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Here are the two ''arithmetic axioms'':

Here are the two ''arithmetic axioms'':

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| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` `( ) ( )` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| Axiom I_1.` ` Distract <--- | ---> Condense ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` ` (( )) ` ` ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

o-----------------------------------------------------------o

| Axiom I_2.` ` ` Unfold <--- | ---> Refold ` ` ` ` ` ` ` ` |

Taking '''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 these axioms to either a root node or else a rooted edge.

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 &nbsp;<math>\vert \,.</math>

For example, consider the reduction that proceeds as follows:

For example, consider the reduction that proceeds as follows:

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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

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o===========================================================o

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` ` ` ` ` ` ` o o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

Regarded as a semiotic process, this amounts to a sequence of signs, every one after the first being the interpretant of its predecessor, ending in a sign that we may regard as the canonical sign for their common object, in the upshot, the result of the computation process.  Simple as it is, this exhibits the main features of all computation, specifically, a semiotic process that proceeds from an obscure sign to a clear sign of the same object, in sum, a case of clarification.

Regarded as a semiotic process, this amounts to a sequence of signs, every one after the first being the interpretant of its predecessor, ending in a sign that we may regard as the canonical sign for their common object, in the upshot, the result of the computation process.  Simple as it is, this exhibits the main features of all computation, specifically, a semiotic process that proceeds from an obscure sign to a clear sign of the same object, in sum, a case of clarification.