Googology Wiki
Googology Wiki

Addition is an elementary binary operation, written \(a + b\) (pronounced "\(a\) plus \(b\)"). It can be informally defined as the total number of objects when \(a\) objects are combined with \(b\) more. Formally, on finite real and cardinal numbers (but not on ordinals), it means the cardinality of a set formed by the union of two disjoint sets with cardinalities \(a\) and \(b\). \(a\) and \(b\) are called the summands, and \(a + b\) is called the sum.

On natural numbers, addition can be defined recursively using the successor function denoted by \(S(n)\):

\begin{eqnarray*} a+0 & = & a \\ a+S(b) & = & S(a+b) \end{eqnarray*}

In googology, it is the first hyper operator, and forms the basis of all following hyper operators.

Addition is commutative on \(\mathbb{N}\) and \(\mathbb{R}\): \(a + b = b + a\) for all natural or real values of \(a\) and \(b\). It is also associative, meaning that \((a + b) + c = a + (b + c)\). Repeated addition is called multiplication.

However, addition is not commutative on ordinals. For any limit ordinal \(\alpha\), \(1+\alpha = \alpha \neq \alpha+1\).

Zero is the additive identity, meaning that \(0 + n = n\) for all \(n\).

In other notations

Notation Representation
Up-arrow notation \(a \uparrow^{-1} b\)
Fast-growing hierarchy \(f_0^b(a)\)
Hardy hierarchy \(H_{b}(a)\)
Slow-growing hierarchy \(g_{\omega+b}(a)\)

Dependency of properties on arithmetics

When we work in ZFC, properties of addition such as commutativity and associativity hold. However, in Robinson arithmetic, the commutativity does not necessarily hold. The statement \(\forall a,b \in \mathbb{N} : a + b = b + a\) is independent of Robinson arithmetic.

See also