Brouwer’s point was that when dealing with infinite sets, use of the law similarly assumes the existence of things we have no right to assume.

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

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

One way to reduce the abstract notion of numbers to logical concreteness was through set theory. A set, or “class” as Russell called it, is simply a group of things. The elements that make up a set can be either explicitly listed or defined by a shared property: the set of black cats; the set of primary colors; the set of the numbers 1, 7, and 23;

he plainly disagreed with the reverence for Wittgenstein’s idea that mathematics, like language, was merely a tool, a set of rules or a syntax that had no inherent meaning in itself.

that it is impossible to derive from the axioms both a proposition and its negation,

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

“There exists some value of x for which the proposition F(x) holds.”

any formal system which contains arithmetic,