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Let <math>S\!</math> be the set of rooted trees and let <math>S_0\!</math> be the 2-element subset of <math>S\!</math> that consists of a rooted node and a rooted edge.  We may express these definitions more briefly as <math>S = \{ \operatorname{rooted~trees} \}</math> and <math>S_0 = \{ \ominus, \vert \}.</math> Simple intuition, or a simple inductive proof, will assure us that any rooted tree can be reduced by means of the axioms of the primary arithmetic either to a root node <math>\ominus</math> or else to a rooted edge <math>\vert~.</math>

Let <math>S\!</math> be the set of rooted trees and let <math>S_0\!</math> be the 2-element subset of <math>S\!</math> that consists of a rooted node and a rooted edge.  Expressed more briefly in various ways:

 

| <math>S\!</math>

| <math>=\!</math>

| <math>\{ \text{rooted trees} \}\!</math>

| <math>S_0\!</math>

| <math>=\!</math>

| <math>\{ \ominus, \vert \} = \{</math>[[Image:Cactus Node Big Fat.jpg|16px]], [[Image:Cactus Spike Big Fat.jpg|12px]]<math>\}\!</math>

 

Simple intuition, or a simple inductive proof, will assure us that any rooted tree can be reduced by way of the arithmetic initials either to a root node [[Image:Cactus Node Big Fat.jpg|16px]] or else to a rooted edge [[Image:Cactus Spike Big Fat.jpg|12px]]&nbsp;.

For example, consider the reduction that proceeds as follows:

For example, consider the reduction that proceeds as follows: