Indescribable numbers: The theorem that made me fall in love with math

Abstraction: Proof that almost all real numbers are indescribable via cardinality

Key points:

• "Indescribable numbers" are real numbers for which no finite string of symbols in any mathematical language can serve as a description.
• Key insight: all mathematical descriptions are finite strings from a finite symbol set, so the set of all descriptions has cardinality Aleph-zero (countable).
• Real numbers have cardinality Aleph-one (uncountable), strictly larger than Aleph-zero; therefore most reals cannot be paired with any description.
• Conclusion: the vast majority of real numbers are indescribable; the "familiar" numbers (integers, rationals, algebraics, pi, e, etc.) are a vanishingly tiny fraction.
• The author independently rediscovered this proof as a first-year EE student; a logic professor confirmed it had been known since the 1940s.
• The proof requires only basic knowledge of cardinality (Cantor's diagonal argument background) and is accessible to early undergraduates.

Connections: Set Theory · Countability · Real Numbers · Infinity

Source: http://blog.ram.rachum.com/post/54747783932/indescribable-numbers-the-theorem-that-made-me-fall-in