Cardinal number

Definition

A cardinal number is a number that indicates quantity, representing the size of a set. They are used to compare the sizes of different sets, such as finite sets, infinite sets, and can even illustrate the concept of different 'sizes' of infinity. Understanding cardinal numbers is essential for grasping deeper concepts in set theory, such as infinite sets and their properties.

5 Must Know Facts For Your Next Test

1. Cardinal numbers can be finite, like 1, 2, or 3, or infinite, such as the cardinality of the set of natural numbers represented by ℵ₀ (aleph-null).
2. Two sets have the same cardinal number if there exists a one-to-one correspondence between their elements.
3. Cantor's work on cardinal numbers showed that not all infinities are equal; for example, the set of real numbers has a greater cardinality than the set of natural numbers.
4. The Continuum Hypothesis proposes that there is no set whose cardinality is strictly between that of the integers and the real numbers.
5. Cardinal numbers play a crucial role in understanding paradoxes in set theory, such as Cantor's Paradox and Burali-Forti Paradox, which challenge naive understandings of size and infinity.

Review Questions

• How do cardinal numbers help differentiate between various sizes of infinity?
  • Cardinal numbers allow mathematicians to categorize different sizes of infinity by assigning distinct values to them. For instance, while both the set of natural numbers and the set of real numbers are infinite, they have different cardinalities; the natural numbers correspond to ℵ₀ (aleph-null), while the reals have a greater cardinality often represented by 2^{ℵ₀}. This understanding helps clarify concepts like the Continuum Hypothesis and shows that not all infinities are equivalent.
• What implications does Cantor's Paradox have on naive set theory regarding cardinal numbers?
  • Cantor's Paradox reveals limitations in naive set theory by demonstrating that the concept of 'the set of all sets' leads to contradictions. According to naive set theory, if we can form such a set, its cardinality would suggest it has more members than any given set. However, Cantor proved that there are larger cardinalities than any given infinite set. This paradox challenges our intuitive understanding of infinity and illustrates the need for more rigorous definitions in set theory.
• Evaluate Gödel's constructible universe in relation to the consistency of the Continuum Hypothesis and its connection to cardinal numbers.
  • Gödel's constructible universe provides a framework within which the Continuum Hypothesis can be shown to be consistent with standard set theory. In this universe, every set can be constructed step by step, allowing for a controlled environment to examine cardinalities. Gödel demonstrated that if certain axioms hold true, then it is impossible to prove or disprove the Continuum Hypothesis using those axioms. This ties back to cardinal numbers as it emphasizes their foundational role in understanding infinite sets and their relationships within set theory.

Related terms

Ordinal Number: An ordinal number describes the position of an element within a well-ordered set, indicating its order rather than quantity.

Set Theory: Set theory is a branch of mathematical logic that studies sets, which are collections of objects, and provides a foundation for much of modern mathematics.

Aleph Numbers: Aleph numbers are used to represent the cardinality of infinite sets, with each successive aleph number representing a larger cardinality.