Teaching the Peano Axioms and really teaching the Dedekind 1888 book “What are and what should be the numbers?” requires some choices on what words will mean.
So far I have used chain of succession from 2 to 5 to include 2 as well as 3, 4 and 5.
Successor in standard treatments means the single number after a reference number, so 3 = 2′, 3 is the successor of two.
In other contexts, though we might want to talk about successors of 2, meaning all numbers after 2. We might also be tempted in other contexts to have successors or some other word include 2 as well.
Nodes are another type of subject instead of numbers. If we are talking about sets of nodes, or linked nodes, we might want to include the starting node.
If we are working with additive identities
x+0 = x
x+y’ = (x+y)’
We want to talk about the case of adding 0. We can call this a succession chain of 0 length or a count-on chain of zero length.
Can we count on by zero? Or does counting on require at least one?
Should a counting on chain start after the anchor or include the anchor?
We want to teach order before addition so that students understand that addition is not some operation that comes from outer space, but arises logically from counting-on chains and thus addition is derived from order.
We can also use a word like followers. This could have a different convention than successors. So successors might not include the anchor number but followers does.
We can also use words like strict to exclude equality and weak to include equality.
A strict successor or strict follower does not include the original number.
Bound is a word used in math where it allows equality. 3 bounds itself. 3 bounds the rational numbers less than or equal to 3. 3 is in fact the least upper bound of this set.
Bounds before math likely meant the exact boundary line. In math, we have weakened bounds to mean numbers beyond the boundary line of the set.
Should we do that with a word like followers where we stretch it to include equality with the anchor number? Or look for a different word?
Nodes are a good thing to think about when thinking of these types of words. But other objects might come to mind.
Team members, tribe members have a leader but the leader is still a member of the team or tribe.
The president of a body is often a member of the body as well. The Speaker of the House is a member of the House. The Chief Justice is a justice and member of the court. So in some cases, the leader is also a member. But we usually don’t include followers to mean that.
However, we also want a sense of direction in our word that includes the anchor number and the numbers that come after it.
We can also invent words like tailons.
The tail that starts with 2 has members. We can call these tailons. In this case, 2 is a tailon.
If we think of the head segment from 0 to 2, we can think of the members as being headers. So 2 is a header.
The elements before 2 we can call preheaders. So preheaders would not include 2. Postheaders would be after 2 and not include 2.
So the postheaders would be the same as the tailons starting from the first successor of the head segment.
For a finite segment from 2 to 5, we need names. The number 2 can be called the start, begging, or anchor. The number 5 can be called the stop or finish or end.
This is a gapless set of nodes. The preheaders can be used to mean the numbers 0 and 1. The tailons can mean the numbers starting at 6. So we can talk of the preheaders of a segment and the tailons of a segment. These are not inclusive.
Other words besides weak and strict are inclusive and exclusive.
So the inclusive tail of 2 includes 2. The exclusive tail of 2 does not include 2.
The head of 2 includes 2. This means the head segment ending in 2.
The exclusive tail of 2 equals the inclusive tail of 3.
The words head and tail could also be overloaded to mean the begin and end numbers of a segment, but this is risky. It seems better to have other distinct words like start and stop or start node and stop node or start number and stop number.
When considering language, consider if we mean nodes instead of numbers.
The tail of nodes starting from 2 includes 2. This is also the inclusive tail of nodes.
We can have the segment of nodes from 2 to 5. We can call this a gapless set of nodes. We can call these linked nodes. 2 is the starting node and 5 the ending node. In this context, we typically want to talk of 2 as being part of the set of linked nodes.
When we talk of a chain of succession from 2 to 5, we are not sure if 2 should be part of the chain. In this context, it seems natural to mean that 2 is part of the chain of succession from 2 to 5. This is despite 2 not being a successor in a linguistic sense of 2.
It is important to have different words that include and some that exclude. It is preferable to have single words instead of phrases.
Short words are preferable to long ones.
For teach order of whole numbers in grades K and grades 1 we want the sentences to flow as so natural the students and the teachers part of the time are not even aware of how contrived they are. They learn how they work, but as they use them, they seem so natural that they don’t realize that careful choices have been made to encode the part of Dedekind’s book from Note 26 to Note 60 or even up to Note 100 or so.
I shall refer to the Dedekind numbered points in his book by D26 for example. Or Dedekind Note 26 or Dedekind 26 or Dedekind Paragraph 26 as the context may indicate.