Незліченна множина

Незліченна множина (англ. Uncountable set) — це нескінченна множина, яка містить занадто багато елементів, щоб бути зліченною. Поняття незліченності тісно пов'язане з кардинальним числом множини: множина є незліченною, якщо її кардинальне число більше ніж кардинальне число множини натуральних чисел.

Характеристики

Є багато еквівалентних характеристик незліченності множини. Множина X є незліченною тоді і тільки тоді, коли виконуються будь-які з наступних умов:

  • Не існує ін'єктивного відображення X у множину натуральних чисел.
  • X не порожня, і для кожної ω-послідовності елементів X існує принаймні один елемент X, не включений у послідовність. Тобто, X не порожня множина і немає сюр'єктивного відображення множини натуральних чисел на X.
  • Потужність X не є ні скінченною, ні рівною (алеф-нуль, потужність натуральних чисел).
  • Множина X має потужність, строго більшу за .

Можна довести, що перші три характеристики є еквівалентними у теорії множин Цермело — Френкеля без використання аксіоми вибору. Еквівалентність третього і четвертого без цього неможливо довести.

Властивості

Якщо незліченна множина X є підмножиною Y, тоді Y також буде незліченною множиною.

Див. також

Джерела

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