6.2 Infinite sets and Dedekind-infinite sets

Infinite sets are mind-bending mathematical concepts that stretch our understanding of numbers. They contain endless elements, yet some can be counted while others can't. It's like trying to count grains of sand on a beach.

Dedekind-infinite sets take this idea further. They're so big that even if you remove some elements, you can still match them up with the original set. It's like having an infinite hotel where you can always make room for more guests.

Infinite Sets

Defining Infinite Sets

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• Infinite set is a set that contains an infinite number of elements
• Cannot be counted or enumerated completely, regardless of how much time is available
• Cardinality, the size of an infinite set, is denoted by the symbol $\aleph$ (aleph)
• Two main types of infinite sets: countably infinite and uncountably infinite

Examples of Infinite Sets

• Natural numbers $\mathbb{N} = \{1, 2, 3, \ldots\}$ form a countably infinite set
  • Each natural number has a unique successor, creating an infinite sequence
  • Can be put into a one-to-one correspondence with the positive integers
• Integers $\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$ also form a countably infinite set
  • Includes both positive and negative whole numbers, as well as zero
  • Can be mapped to the natural numbers by interleaving positive and negative integers (e.g., 0, 1, -1, 2, -2, ...)

Dedekind-Infinite Sets

Defining Dedekind-Infinite Sets

• Dedekind-infinite set is an infinite set that contains a proper subset that can be put into a one-to-one correspondence with the original set
• Proper subset is a subset that is not equal to the original set
• One-to-one correspondence means that each element in one set can be paired with exactly one element in the other set, with no elements left unpaired

Properties of Dedekind-Infinite Sets

• All Dedekind-infinite sets are infinite, but not all infinite sets are Dedekind-infinite
• Removing a finite number of elements from a Dedekind-infinite set results in a set that is still Dedekind-infinite
• Union of two Dedekind-infinite sets is also Dedekind-infinite
• Cartesian product of two Dedekind-infinite sets is Dedekind-infinite

Types of Infinite Sets

Countably Infinite Sets

• Rational numbers $\mathbb{Q}$ form a countably infinite set
  • Can be expressed as fractions $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$
  • Can be put into a one-to-one correspondence with the natural numbers using a diagonalization argument (e.g., $\frac{1}{1}, \frac{1}{2}, \frac{2}{1}, \frac{1}{3}, \frac{2}{2}, \frac{3}{1}, \ldots$)
• Algebraic numbers, roots of polynomials with integer coefficients, form a countably infinite set

Uncountably Infinite Sets

• Real numbers $\mathbb{R}$ form an uncountably infinite set
  • Includes rational and irrational numbers, which cannot be expressed as fractions
  • Cannot be put into a one-to-one correspondence with the natural numbers
  • Cardinality of the real numbers is denoted by $\mathfrak{c}$, the cardinality of the continuum
• Cantor's theorem proves that the power set of any set has a strictly greater cardinality than the original set, implying the existence of uncountably infinite sets