Matematika

We start learning to count and perform various operations with numbers when we are still very young. But the axiomatic description of various number systems is only a subject of the curriculum of specialized universities at most. So we learn something at a very basic level that perhaps has nothing to do with thoroughly transparent and understood logical deductions throughout our lives. It is just a mere practice behind which we do not see the deeper meaning. After all, it is precisely the axiomatic deduction and the definitions and theorems deduced from them that reveal the inner meaning of mathematics. The introduction to geometry has basically been taught since ancient times according to Euclid's axioms, but this is not the case with numbers. The reason for this is that mathematicians consider these axioms to be too abstract for a small child to understand. Of course, this contradicts the requirement expected of axioms, that an axiom should be so simple, almost self-evident, that we do not even try to doubt its truth. After all, the proof of every statement and theorem is reduced to simpler, already proven statements, and then ultimately the truth of every statement in mathematics depends on the truth of the axioms. No one deals with proving them, and there is no mathematical method for it. Thus, the truth of the axioms is of central importance.

The question is not easy, of course. There have always been problems with numbers. They went through a lot of development, until the axioms used in mathematics today were born. But as I wrote, understanding these axioms requires advanced mathematical training, so simplicity and self-explanatoryness are out of the question, especially starting the teaching of numbers in lower school with this. So there is a good reason why they are not taught here.

But this is not the true face of mathematics, and the set of natural numbers is not as it was previously imagined and as it was included in axioms.

Two sets have the same cardinality if there is a one-to-one correspondence between the elements of the two sets, i.e. their elements can be arranged in pairs. In the case of countable sets, this is equivalent to the set having infinite cardinality and its elements being indexed by the natural numbers, i.e. it is an infinite sequence. Galileo already showed that the natural numbers and the square numbers are of the same number.

As a first step, let us consider the sequence of power sets of natural numbers. The sequence of power sets of natural numbers is defined in the following image.

Let the power set of 0 be H(0) = {{}}.

Let the power set of n be H(n), and from this we form the power set of H(n+1) by adding to the power set of H(n) the extensions of the power set of H(n) with the number n, that is, we add the number n to each element set of H(n), so that the cardinality of the power set doubles for each new number. Thus, the following four members:
H(1) = {{},{0}}
H(2) = {{},{0},{1},{0,1}}
H(3) = {{},{0},{1},{0,1},{2},{0,2},{1,2},{0,1,2}}
H(4) = {{},{0},{1},{0,1},{2},{0,2},{1,2},{0,1,2},{3},{0,3},{1,3},{0,1,3},{2,3},{0,2,3},{1,2,3},{0,1,2,3}}
And this process can be continued indefinitely, so this is a completely ordinary infinite series, so there is no reason to doubt that the cardinality of this series is different from the natural numbers. In contrast, Georg Cantor's theorem states that the cardinality of a given set is always greater than the cardinality of its power set.

So what is the reason for the discrepancy that led Cantor to a different conclusion? In his own proof, Cantor did not follow the path I have presented here, but rather used tricky set operations to demonstrate a logical contradiction between the set and its power set. And from what has already been described, the source of the discrepancy is clear, which is that Cantor interpreted the natural numbers as a set, while I only referred to their nature as a sequence. While I, as we have seen, was talking about the series of power sets of natural numbers, Cantor was talking about the power set of the set of natural numbers. Namely, the concepts of series and set are different, and it is not clear how to treat natural numbers. As a set or as a series. As we have seen in the definition of identical cardinality, in the case of infinite discrete series, indexability with natural numbers fully ensures identical cardinality with natural numbers, while in the case of other sets we have to use other types of functions and procedures, since they cannot be indexed in this way.

How can we deal with this problem? To do this, we need to go back to the axioms of natural numbers. It is enough to examine the simpler, 130-year-old Peano axioms, but presented here in a slightly modified form.

AX1. The natural number 0 is the initial and smallest member of the sequence of natural numbers.

AX2. The natural number 1 is the member of the sequence of natural numbers following 0, the unity of the natural numbers.

AX3. For every natural number, there is one and only one subsequent natural number that is not identical to any other natural number and is one (1) greater than the preceding one.

AX4E. There exists a set with series of natural numbers that contains only natural numbers.

The first three axioms define the sequence of natural numbers and are fully sufficient for number theory theorems to be reduced to the axioms by complete induction proofs. The second axiom defines the unity that allows for the definitions of arithmetic operations.

The fourth axiom is confusing, it contradicts the other axioms. This has not caused anyone any trouble so far, because axioms do not need to be proven, they must be accepted as dogma, they must be acknowledged. But it is obviously a source of errors if the axioms are self-contradictory, and moreover, a source of undetectable errors, since the proof procedure does not extend to examining the axioms. Let us first look at the proof of self-contradiction without using the last axiom!

1. For every natural number, there exists a subset that contains it as its largest element.

2. Every subset of natural numbers has a largest element and the next natural number is not an element of the subset

3. It is not possible to choose a subset that contains all natural numbers.

4. If there were such a set containing all natural numbers that contains only finite natural numbers, then it must be a choosable subset.

So we have arrived at a statement from the first three axioms that contradicts the fourth axiom.∎

The acceptance of the last axiom AX4E was never self-evident, it required the high level of abstraction developed in higher education. Now, however, we discard this axiom and try to construct a set from the sequence of natural numbers defined so far in a different way.

So is there a set that contains all natural numbers? Yes. This set can be obtained by the union of a series of sets formed from natural numbers. $\underset{\mathrm{n=}0}{\overset{\infty }{\cup }}\text{{}n\text{}}$

The union of an infinite divergent series gives two possible results. Either we treat it as a meaningless operation, or the set includes the limit of our divergent series, the infinitely large natural numbers in an uncountably infinite cardinality.

Here we have skipped two steps that we did not define before. One concerns the limit value of infinite sequences, the other concerns the limit value of a divergent sequence. We will not deal with the former now, we only note that many operations on sequences are interpreted (often without special notation, Π, Σ, ∪, ∩, infinite decimal fractions) as meaning the limit value of the sequence or the operation performed on it. For the latter, we need to define another axiom, which is actually not unknown in mathematics, it is just known not as an axiom, but as the limit value of a divergent sequence. We also discard the previously described axiom AX4E.

AX4. Summing one ≺∞ times gives the value ≺∞.  $\sum _{\mathrm{n=0}}^{\mathrm{\prec \infty }}\text{1 = ≺∞}$.

Thus, the union of sets formed by natural numbers is $\underset{\mathrm{n=0}}{\overset{\infty }{\cup }}\text{{}\sum _{\mathrm{i=1}}^n\text{1} = {0,1,2,…,∞} = <span><em>𝕀</em></span>}$.

However, the last element, ∞ (infinite), is part of the infinite chain, despite the fact that the smaller, larger relation cannot be interpreted on it, arithmetic operations do not work, and this chain marked with the symbol ∞ denotes a member with uncountably infinite cardinality, so the entire set 𝕀 is also uncountably infinite cardinality. Namely, the axiom AX4 has the same meaning even if we start the summation with an arbitrary finite natural number. So it is true that n+∞=∞, for an arbitrary finite natural number n.

Several notable series similar to the limit values of the series described as axiom AX4 are known, for example:

Σn = ∞

$\sum _{\mathrm{n=1}}^{\infty }\frac{1}{n}\text{= ∞}$

$\sum _{\mathrm{n=1}}^{\infty }\frac{1}{2^n}\text{= 1}$

The basic idea that made it necessary to define the infinite sum as an axiom is to clearly understand the essential difference between an operation performed arbitrarily finitely many times and an operation performed infinitely many times. In the former case, the rules of arithmetic operations defined for finite numbers apply, in the latter case they do not. Thus, a statement true for all natural numbers is always understood for finite natural numbers, and is not interpreted for infinitely large natural numbers, which are new, little-discussed objects in mathematics. In addition, mathematics treats operations to be performed infinitely many times as limits of sequences. That is, the quantifier all does not mean everything in all cases. Since we do not prove axioms, they should only contain the simplest possible statements, ones whose insight is self-evident. A complex science like mathematics cannot be founded if the foundations are also uncertain, and the meaning of the axioms is confused.

Since the new axiom AX4 does not axiomatically define the existence of the set of natural numbers either, but with the aforementioned infinite union formation this set 𝕀 can be created, that is, it exists, it is therefore unnecessary to axiomatically record the existence of this set, but it must be noted that it has an uncountably infinite cardinality, unlike the sequence of natural numbers. At the same time, we must also see that the set thus created is not identical to the set of natural numbers, it is not something that every mathematician wants to see, and in fact, it is not possible to perform calculations with this set, since the infinitely large numbers in it do not allow this. So we must keep trying.

Definition. Let the symbol ≺∞ denote a finite number that is greater than every finite number but less than every infinitely large number. This number cannot be defined in more detail, but it is still of great importance. We can call it the horizon of finite numbers. And a new axiom:

AX5. ≺∞ alkalommal összegezve az egyet, az ≺∞ értéket ad eredményül.  $\sum _{\mathrm{n=0}}^{\mathrm{\prec \infty }}\text{1 = ≺∞}$.

This introduces a new partial way of finding the limit, which takes into account all finite terms but does not include the limit or limits. This corresponds to the familiar approach of considering the neighborhood of a given point but not the given point. So, for example, $\sum _{\mathrm{n=1}}^{\mathrm{\prec \infty }}\frac{1}{2^n}\text{= ≺1}$, i.e. this limit gives the neighboring rational number less than 1.

Thus, the traditional set of natural numbers $\underset{\mathrm{n=0}}{\overset{\mathrm{\prec \infty }}{\cup }}\text{{}\sum _{\mathrm{i=1}}^n\text{1} = {0,1,2,…}}$ = ℕ.

Rational numbers on the interval [0,1] $\underset{\mathrm{n=1}}{\overset{\mathrm{\prec \infty }}{\cup }}\underset{\mathrm{i=0}}{\overset{n}{\cup }}\text{{i/n}}$ = ℚ ∩ [0,1]

Real numbers on the interval [0,1] $\underset{\mathrm{n=1}}{\overset{\infty }{\cup }}\underset{\mathrm{i=0}}{\overset{n}{\cup }}\text{{i/n}}$ = ℝ ∩ [0,1]

So far we have described 5 axioms, introduced the concepts of infinity, the symbol of the finite horizon, and two types of limit formation, as well as limit operations on sequences (union, sum). But the use of natural numbers can be started with just 3 axioms, and the further steps can be learned when the children are mature enough.

But let's return to the initial problem. The sequence of natural numbers and the sequence of power sets of natural numbers have the same cardinality, but this is not true for the sets formed from them. Among the subsets of the set ℕ are, for example, the even numbers, or the subsets of odd numbers, whose ordinal number there is not in ℕ. However, it is in the previously defined set 𝕀. Written as a binary number, following the example of the series H(n):
 ...1010     the ordinal number of the set of even numbers in the set 𝕀,
 ...0101     the ordinal number of the set of odd numbers in the set 𝕀.

The cardinality of our sets: |P(ℕ)| = |𝕀| = |ℝ|

(Ferenc Takács, January 2025)