Counting on no cycles

If we start counting on from i, we never come back to i.  This is the no-cycles rule or property of counting on.

This is connected to order.

If we count on from i to j by one or more then i is less than j.

If we including counting on by zero, then counting on from i to j shows that i is less than or equal to j.

Counting on is the basis of order.  This again highlights that counting on by one and successor are the same.

Counting on encounters numbers along the way and these are less than the numbers encountered later in such a path.

The initial segments are found by counting on from 0.

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About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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