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The axioms of the primary arithmetic are shown below, as they appear in both graph and string forms, along with pairs of names that come in handy for referring to the two opposing directions of applying the axioms.

The axioms of the primary arithmetic are shown below, as they appear in both graph and string forms, along with pairs of names that come in handy for referring to the two opposing directions of applying the axioms.

<p>[[Image:Logical_Graph_Figure_14_Banner.jpg|center]]</p>

<p>[[Image:Logical_Graph_Figure_14_Banner.jpg|center]]</p><br>

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<p>[[Image:Logical_Graph_Figure_15_Banner.jpg|center]]</p><br>

<p>[[Image:Logical_Graph_Figure_15_Banner.jpg|center]]</p>

Taking '''S''' to be the set of rooted trees and '''S'''₀ to be the set that has the root node and the rooted edge as its only two elements, writing these facts more briefly as '''S''' = {rooted trees} and '''S'''₀ = {O, |}, 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 "@" or else to a rooted edge "|".

Let <math>S\!</math> be the set of rooted trees and let <math>S_0 \subset S</math> be the 2-element subset consisting 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>.

For example, consider the reduction that proceeds as follows:

For example, consider the reduction that proceeds as follows: