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Natural Numbers and Peano’s Axioms

SUMMARY
This class covers natural numbers and how they are defined using Peano’s axioms: a series of mathematical principles that establish their fundamental properties. It also explains how symbols are used to represent the successors of natural numbers, their symbolic representation, and the use of the principle of mathematical induction for conducting inductive proofs.

LEARNING OBJECTIVES

Understand Peano’s axioms for the formulation of natural numbers.
Understand the formulation of the symbolic representation of natural numbers.

INDEX

The Peano’s axioms
The principle of induction in natural numbers
Commentary on demonstrations

The Peano’s Axioms

The Natural Numbers, also known as positive integers, are those we use for counting and measuring. They naturally arise in the operation of counting, which is the simplest arithmetic operation. These numbers are defined through the Peano’s axioms, a set of mathematical principles that establish how these numbers function.

“ $1$ ” is a natural number.
If $n$ is a natural number, then its successor $S(n)$ is also a natural number.
“ $1$ ” is not the successor of any natural number.
If $S(n) = S(m)$ , then $n=m$ .
If $1$ belongs to some set $A$ ; and if, given any $k$ in $A$ , $S(k)$ is also in $A$ , then $A$ is the set of natural numbers denoted by $\mathbb{N}$ .

When studying Peano’s axioms, we realize that the symbol “ $1$ ” is just a representation used to indicate a specific natural number. This number is the one that meets these properties. Just as $1$ represents the “first natural”, we also use symbols (which are familiar to us) to represent its successors.

$2=S(1)$
$3=S(2)$
$4=S(3) \\ \vdots$

and so on. In this way, the symbols $1$ , $2$ , $3$ , etc… are abstract entities representing the different successors of $1$ . The collection of all these objects are the natural numbers, which we represent as:

$\mathbb{N}=\{1,2,3,4,\cdots \}$

It is also said that natural numbers are arranged in a sequence, the sequence of natural numbers:

$1,2,3,4,5,6,7,8,9,10,11,12, \cdots$

The Principle of Induction in Natural Numbers

An important aspect of natural numbers is that there is always a number after each one, which means there are infinite. We can intuit this from the fifth axiom, or principle of induction, which is expressed as follows:

If a property holds for $1$ ; and if given that it holds for any natural number $k$ , it also holds for the next $S(k)$ ; then such property holds for all natural numbers.

The principle of induction provides, in addition to a foundational basis, a useful tool to prove if a property holds for natural numbers. To understand this, let’s see a simple example:

EXAMPLE: Through the principle of induction, it can be demonstrated that every natural number is different from its successor.

While this is obvious, it helps to understand the procedure when proving through induction.

Proof:

It’s clear that $1$ is different from $S(1)=2$ . This is the initial step, where we verify that the property holds for the first element.
Assume the property holds for any $k$ , that is, $k\neq S(k)$ ; what we will do is prove that from this it is also verified for $S(k)$ (i.e., it is also verified that $S(k)\neq S(S(k))$ . This is the inductive step. If these two steps are completed, then it is said that the induction is complete and the property holds for all natural numbers.
[1] To start, note that $S(k) \neq k,$ is equivalent to saying $\neg [k=S(k)]$ .
[2] But since both $k$ and $S(k)$ are natural numbers, by axiom 2 we can say they both have successors: $S(k)$ and $S(S(k))$ , respectively. Both are also natural numbers.
[3] Then, by axiom 4 we can say that: $S(k) = S(S(k))$ implies that $k = S(k)$ . We can express this as:
$\left[ S(k) = S(S(k)) \right] \rightarrow \left[k = S(k)\right]$
which by contrapositive of the implication, is equivalent to saying:
$\neg \left[k = S(k)\right] \rightarrow \neg \left[ S(k) = S(S(k)) \right]$
[4] Finally, applying a modus ponens between this last expression and the one obtained in step [1], we get
$\neg \left[ S(k) = S(S(k)) \right]$
which is the same as saying
$S(k) \neq S(S(k))$
And therefore, we have shown that if it is verified that $S(k) \neq k,$ , then it is also verified that $S(k) \neq S(S(k))$ ; as it is also evident that $1\neq 2$ , the induction is complete and it can be written that
$\left(\forall n\in\mathbb{N}\right)\left(n \neq S(n)\right)$

Comment on Proofs

While the property stated in the example is quite obvious, it’s common in mathematics for proofs not to maintain this obviousness. This proof we just saw is an example of what is typically done when working mathematically. To support your understanding of the deduction techniques inherent in mathematics, I recommend you review the materials designed for the mathematical logic course.

Natural Numbers and Peano’s Axioms

The Peano’s Axioms

The Principle of Induction in Natural Numbers

Comment on Proofs

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