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

how one could possibly prove or disprove the proposition that somewhere in the infinite decimal expansion of pi the sequence 7777 occurs.

In first-order logic, “some” or “all” statements can be made about objects (all cats, some numbers, some sets), but not about properties or sets of those same objects, which rules out such propositions as “there is some color that two cats share” or “some sets of cats contain a black cat.”

be 2ℵ0, which will thus equal one of those larger alephs.

Euclid’s argument ran like this: Assume there is a largest prime number, p. You can then create a larger number by multiplying together every prime number up to and including p, and adding 1. But that number, when divided by any of those primes, will leave a remainder of 1—so it, too, is prime (or, possibly is divisible by a prime larger than p, li

“One can, assuming the consistency of classical mathematics, even give examples of propositions (and in fact of the kind of Goldbach or Fermat) that really are contentually true but are unprovable in the formal system of mathematics,”

line, the so-called “continuum”—which is precisely equivalent to the set of all real numbers, which includes “irrational” numbers like pi and the square root of 2 that can be expressed only as an infinite series of decimals—are not countable by any possible scheme: they constitute a larger infinity than that of the counting numbers.

The scheme he devised made it possible to convert any string of symbols used to express a mathematical or logical formula in Principia Mathematica into a single, unique integer.

But Brouwer completely rejected the idea of applying the law to infinite sets.

there is no possible series of proof steps, starting from the axioms of the system and proceeding by its valid laws of inference, by which formula number g can be derived. But formula number g is that proposition G itself. G in other words states: “G is unprovable.”