Professor David E. Joyce of Clark University has written up a very valuable set of notes on Richard Dedekind’s book, “Was sind und was sollen die Zahlen?” (What are and should be the numbers?) (Zahlen is capitalized in German because it is a noun.)
This is the second essay in the book and has the somewhat different title in the English translation “The Nature and Meaning of Numbers.” The Joyce notes have paragraph numbers corresponding to the paragraph numbers in Dedekind.
Note that Z_n is the set of numbers from 1 to n including n. Joyce at one point incorrectly states Z_n does not include n. See page 17 the discussion on paragraphs 81 to 118. However, at other points he has Z_n does include n.
So Z_2 is the set consisting of exactly 1 and 2. Z_n is a set. Joyce calls Z_n an initial segment.
Dedekind defines Z_n in numbered paragraph 98. In numbered paragraph 99, he points out that n is an element of Z_n.
Note that closed sets under successor for the natural numbers are what we call tails or one could call final segments. So the tail 2 is 2 and all further numbers. This can be called the final segment for 2. Dedekind uses the notation 2_0 for this final segment of 2. This includes 2.
At an earlier stage, Dedekind defines chains, which are closed sets under a transformation. These can be interpreted as closed for some other function than successor. If one uses predecessor, one has to make appropriate changes for the first natural number having no predecessor. This first natural number is 1 for Dedekind and the Joyce Notes and is 0 for Pre-Algebra New Math Done Right Peano Axioms (PANMDRPA).
(In Pre-Algebra New Math Done Right Peano Axioms we use H(n) to mean 0 through n including 0 and n. So 0, 1, and 2 is H(2).)
One of the proofs concentrates on learning mathematical induction instead of complicated algebraic formula. This is farther in the Joyce pdf. Skip the first one that is quadratic and go on to Theorem 2 on page 2. That proof has no algebraic formulas to learn.
Pre-service teachers in college in one study complained that complicated algebra formulas made learning mathematical induction harder. This case has no such formulas. That allows focusing on the mathematical induction.
As one sees in Joyce, the math induction without algebra can be quite easy. This is one of the harms that not teaching the Peano Axioms has had. It has created a difficult to surmount algebraic barrier to learning mathematical induction. As the Joyce proof shows, this is completely unnecessary to learn math induction.
Simple mathematical induction proofs are in the book “Pre-Algebra New Math Done Right Peano Axioms.” There are no quadratic or higher polynomials in that book. But it has many proofs by mathematical induction. Thus it is a good place to learn proofs by mathematical induction.
Note that Joyce and Dedekind use 1 as the start of the number sequence. Pre-Algebra New Math Done Right Peano Axioms (PANMDRPA) starts at zero.
The Joyce notes will be indispensable for most people to understand Dedekind. Even with them some parts require re-reading several times. This is particularly the case starting with “chains”, or what we would call closed sets under some function. That function is usually successor but could be predecessor with appropriate modifications for the first natural number having no predecessor.