Professor R. James Milgram of the Stanford Math Department is extensively involved in math education at the elementary school level.
Milgram stresses that in Russia, the elementary school math program is better devised than in the US. The math program in Russia is based on understanding the Peano Axioms.
In first grade, the Peano Axioms are taught uses pictures according to Professor Milgram.
One can go to the following link and download the papers one wishes.
The following has some actual lessons for grades 2 to 3 and also some more advanced sections on proofs of theorems using the Peano Axioms as well as definitions of addition and multiplication.
This presentation basically assumes that order has been done before addition. So one can make statements about the set of the first n integers. The Recursion Theorem is in effect implicitly done in the definition of addition and then in the definition of multiplication. This in effect glosses over the explicit development of the Recursion Theorem.
Milgram’s discussion of Russian math education is a tour-de-force. So are his definitions of Peano Addition and Multiplication of natural numbers and of their properties. He proves the addition laws of associativity and commutativity. He proves the multiplication laws of associativity, commutativity and of distributivity over addition.
Pre-Algebra New Math Done Right Peano Axioms covers the Peano Axioms and addition definition and proof. It also informally covers part of the order of natural numbers, set theory basics and the recursion theorem briefly. This is in approximately 390 pages. This has almost 200 examples and over 400 exercises. Number line and bead string versions of the Peano Axioms are given. Many ways of defining addition are discussed. The close link of order to addition is emphasized.
If the Milgram versions are too fast or more on set theory and order from Peano Axioms are needed, then Pre-Algebra New Math Done Right Peano Axioms will help cover that. It is available at ebook vendors. See e-book at the top of the page.