# On what represent natural numbers

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What is a number? Consider, for instance, some examples of natural numbers. What represents “0”? A totality? No, at all, rather the absence of an element or multiplicity. What represents “1”? A totality? Usually, no, not even a multiplicity. “1” represents an individuality or element, not a multiplicity.

ON WHAT REPRESENT NATURAL NUMBERS
What is a number? Consider, for instance, some examples of natural numbers. What represents “0”? A totality? No, at all, rather the absence of an element or multiplicity. What represents “1”? A totality? Usually, no, not even a multiplicity. “1” represents an individuality or element, not a multiplicity. Most natural numbers represent abstract multiplicities. The cardinality of natural numbers does not represent, cannot represent a totality.

Beyond 0 and 1, the natural numbers represent, orderly, abstract multiplicities, more precisely, different quantities of abstract multiplicities. For instance, “3” represents a multiplicity that is greater with one element than the multipliplicity represented by “2”. Thus, most natural numbers represent, in order, the quantity of the abstract multiplicities.

If we unify a multiplicity of points in a line, then the result, that totality, is no more a multiplicity, but a unity, an individuality a “1”. The totality is no more than a kind of quantity of a multiplicity, the elements of the multiplicity considered as a whole, or the multiplicity considered as a whole. However, the multiplicities can be compared quantitatively, that is, the natural numbers are a ring of increasing multiplicities.

That is, the quantity of the multiplicity can be considered in many senses: i) the quantity of the multiplicity in comparation with other multiplicities, and this is the most important sense, for the consideration of natural numbers; ii) and in the Kant’s sense, considering the quantity of the elements, of the same multiplicity, which are taken in consideration.

That is, the sense of the natural numbers’ totality depends on the more fundamental concepts of individuality or element and multiplicity. In truth, when we speak of multiplicities comparatively, the concept of totality is not really needed. Two totalities cannot be compared quantitatively, without to think at the multiplicities or individualities which they represent, because they haven’t an absolute sense.

The multiplicity is more properly used in the logical context of the quantification of the extensions of terms, not in the context of the explaining of the concepts of natural numbers. For instance, when we ask how many individuals of this class have a property X? One, few, all? But when we consider the natural numbers, we understand that most of them represent multiplicities, and their differences consist in the different quantities of their multiplicities, comparatively.

There can be no the totality of all the natural numbers.
There can be no a greatest natural number.

When we consider comparatively the totalities, when we want to measure them, you know that, we will arrive at the fundamental problem of exact measure of the quantity of the abstract multiplicity, that is, we will need to consider how many elements contains each multiplicity, and we will need a symbolism and a way of notation of the different quantities of multiplicities. Even a non-multiplicity can be considered as a totality.

But, …you know, beyond of appearances, a totality means no more than either an element, or a multiplicity, a relative, or an absolute? multiplicity…
However, in realm of natural numbers, there can be no an absolute totality, in the sense of a greatest natural number.

The can be a finite multiplicity of natural numbers, BUT THERE CAN BE NO AN ACTUAL TOTALITY OF ALL THE NATURAL NUMBERS.

My core conclusions:
A set, that is, a multiplicity cannot be its sub-multiplicity or individual element, but a multiplicity can be only identical whit itself. An individuality or the emptiness do not represent a multiplicity. A multiplicity can contain only elements or sub-multiplicities. A multiplicity cannot be its sub-multiplicity. Without separation, discontinuity, difference there can be no multiplicity. Unifying or destroying the discontinuity, which underlies the multiplicity, we destroy the multiplicity, creating an element, the natural number one.

That is, i.e. from the number 2 we create the number 1, because a continuous entity is an element, a number one, though that is two times bigger than other element. Also, when we consider the abstract multiplicity, we are interested mainly of the quantity or how many elements they have, not if one subset or sub-multiplicity has a greater cardinality. The multiplicity is mainly about elements, not about their dimension
(i.e. length, volume etc.).

A natural number represent abstractly either the absence of an element, or the presence of an element, or the presence or existence of different multiplicities of elements. Without discontinuity, separation and difference in being, there can be no plurality, multiplicity of separated, different parts, and the system of natural number has as great utility to make possible the measure or counting of the quantity of different multiplicities.

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