How Math Explains the World by James D. Stein Jr

Author:James D. Stein, Jr.
Language: eng
Format: epub
ISBN: 9780061828683
Publisher: HarperCollins

Proofs of Gödel’s

There are many different ways to go about demonstrating Gödel’s theorem. I have elected to go with demonstrating here that it is plausible and have given a reference to formal proof in the notes to this chapter that gives the flavor of Gödel’s original proof.4

Gödel took the liar’s paradox, and modified the sentence This statement is false (which, as we have seen, lies outside the realm of statements that can be judged to be true or false) to This statement is unprovable. That was Gödel’s starting point, but by a technique known as Gödel numbering, which is described in the proof, he linked unprovability of statements to unprovability of statements about integers in the Peano axioms framework. If the statement This statement is unprovable is unprovable, then it is true, and the link Gödel established with arithmetic showed that there exist unprovable statements in number theory. If the statement This statement is unprovable is provable then it is false, and Gödel’s proof linked this conclusion to the inconsistency of the Peano Axioms.5

What exactly is meant by the word unprovable? It means just what it says, that there is no proof that will determine the truth or falsity of the statement. Needless to say, the existence of unprovable statements raises some questions. There are two schools of thought on the subject. Recall that the uncertainty principle is interpreted by most physicists to mean that conjugate variables do not have specifically defined values, not that humans are just not good enough to measure the specifically defined variables. One group of mathematicians interpret unprovability in the same fashion—it isn’t that we aren’t bright enough to prove that a statement is true or false, it’s that if logic is used as the ultimate arbiter, it is inadequate to the task. Others view an unprovable statement as one that is inherently true or false, but the system of logic used just doesn’t reach far enough to discern it.

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