After the Peano Axioms are stated, we need to define order. Dedekind in 1888 did order before addition. Edmund Landau in 1930 did addition before order. The addition before order approach obscures that the Peano Axioms are about maintaining your place in the count, and so are about indexing and order.

Addition is a method for keeping your place in the count more efficiently. So is addition in place value notation. This logical order is covered up by doing addition first. It is possible to do order based on addition precisely because addition is just a method for keeping track of one’s place in the count, i.e. order.

So addition is an order concept and that is why addition implies order. Addition is a method to implement order, so it implies order.

The Peano Axioms give us a local order from one link to the next. We then need order for two or more links. Intuitively, if there is a chain of succession from m to n, then we say that m is less than n. We can also call a chain of succession a lineage.

A finite chain of succession can be defined as a set of numbers so that exactly one has no predecessor in the chain and exactly one has no successor in the chain and otherwise, each other number has both a predecessor and successor in the set. We can call a point with a successor and predecessor in the set an internal member, point or number. The point with no successor in the set, we can call the terminal point or final point in the chain.

A chain of succession need not have a terminal point, in that case we can call it a tail set.

If we use duck versions of the Peano Axioms, we can use family line or lineage to mean a finite chain of succession. A tail lineage is one that does not end.

In the duck version, we have the ducks stand for numbers. We can think of each duck as having its number on its chest and back. Zero is a mom duck but not a baby duck. All other ducks are both moms and babies. Each baby has one mom and each mom has one baby.

If a set contains the baby of each mom in the set, the set is closed under succession, or just closed. Lineage chains as defined above are not closed, but tail lineages are.

Lineage chains can be linked to form a longer lineage chain. So if for two chains, the end of one is the start of the next, then the two together form a single lineage chain with a single start and end. This gives us transitivity, since this implies the start of the big chain is less than the end of the big chain by definition of less than. So if a is less than b and b less than c, then there is a chain from a to c and so a is less than c.

This approach is discussed further in the book Pre-Algebra New Axioms Done Right Peano Axioms, now on summer sale at 2.99 for the e-book.

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