The geometry of Euclid has been a model of certainty for a couple of millennia. Hobbes or some other writer a couple of centuries back describes in a famous passage how an open book showing a proof of what we call the Pythagorean theorem, the one about the square of the hypotenuse equaling the square of the other two sides riveted him. Here, he thought, was the embodiment of God's thinking. Here, man had perceived ultimate truth, had grasped Reality with a capital R, was thinking the way God must also think. Modern thinkers say," Not so fast! Cool it a little." There are some caveats and conditions to Euclid's work that one might need to understand if one desires to have everything work out. As far back as the middle of the 1800's, Russian and German mathematicians were realizing that fundamental truths on a flat plane didn't always hold true on curved surfaces.
The basic structure of plane geometry was attractive, though. Start with some axioms that seem obviously true, self-evidently (!!! Beware!) true and build mathematical theory on those axioms. You can't go wrong.
Along came Kurt. He was not a very social guy or all that well developed at schmoozing. In fact, he was famously, seriously introverted and beyond, up to perhaps a little off. Take a look at his picture and you may be able to see what I mean. Take a look at his story by Rebecca Goldstein and see that he barely talked to anybody at Princeton except Einstein. But all that aside, he proved that any set of axioms rich enough to capture the truths of arithmetic would be either incomplete or contradictory. That is, there would be truths that could not be derived from the axioms (incomplete) or contradictions could be derived from them. Later work by Alan Turing, a British computer scientist and logician showed a same thing true in computing.
Godel's work on "formally undecidable truths" has excited attention in many other subjects than mathematical logic. (A key word here is "formally", that is by strict mathematical or logical operations.) This section on Franzen's book on Godel emphasizes ideas people have assumed Godel proved when he fact he did not:
"Among the many expositions of Godel's incompleteness theorems written for non-specialists, this book stands apart. With exceptional clarity, Franzen gives careful, non-technical explanations both of what those theorems say and, more importantly, what they do not. No other book aims, as his does, to address in detail the misunderstanding and abuses of the incompleteness theorems that are so rife in popular discussions of their significance. As an antidote to the many spurious appeals to incompleteness in theological, anti-mechanist and post-modernist debates, it is a valuable addition to the literature" - John W. Dawson, Jr., author of Logical Dilemmas: The Life and Work of Kurt Godel. Godel's Theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths... It plays a part in modern linguistic theories, which emphasize the power of language to come up with new ways to express ideas. And it has been taken to imply that you'll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself. - An Incomplete Education, by Judy Jones and William Wilson"
Franzen goes on to say
