Changes

===Primary arithmetic as semiotic system===

===Primary arithmetic as semiotic system===

Though it may not seem too exciting, logically speaking, there are many good reasons for getting comfortable with the system of forms that is represented indifferently, topologically speaking, by rooted trees, by well-formed strings of parentheses, and by finite sets of non-intersecting simple closed curves in the plane.

Though it may not seem too exciting, logically speaking, there are many reasons to make oneself at home with the system of forms that is represented indifferently, topologically speaking, by rooted trees, by balanced strings of parentheses, and by finite sets of non-intersecting simple closed curves in the plane.

:* One reason is that it gives us a respectable example of a sign domain on which to cut our semiotic teeth, non-trivial in the sense that it contains a [[countable]] [[infinity]] of signs.

:* One reason is that it gives us a respectable example of a sign domain on which to cut our semiotic teeth, non-trivial in the sense that it contains a [[countable]] [[infinity]] of signs.

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><br>

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

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

<br>

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

<br>

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>.

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>.