Journey to the Edge of Reason: The Life of Kurt Gödel

If the Continuum Hypothesis is both consistent with the axioms of set theory (thus not disprovable) and independent of them (thus not provable), that would affirm the inadequacy of existing theory, as well as offering a stunning instance of the kind of undecidable proposition that, as his Incompleteness Theorem demonstrated, will exist within any c

every even number greater than 2 is the sum of two primes.

no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer n greater than 2.

the formula number for the resulting version of the formula is none other than . . .

Russell had made an analogous point with his whimsical example, “The present King of France is bald.” The sentence would appear to be false, since there is no King of France at present, but that would not warrant applying the Law of the Excluded Middle to conclude, “The present King of France is not bald.”

the axioms represent a truth that the human mind can perceive as such, even though they cannot be formally derived from any rules or any process that a machine can imitate, then a way around such skeptical resignation is opened.9 What he called “intuition” in this context had nothing to do with Brouwer’s “intuitionism”; he was using the term to sug

Some sets, Russell observed, could even be members of themselves. For example, the set of all sets containing more than two elements would certainly be such a set,

all subsets that can be formed from the infinite set of counting numbers is also one of those larger alephs—in fact, it is exactly equal to the cardinality of the continuum.

Imagine any list of real numbers between 0 and 1; then create a new number by selecting one digit in turn along a diagonal line running through them: 0 . 3 4 9 0 1 1 7 . . . 0 . 1 5 8 0 2 2 8 . . . 0 . 9 6 7 1 4 0 5 . . . 0 . 2 3 1 4 1 5 9 . . . 0 . 7 7 4 1 0 6 3 . . . 0 . 8 3 1 1 9 7 5 . . . If you then alter that new number (0.357407. . .in this