter what your age is.
The book is mostly a progression of logical conundrums. You are started out on the island of knights and knaves. These two types of people are visually indistinguishable, but knights always tell the truth while knaves always lie.
You are then presented with various scenarios where the objective is for you to ask one yes/no question from which you obtain some meaningful information without knowing whether the person you are asking is a knight or a knave. The obvious example of this, you meet one person on the island, and you want to ask them one yes/no question that allows you to determine whether they are a knight or a knave. Obviously, this would be pretty handy under the circumstances. Can it be done? Yes. Ask them, "If I were to ask you if you were a knight, would you say 'yes'?" A knight will always answer this question "Yes" and a knave "No". If you can follow the logic through to conclude this, you are on your way!! It's very easy to follow through for the knight, but the knave is a bit more tricky, but this example indicates the importance of case analysis and the use of hypotheticals in your questions to induce lying about lying.
The situations in the book steadily grow more complex. For example, later you find yourself on a similar island where the natives use the words "boo" and "da" for "yes" and "no". The problem is, you don't know which is which!
At the end of the book, you are presented with the ultimate level of complexity where not only do half the people always lie and half tell the truth, and not only do they use the words "boo" and "da" for "yes" and "no" (without you knowing which is which), but half of the population is also insane which means that whatever is true, they BELIEVE the opposite. So an insane liar always inadvertantly tells the truth because what they believe is false...and then they lie about it.
Sound tricky?
Yeah, that's the point.
Nonetheless, the book is a nice progression, and you definitely get better and better and following the logic through and thinking in these terms, which makes this book GREAT mental exercise! Some of the best I have found, in fact.
I will leave it to Dr. Smullyan to discuss to connection of these exercises with Godel's work.
One final comment, an earlier review is very wrong on the point of implication: an implication of the form a->b, is ALWAYS true when a is false. This is elementary formal logic -- a subject in which Dr. Smullyan was a renowned expert.
Interestingly, this strikes many people as highly objectionable as you will be able to tell from the comments to this review. The crux of the issue is that when I utter a statement of the form p->q, I am not asserting anything about the truth of either p or q. I am only asserting something about the relationship between the two. Namely, I am asserting that whenever p is true, q will be true as well. So the only way I can possibly be wrong about that -- that is, the only way the statement p->q can be false -- is if p is true and q is false.
And that is how we define the truth value of the statement p->q in mathematics: false if p is true and q is false, true in all other cases.
Ultimately, this works well for mathematics because mathematical statements are atemporal. For example, the statement "It is raining outside" might be true at some times, false at others. But that is not a mathematical statement. Mathematical statements like "2+2=5" are either always true or always false and we just have to figure out which.
For domains outside of mathematics, other conventions might be more appropriate...
Regardless, a wonderful book.
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