One way to understand Gödel's theorem (in combination with his 1929 "completeness theorem") is that no system of logical axioms can produce all truths about numbers because no system of logical axioms can pin down exactly what numbers are. My fourth-grade teacher used to ask the class to define a peanut butter sandwich, with comic results. Whatever definition you propose (say, "two slices of bread with peanut butter in between"), there are still lots of non-peanut-butter-sandwiches that fall within its scope (say, two pieces of bread laid side by side with a stripe of peanut butter spread on the table between them). Mathematics, post-Gödel, is very similar: There are many different things we could mean by the word "number," all of which will be perfectly compatible with our axioms. Now Gödel's undecidable statement P doesn't seem so paradoxical. Under some interpretations of the word "number," it is true; under others, it is false.
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How can this be, when Gödel cuts the very definition of "number" out from under us? Well, don't forget that just as there are some statements that are true under any definition of "peanut butter sandwich"—for instance, "peanut butter sandwiches contain peanut butter"—there are some statements that are true under any definition of "number"—for instance, "2 + 2 = 4." It turns out that, at least so far, interesting statements about number theory are much more likely to resemble "2 + 2 = 4" than Gödel's vexing "P." Gödel's theorem, for most working mathematicians, is like a sign warning us away from logical terrain we'd never visit anyway.