Cardinal arithmetic

We can define arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers. It can be shown that for finite cardinals, these operations coincide with the usual operations for natural numbers. Furthermore, these operations share many properties with ordinary arithmetic.

Successor cardinal

If the axiom of choice holds, then every cardinal κ has a successor, denoted κ⁺, where κ⁺ > κ and there are no cardinals between κ and its successor. (Without the axiom of choice, using Hartogs' theorem, it can be shown that for any cardinal number κ, there is a minimal cardinal κ⁺ such that \(κ^+≰κ\). For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the successor ordinal.

Cardinal addition

If X and Y are disjoint, addition is given by the union of X and Y. If the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality (e.g., replace X by X×{0} and Y by Y×{1}).

\( |X|+|Y|=|X\cup Y|.\)

• Zero is an additive identity κ + 0 = 0 + κ = κ.
• Addition is associative (κ + μ) + ν = κ + (μ + ν).
• Addition is commutative κ + μ = μ + κ.
• Addition is non-decreasing in both arguments: \( (\kappa \leq \mu )\rightarrow ((\kappa +\nu \leq \mu +\nu ){\mbox{ and }}(\nu +\kappa \leq \nu +\mu ))\).

Assuming the axiom of choice, addition of infinite cardinal numbers is easy. If either κ or μ is infinite, then

\( \kappa +\mu =\max\{\kappa ,\mu \}\,\).

Subtaction

Assuming the axiom of choice and, given an infinite cardinal σ and a cardinal μ, there exists a cardinal κ such that μ + κ = σ if and only if μ ≤ σ. It will be unique (and equal to σ) if and only if μ < σ.

Cardinal multiplication

The product of cardinals comes from the Cartesian product:

\( |X|\cdot |Y|=|X\times Y|\)

• κ·0 = 0·κ = 0.
• κ·μ = 0 → (κ = 0 or μ = 0).
• One is a multiplicative identity: κ·1 = 1·κ = κ.
• Multiplication is associative (κ·μ)·ν = κ·(μ·ν).
• Multiplication is commutative κ·μ = μ·κ.
• Multiplication is non-decreasing in both arguments: κ ≤ μ → (κ·ν ≤ μ·ν and ν·κ ≤ ν·μ).
• Multiplication distributes over addition: κ·(μ + ν) = κ·μ + κ·ν and (μ + ν)·κ = μ·κ + ν·κ.

Assuming the axiom of choice, multiplication of infinite cardinal numbers is also easy. If either κ or μ is infinite and both are non-zero, then

\( \kappa \cdot \mu =\max\{\kappa ,\mu \}\).

Division

Assuming the axiom of choice and, given an infinite cardinal π and a non-zero cardinal μ, there exists a cardinal κ such that μ · κ = π if and only if μ ≤ π. It will be unique (and equal to π) if and only if μ < π.

Cardinal exponentiation

Exponentiation is given by

\( |X|^{|Y|}=\left|X^{Y}\right|\),

where XY is the set of all functions from Y to X.

• κ⁰ = 1 (in particular 0⁰ = 1), see empty function.
• If 1 ≤ μ, then 0^μ = 0.
• 1^μ = 1.
• κ¹ = κ.
• κ^{μ + ν} = κ^μ·κ^ν.
• κ^{μ · ν} = (κ^μ)^ν.
• (κ·μ)^ν = κ^ν·μ^ν.

Exponentiation is non-decreasing in both arguments:

• (1 ≤ ν and κ ≤ μ) → (ν^κ ≤ ν^μ)
• (κ ≤ μ) → (κ^ν ≤ μ^ν).

2^|X| is the cardinality of the power set of the set X and Cantor's diagonal argument shows that 2^|X| > |X| for any set X. This proves that no largest cardinal exists (because for any cardinal κ, we can always find a larger cardinal 2^κ). In fact, the class of cardinals is a proper class. (This proof fails in some set theories, notably New Foundations.)

All the remaining propositions in this section assume the axiom of choice:

• If κ and μ are both finite and greater than 1, and ν is infinite, then κ^ν = μ^ν.
• If κ is infinite and μ is finite and non-zero, then κ^μ = κ.

If 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then:

• Max (κ, 2^μ) ≤ κ^μ ≤ Max (2^κ, 2^μ).

Using König's theorem, one can prove κ < κ^cf(κ) and κ < cf(2^κ) for any infinite cardinal κ, where cf(κ) is the cofinality of κ.

Roots

Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 0, the cardinal ν satisfying \( \nu ^{\mu }=\kappa \) will be \(\kappa \).

Logarithms

Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying \( \mu ^{\lambda }=\kappa \). However, if such a cardinal exists, it is infinite and less than κ, and any finite cardinality ν greater than 1 will also satisfy \( \nu ^{\lambda }=\kappa \).

The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2^μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces, though they lack some of the properties that logarithms of positive real numbers possess.