8.2 Cardinal arithmetic operations

Cardinal arithmetic operations extend basic math to infinite numbers. Addition, multiplication, and exponentiation now apply to infinite cardinals. These operations follow familiar rules like commutativity and associativity, but with some unique twists for infinite sets.

Infinite sums and products take these ideas even further. They allow us to work with endless sequences of cardinals. König's theorem gives us important limits on these operations, helping us understand how big these infinite results can get.

Cardinal Arithmetic Operations

Defining Cardinal Arithmetic

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• Cardinal arithmetic operations generalize arithmetic operations to infinite cardinal numbers
• Cardinal addition extends the concept of adding natural numbers to infinite cardinals
• Cardinal multiplication extends the concept of multiplying natural numbers to infinite cardinals
• Cardinal exponentiation extends the concept of exponentiation to infinite cardinals

Properties of Cardinal Arithmetic

• Commutative property holds for cardinal addition and multiplication ($\aleph_0 + \aleph_1 = \aleph_1 + \aleph_0$, $\aleph_0 \cdot \aleph_1 = \aleph_1 \cdot \aleph_0$)
• Associative property holds for cardinal addition and multiplication ($(\aleph_0 + \aleph_1) + \aleph_2 = \aleph_0 + (\aleph_1 + \aleph_2)$, $(\aleph_0 \cdot \aleph_1) \cdot \aleph_2 = \aleph_0 \cdot (\aleph_1 \cdot \aleph_2)$)
• Distributive property holds for cardinal multiplication over addition ($\aleph_0 \cdot (\aleph_1 + \aleph_2) = \aleph_0 \cdot \aleph_1 + \aleph_0 \cdot \aleph_2$)
• Absorption laws state that for any two infinite cardinals $\kappa$ and $\lambda$, $\kappa + \lambda = \max(\kappa, \lambda)$ and $\kappa \cdot \lambda = \max(\kappa, \lambda)$ ($\aleph_0 + \aleph_1 = \aleph_1$, $\aleph_0 \cdot \aleph_1 = \aleph_1$)

Examples of Cardinal Arithmetic

• If $A$ is a set of even numbers and $B$ is a set of odd numbers, then $[|A|](https://www.fiveableKeyTerm:|a|) + |B| = |\mathbb{N}| = \aleph_0$
• If $C$ is the set of all functions from $\mathbb{N}$ to $\{0, 1\}$, then $|C| = 2^{\aleph_0} = \mathfrak{c}$ (the cardinality of the continuum)
• For any infinite cardinal $\kappa$, $\kappa \cdot \kappa = \kappa$ ($\aleph_0 \cdot \aleph_0 = \aleph_0$, $\aleph_1 \cdot \aleph_1 = \aleph_1$)
• For any two infinite cardinals $\kappa$ and $\lambda$, $\kappa^\lambda = \max(\kappa, 2^\lambda)$ ($\aleph_0^{\aleph_1} = 2^{\aleph_1} = \aleph_2$, assuming the Continuum Hypothesis)

Infinite Cardinal Operations

Infinite Sums and Products

• Infinite sums and products are generalizations of finite sums and products to infinite cardinals
• The infinite sum of a sequence of cardinals $(\kappa_i)_{i \in I}$ is defined as the cardinality of the disjoint union of sets with cardinalities $\kappa_i$ ($\sum_{i=0}^\infty \aleph_i = \aleph_\omega$)
• The infinite product of a sequence of cardinals $(\kappa_i)_{i \in I}$ is defined as the cardinality of the Cartesian product of sets with cardinalities $\kappa_i$ ($\prod_{i=0}^\infty \aleph_i = \aleph_\omega$)
• The infinite sum and product of a constant sequence of cardinals $\kappa$ are both equal to $\kappa$ ($\sum_{i=0}^\infty \aleph_0 = \aleph_0$, $\prod_{i=0}^\infty \aleph_0 = \aleph_0$)

König's Theorem

• König's theorem states that for any infinite cardinal $\kappa$, the sum of $\kappa$ many cardinals, each smaller than $\kappa$, is less than or equal to $\kappa$ ($\sum_{i < \kappa} \lambda_i \leq \kappa$ if $\lambda_i < \kappa$ for all $i < \kappa$)
• Corollary 1: The infinite sum of a sequence of cardinals $(\kappa_i)_{i \in I}$ is always less than or equal to $|I| \cdot \sup_{i \in I} \kappa_i$ ($\sum_{i=0}^\infty \aleph_i \leq \aleph_0 \cdot \aleph_\omega = \aleph_\omega$)
• Corollary 2: The infinite product of a sequence of cardinals $(\kappa_i)_{i \in I}$ is always less than or equal to $(\sup_{i \in I} \kappa_i)^{|I|}$ ($\prod_{i=0}^\infty \aleph_i \leq \aleph_\omega^{\aleph_0} = \aleph_\omega$)
• König's theorem and its corollaries provide upper bounds for infinite sums and products of cardinals

Examples of Infinite Cardinal Operations

• The set of all finite sequences of natural numbers has cardinality $\sum_{n=0}^\infty \aleph_0 = \aleph_0$
• The set of all infinite sequences of natural numbers has cardinality $\prod_{i=0}^\infty \aleph_0 = \aleph_0^{\aleph_0} = \mathfrak{c}$ (the cardinality of the continuum)
• If $(\kappa_i)_{i \in I}$ is a sequence of cardinals with $|I| = \aleph_0$ and $\kappa_i < \aleph_1$ for all $i \in I$, then $\sum_{i \in I} \kappa_i \leq \aleph_1$ by König's theorem