Leon Henkin On Mathematical Induction 1960 is available for free download at Jstor.
Stable Link: http://www.jstor.org/stable/2308975
You can click on the download pdf at the above link.
This article is easier in some parts than others. As it goes on, it becomes harder to sustain. Despite this, it is valuable for its contents and for a snapshot of what the advanced state of thinking about the Peano Axioms, definition by induction of addition, and recursion was in 1960. This was a key point in the history of New Math. The state of Peano Arithmetic as represented by this article was still being worked out. There was confusion because of the Kalmar definition of addition in Edmund Landau’s “Foundations of Analysis”.
The New Math in education movement of the 1960s was not ready to preach and teach addition of natural numbers by Peano Axioms. They still had uncertainty about what they knew. The Henkin article gives us a clue as to this state of uncertainty by the math establishment on Peano Arithmetic as of 1960.
It was this state of uncertainty about Peano Arithmetic and the lack of a good understanding of how to do it that led to it being left out of 1960s New Math. But that left out the whole point of New Math. New Math was really invented by Dedekind in 1888 in his book, “Was Sind und was Sollen die Zahlen?” (What are and what should be the numbers?).
Dedekind did order first, then recursion theorem then addition as a recursive function with two inputs. Peano dropped the order first and the recursion theorem. Landau in 1930 with the Kalmar definition kept addition first and tried to get around the recursion theorem with the Kalmar definition. Whether that was successful is disputed by Leonard Blackburn.
Current recursion theorems before order or addition have difficult proofs. These end up using ideas in Dedekind such as intersections of sets satisfying a minimal condition. This then leads to the set desired. So all naturals from 2 onwards including 2 are the tail 2.
We can define a set as closed under succession if it contains the successor of each element in the set. The closed sets under succession are the tails. This is obvious. But to get that, we define the tail 2 as the intersection of all closed sets that contain the number 2. This is the Dedekind 1888 definition. The recursion theorem approaches are basically using this strategy one way or another. However, “or another” contains some seemingly difficult to understand proofs that replicate some of the lemmas Dedekind proves one by one in his book.
1960s era New Math authors did not feel sufficiently comfortable with their knowledge or their ability to teach it to include Peano Axioms in New Math. But that left out the whole point of the set theory, ordered pairs, and functions that Dedekind used in 1888 to get to addition and prove it was commutative and associative. Thus 1960s New Math threw out the punch line. This is why it failed.
New Math 1960s style was not focused on getting to place value notation and place value algorithms based on the Peano Axioms as the starting base. This is and should be the goal of elementary education in math. At least according to New Math Done Right it should be.