Natural numbers

Natural numbers are the ordered set of numbers that humanity has used to count, which are assigned ordinal names to name each particular number:

And so we could continue listing them until we get bored, since the set of natural numbers is infinite

However, there is no universal agreement on whether zero should be considered a natural number. Historically, 0 as a number had a much later origin than the rest of numbers. The Babylonians, in the 7th century BC. they had a symbol for zero, but just to leave no gaps when representing quantities in their base 60 positional numbering system in cuneiform script

The Hindu mathematician Brahmagupta, already considered it a number more, in the 7th century; and possibly, the Olmec and Mayan civilizations already used it as a number, centuries earlier. On the other hand, the Greeks, from which the mathematics of Western culture derive, had no concept of zero; and for that reason, there is no way to represent it in Roman numerals, a culture that drank from Greek, which prevailed in Europe until the base 10 decimal system (of Indian origin and adopted by the Arabs) began to slowly prevail from the 13th century onwards

For this reason, you can define natural numbers in two ways:

The axioms of Peano

The Peano axioms (also known as Peano's postulates) were a proposal by the Italian mathematician Geuseppe Peano in 1889, in order to axiomatically formalize the natural numbers, based on the set theory developed by Georg Cantor

Axioms, which is still used in the present

Natural numbers are defined as a set (called \mathbb{N}_0), an element (which assumes the role of zero and that we will denote as 0) and a “next” (or “successor”) element that is an application denoted by S so that it fulfills:

1. The zero is a natural number
2. The next natural number is also a natural number
3. There is No natural number whose next is zero
4. If the next of two natural numbers are equal, then the two numbers are equal
5. If S is a set of natural numbers such that zero is of S and whenever a natural number is of S also its next one is in S, then S is the set of natural numbers

Using a more formal or algebraic language, these five axioms can be stated like this:

1. 0\in \mathbb{N}_0
2. \exists s| \mathbb{N}_0\rightarrow \mathbb{N}_0 and in addition satisfies the axioms following
3. \not\exists n\in \mathbb{N}_0 |s(n)=0
4. s(n)=s(m)\Rightarrow n=m (s is injective)
5. S\in \mathbb{N}, 0\in S |\forall n\in S=s(n)\in S\Rightarrow S=\mathbb{N}_0

The method of induction

From the concept of induction, the idea of mathematical induction appears, which allows demonstrations by the induction method

You can express the natural numbers as a triplet (\mathbb{N}_0,0,s) with \mathbb{N}_0 a set, 0\in \mathbb{N}_0, s| \mathbb{N}_0\rightarrow\mathbb{N}_0\backslash\{0\} an injective application, and in such a way that the fifth of the previous axioms is fulfilled, which is called the axiom of induction or the principle of induction

To do this, S is taken as the set of natural numbers that satisfy a certain property that wants to be proved, it is verified that zero fulfills the property (that is 0\in S), and that if a number n fulfills it, the next will also fulfill it (that is n\in S\Rightarrow s(n)\in S); and as a consequence of the axiom of induction, all natural numbers fulfill the property (S=\mathbb{N}_0)

In practice, the principle of induction is usually applied in terms of properties rather than in terms of sets. To carry it out, we will carry out the following steps:

Suppose that for each natural number n\geq n_0 it has a certain property P_n which may or may not be true. We assume that:

1. P_n it is certain
2. If for some n\geq n_0 the property P_n is true then P_{n+1} also what is

Then P_n it is certain for all n\geq n_0

The demonstration will be over, because we will have been able to test the property for all natives

Induction Example: The Gaussian Sum

The Gaussian sum is a practical example of the induction method

In 1789. Carl Friedrich Gauss (who would become a great mathematician and physicist), while in school, at the age of nine, his teacher wanted to keep the children busy for a while, he ordered them to add the first hundred numbers. He had barely finished assigning homework when Gauss got up and handed over his whiteboard. On the board there was a single number: 5050. It turned out that 5050 was precisely the sum of the numbers from one to one hundred. How had he found the solution so quickly?

Gauss realized something curious. The sum to be made was \sum\limits_{k=1}^{100} k, which can take a long time if done in that order, but if you add the first and last numbers, you get 101. The same happens if you add the second with the penultimate and the third with the penultimate, and so on. You can see that all these sums have the same result: 101

Since there are obviously 50 pairs whose sum is 101, the result of adding from one to one hundred is 50\times 101=5050. This way of dealing with the problem is an excellent example of an elegant solution

This solution is not only valid for numbers from one to one hundred. In general, the sum of the first n numbers (where n is an even number) is the last number plus one times the number of pairs, that is \sum\limits_{k=1}^{n} k=\frac{n\cdot(n+1)}{2}

Now we are going to prove that \sum\limits_{k=1}^{n} k=\frac{n\cdot(n+1)}{2} is true for all natural numbers using the induction method:

We take it as a P_n the left side of the equality, we check that for P_1 is true: P_1=\sum\limits_{k=1}^{1} k=1

Now compare it with the right side \frac{1\cdot(1+1)}{2}=\frac{1\cdot 2}{2}=\frac{2}{2}=1. Since 1=1, we have that the equality was true for P_1

We assume that it is true up to n: \sum\limits_{k=1}^{n} k=\frac{n\cdot(n+1)}{2}

Now we check that it is true for n+1:

\sum\limits_{k=1}^{n+1} k=\frac{(n+1)\cdot((n+1)+1)}{2}=\frac{(n+1)\cdot(n+2)}{2}=\frac{n^2+2\cdot n+n+2}{2}=\frac{n^2+3\cdot n+2}{2}
(\sum\limits_{k=1}^{n} k)+(n+1)=\frac{n^2+3\cdot n+2}{2}

We apply the induction hypothesis: (\frac{n\cdot(n+1)}{2})+(n+1)=\frac{n^2+3\cdot n+2}{2}

\frac{n\cdot(n+1)+2\cdot(n+1)}{2}=\frac{n^2+3\cdot n+2}{2}
\frac{n^2+n+2\cdot n+2}{2}=\frac{n^2+3\cdot n+2}{2}
\frac{n^2+3\cdot n+2}{2}=\frac{n^2+3\cdot n+2}{2}

It is fulfilled, then P_n it is certain for all n\geq n_0 and therefore is true for all natural numbers

Proposition: principle of good ordination

The principle of good management that if (\mathbb{N}_0, \leq) is a well-ordered set (that is to say, any subset of \mathbb{N}_0 non empty) has minimal

Demonstration: principle of good ordination

We are going to prove by reduction to absurdity

Suppose that there is a subset A\subset\mathbb{N}_0, A\not=\emptyset, which has no minimum

Then we define the set S=\{n\in\mathbb{N}_0|n\leq a, \forall a\in A\}

We must realize that if n\in A\cap S, we would have to n=\text{min}(S) and we've assumed that S has no minimum, therefore A\cap S=\emptyset

Now let's test it for the whole set using the induction axiom:

We found that 0\in S
0+a=a then 0\leq a, \forall a\in A

\text{If }n\in S, as A\cap S=\emptyset then n < a, \forall a\in A, (by defining order in \mathbb{N}_0) in each case there will be a natural number n_a \geq 1 such that n+n_a=a, which is done whenever s(n)=n+1\leq n+n_a=a, and therefore s(n)\in S

In compliance with these conditions, the axiom of completeness assures us that S=\mathbb{N}_0

Which is absurd, for it A\cap S=\emptyset then A=\emptyset, which is clearly a contradiction of the original assumption A\not=\emptyset

A model of the natural numbers

We're going to see a model of the natural

A theory set of axioms is consistent when there appears some kind of contradiction that makes the axioms do not make sense

Peano's axioms introduce natural numbers from set theory. After exposing the axioms, some additional definitions (the sum, the product and the principle of good management) have been given. But are Peano's axioms consistent? That is, is there a model based on the assembly model that meets Peano's axioms? Let's see how to build a model of natural numbers from set theory

Note if we had a model, we would be reducing the consistency of numbers to that of set theory, and the set theory is consistent? At some point we have to stop descending, and you have to assume something as true without the possibility that it will be demonstrated. In 1930, He tested his famous theorems of incomprehensibility, which in a simplified way state the following:

• First theorem of incompleteness
  In any consistent formalization of mathematics that is strong enough to define the concept of natural numbers, an affirmation can be constructed that cannot be demonstrated or refuting within that system
• Second theorem of incompleteness
  Any consistent set may be used to prove itself

These results were very negative for the philosophical approach to mathematics, proposed by David Hilbert, known as Hilbert's formalization program. He had proposed that the consistency of more complex systems, such as actual analysis, had to be tested in terms of simpler systems. Finally, the consistency of all mathematics could be reduced to basic arithmetic

With his first theorem, Godel demonstrated that arithmetic is incomplete, and in this way it will be impossible for it to be used to demonstrate the consistency of any axiom system. The paradise devised by Hilbert did not exist, it was a utopia

Let's stop the logic and go back to the natural numbers. A model of Peano's axioms would be a triplet (\mathbb{N},0,s) with \mathbb{N} a set, 0\in \mathbb{N}, s|\mathbb{N}\rightarrow\mathbb{N} an application that would satisfy these axioms. It is not difficult to see that if we have two models, they should be the same, since given two models (\mathbb{N}_A, 0_A, s_A) and (\mathbb{N}_B, 0_B, s_B) Peano's axioms, the application f|\mathbb{N}_A\rightarrow \mathbb{N}_B defined by:

\begin{cases}f(0_A)=0_B \\ f(s_A(n))=s_B(n) \end{cases}

it's a bijection. Since we have to, there can only be one model of Peano's axioms, but we haven't defined any yet!

Logicians Gottlob Fregge and Bertrand Russell built models of Peano's axioms based on the intuitive idea that a natural number is the cardinal of an ensemble. That is, each natural number is the "essence" shared by the bijective (and finite) sets whose cardinal number is that cardinal number. Then a natural number would be defined as the equivalence case of those sets; in addition, zero corresponds to the empty set (actually, its equivalence class) and s applied to an assembly is to add an element that is not in the set (for each set of the equivalence class). However, this idea has serious logical difficulties, because, according to the laws of Zermelo-Fraenkel's axiomatics on which the current mathematics is based, put informally: "the whole of all sets is not a whole", the equivalence classes of bijective sets also do not form a set of

The model commonly used is that of the Hungarian nationalized American mathematician, János von Neumann, who not only made important contributions in many fields of mathematics, but in computer science with von Neumann's architecture, currently used in computers and in the manufacture of the first atomic bomb

The model devised by von Neumann begins with the definition of 0 as the empty set (\emptyset), and also defines an operator s acting on the sets by s(A)=A\cup(A)

Defines the set of natural numbers \mathbb{N}_0, such as the intersection of all closed sets under the action of s (that is, of all C sets such that s(C)\subset C) containing the empty set. Each natural number (understood as a set), is the set of natural numbers less than him:

\scriptsize\begin{cases}0=\emptyset \\ 1=s(0)=\emptyset\cup\{\emptyset\}=\{\emptyset\}=\{0\} \\ 2=s(1)=\{\emptyset\}\cup\{\{\emptyset\}\}=\{\emptyset,\{\emptyset\}\}=\{0,1\} \\ 3=s(2)=\{\emptyset,\{\emptyset\}\}\cup\{\{\emptyset,\{\emptyset\}\}\}=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}=\{0,1,2\} \\ \cdots \\ n=s(n-1)=\cdots=\{1,2,3,\cdots,n\} \end{cases}

The set \mathbb{N}_0, along with 0 and the "next" function s|\mathbb{N}_0\rightarrow\mathbb{N}_0, satisfies Peano's axioms

Operations of the natural numbers

We're going to see the operations of the natural

With the axioms of Peano and with the zero as the first element of \mathbb{N}_0, we introduce the notation of others:

Operation sum

The sum or (addition) in \mathbb{N}_0 it is an operation that +|\mathbb{N}_0\rightarrow \mathbb{N}_0 which is recursively defined as:

\begin{cases}a+0=a \\ a+s(b)=s(a+b) \end{cases}

Properties of the sum

Dice a, b, c\in\mathbb{N}_0 comply with:

• Associative property for addition
  a+(b+c)-(a+b)+c (as a result of the previous property, there is no need to enter parentheses and a+b+c can be written)
• Commutative property for addition
  a+b=b+a
• Neutral element of the sum
  0\in\mathbb{N}_0,\forall a\in\mathbb{N}_0|a+0=a; \forall a\in\mathbb{N}_0, \exists b\in\mathbb{N}_0|a+b=0
• Cancellation property (simplification) in the sum
  \text{If }a+c=b+c\rightarrow a=b

Operation product

The product or (multiplication) in \mathbb{N}_0 it is an operation that \cdot|\mathbb{N}_0\rightarrow \mathbb{N}_0 which is recursively defined as:

Properties of the product

Dice a, b, c\in\mathbb{N}_0 comply with:

• Associative property for the product
  a\cdot(b\cdot c)=(a\cdot b)\cdot c (as a result of the above property, there is no need to indicate parentheses and can be written a\cdot b\cdot c)
• Commutative property for the product
  a\cdot b=b\cdot a
• Neutral element of the product
  1\in\mathbb{N}_0,\forall a\in\mathbb{N}_0|a\cdot 1=a; \forall a\in\mathbb{N}_0,a\not=0, \exists b\in\mathbb{N}_0|a\cdot b=1
• Cancellation property (simplification) on the product
  \text{If }a\cdot c=b\cdot c\text{ con }c\not= 0\rightarrow a=b

Operation sum and product

In addition, sum and product share the following property:

Property of the sum and the product

• Distributive property of the product regarding the sum
  \text{If }a\cdot (b+c)=(a\cdot b)+(a\cdot c)

Note if we had chosen to represent the natural ones without zero, in the axioms of Peano it would be enough to change \mathbb{N}_0 by \mathbb{N} and 0 by 1. Since the first axiom only serves to ensure that \mathbb{N}_0 is not the empty set, and the name given to the first element is not relevant in its definition. When it becomes relevant, it is when the sum and product operations are defined. If the natural ones are built starting at 1, the definition of the sum is started with a+1 s(a), and that of the product by a\cdot 1=a