Start Learning Numbers 2 | Natural Numbers (Successor Map and Addition) [dark version]

TL;DR

Natural numbers are characterized by a nonempty set with a distinguished element zero and an injective successor map whose image excludes zero.

Briefing Cornell Notes

Briefing

Natural numbers can be defined rigorously not by how they look on a number line, but by three structural properties: a nonempty set with a distinguished element called zero, an injective successor map that never outputs zero, and an “induction” minimality condition that forces every subset containing zero and closed under successors to be the whole set. Together, these conditions pin down the natural numbers as the smallest infinite set and justify why every natural number can be reached by repeatedly applying the successor operation starting from zero. The injectivity of the successor map rules out “loops” in the ordering picture, while the fact that zero is not in the successor map’s range prevents cycles and ensures the structure extends indefinitely.

Once this foundation is in place, arithmetic—specifically addition—gets defined in a way that works for infinitely many inputs without listing them all. Addition is treated as a function from pairs of natural numbers to a natural number, written M + N. The definition starts with two base/step rules: adding zero changes nothing (M + 0 = M), and adding one moves to the successor (M + 1 = s(M)). From there, the general pattern emerges: adding a successor on the right corresponds to applying the successor to the result of adding the previous number. In the transcript’s notation, this yields a recursive description where M + (n+1) is the successor of M + n.

To make the definition concrete, the method is illustrated with an example like 2 + 5. Since 5 is the successor of 4, the rule rewrites 2 + 5 as s(2 + 4). But 2 + 4 is already known from earlier steps (in the usual arithmetic sense), so the process reduces the new computation to a previously determined case. That “work backwards until reaching a known value” is exactly what makes the definition recursive: each new output is built from earlier ones.

Because recursive definitions can be risky if they don’t actually determine a unique function, the transcript invokes a general existence-and-uniqueness result often called the recursion principle (attributed there to “Dakin’s principle of recursive definition”). In its abstract form, given any set A, a chosen element a in A, and a function H from A to A, there exists a unique function f from natural numbers into A such that f(0) = a and f(s(n)) = H(f(n)). Interpreted in the addition setting, this guarantees that the recursive rules for addition determine exactly one well-defined addition function on the natural numbers.

The payoff is that once the successor-based structure and the recursion principle are secured, the usual calculation rules for natural numbers can be derived and proved systematically in later work.

Cornell Notes

The natural numbers are defined using a set N0 with a special element 0 and a successor map s. The successor map is injective (different numbers have different successors) and zero is never a successor (0 is not in the image of s). A minimality condition says any subset M of N0 that contains 0 and is closed under successors must equal N0; this is what underwrites mathematical induction. Addition is then defined recursively: M + 0 = M and M + 1 = s(M), with the general idea that M + (n+1) is the successor of M + n. A recursion principle ensures these rules produce a unique, well-defined function for addition.

What three properties pin down the structure of the natural numbers in this approach?

They start with a set N0 that is nonempty and contains a distinguished element called zero. Next comes a successor map s from N0 into N0 that is injective (no two different natural numbers share the same successor), and zero is not hit by the successor map (zero is not in the range/image of s). Finally, N0 is minimal among infinite sets: if a subset M contains zero and contains the successor of every element it already has, then M must be all of N0. This closure-and-minimality condition is what yields induction.

Why does injectivity of the successor map matter for the “number line” intuition?

Injectivity prevents “loops” in the ordering picture. If s were not injective, two different natural numbers could have the same successor, collapsing distinct points and breaking the clean linear progression. With injectivity, the successor operation moves forward in a consistent one-to-one way, matching the idea of natural numbers arranged in order.

How does the definition of addition avoid listing infinitely many cases?

Addition is defined by recursion rather than by enumerating all pairs (M, N). The rules begin with base cases: M + 0 = M (adding zero changes nothing) and M + 1 = s(M) (adding one moves to the next number). Then the general step is that adding a successor on the right corresponds to applying the successor to the previous result: M + (n+1) = s(M + n). This generates every needed value from earlier ones.

What does the example 2 + 5 illustrate about recursive definitions?

It shows how new computations reduce to previously determined ones. Since 5 is the successor of 4, the rule rewrites 2 + 5 as s(2 + 4). Once 2 + 4 is known, the successor gives the final answer. The computation effectively “walks backward” through the recursion until it reaches a case already fixed by the base rules.

What does the recursion principle guarantee, and why is it necessary?

Recursive rules could be ambiguous or inconsistent unless they determine a unique function. The recursion principle states: for any set A, any element a in A, and any function H: A → A, there exists a unique function f: N0 → A such that f(0) = a and f(s(n)) = H(f(n)). This ensures that the recursive specification (like the one used for addition) actually defines exactly one function on natural numbers.

Review Questions

How do the injectivity and “zero not in the successor range” conditions work together to prevent cycles in the natural-number structure?
Write the recursive rule for addition in terms of M + (n+1). How does it relate to the successor map s?
What does the recursion principle require (data like A, a, and H), and what does it conclude about existence and uniqueness?

Key Points

1
Natural numbers are characterized by a nonempty set with a distinguished element zero and an injective successor map whose image excludes zero.
2
The minimality/closure condition—subsets containing zero and closed under successors must equal the whole set—yields mathematical induction.
3
Addition is defined as a function on pairs of natural numbers using base rules (M + 0 = M) and a successor step (M + 1 = s(M)).
4
The general recursive pattern is M + (n+1) = s(M + n), so each new value depends on a previous one.
5
Recursive definitions work by reducing a target computation to earlier cases until reaching a base case already fixed by the rules.
6
A recursion principle (existence and uniqueness) ensures that such recursive specifications determine exactly one function, preventing ambiguity.
7
With the successor-based structure and recursion principle in place, later arithmetic rules can be derived and proved systematically.

Highlights

The successor map’s injectivity and the fact that zero is never a successor prevent loops and force an infinite, linearly ordered structure.

Mathematical induction emerges from the idea that any subset containing zero and closed under successors must be the entire natural-number set.

Addition is defined without infinite case lists: M + 0 = M and M + (n+1) = s(M + n).

A general recursion principle guarantees that recursive rules define a unique function, making the foundation mathematically sound.

Topics

Natural Numbers
Successor Map
Mathematical Induction
Recursive Addition
Recursion Principle