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About natural numbers and numerals
There are essentially two definitions of the natural numbers, Peano axiomatic one and the von Neumann set model. Both definition do not consider numerals as natural names for the natural numbers and written form of them, their notation.
All languages [1], I know, have numerals k for numbers 0 till ten. Numerals for 11-19 are formed as (k + 10) e.g. fourteen, mostly. Because they were used often, they are corrupted as eleven is. Multiplets of tens are expressed by one numeral formed as (k*ten) e.g. forty. Hundreds and thousands are counted separately, only kilomultiplets of thousands have their own numerals. Numbers between these pivots are expressed as linear combinations of basic numerals.
There exist exceptions, for example in Hindí language [2], where corruptions and exceptions appear till one hundred. Ancient Egyptians had for decimals specific names and hieroglyphs [3].
Notations of numbers had different forms: In the primitive form, one cut on a stick corresponded to each counted object. Egyptians introduced specific signs for powers of 10 till 107, but numerals till 9 expressed primitively by the corresponding number of signs. Phonicians introduced letters for 1-9, 10-90 and 100-900. It shortened the notation considerably. This system has been taken over by Hebrews and Greeks. Romans used their own system. Specific symbols were reduced on I, V, X, L, C, D, and M and the number of necessary symbols in one number by using a position system IV = one hand without one, IX = two hands without one. At last, we have Indian-Arabic decadic position system.
It should be mentioned the Mayan score system with position notation, where zero with a numeral signified multiplication by 20 (quatre-vingt in French) and the Babylonian hexadecimal system (German Schock, Czech kopa), where powers of three scores were expressed by size of their symbol (compare dozen - gross - great gross).
The names of numbers, numerals are generated by a modular system which is based on our fingers. We count sets by grabbing them with our hands and it is the natural way we speak and think about numbers. The definition of the natural numbers should express this fact. Therefore I propose the following definition:
The natural numbers are generated by a serie of modular operations, comparing of two sets, the compared set {n} and the modular set {m}.
The empty set {0} is from obvious reasons unsuitable as the modular set {m}.
The set {1} as the modular set {m} generates the natural number 0, only, since
{n} (mod {1} 0.
The set {2} generates the natural numbers 0 and 1.
Using a great inough modular set {m} we obtain in one modular operation all natural numbers. But it is inconvenient since we do not have inough simple symbols and numerals for them. Therefore, we must use a serie of modular comparisons, resulting in a serie of modular identities, which position notation leads to the modular equalities:
{135} (mod {10}) = 135
{135} (mod {4}) = 2013
It is obtained by the serie of consecutive divisions with the modulo rests
135 : 4 = 33 + 3
33 : 4 = 8 + 1
8 : 4 = 2 + 0
2 : 4 = 0 + 2
The resulting number modulo 4 is formed as the position combination all modular rests in inversed order = 2013.
Although the set {1} seems to be a natural base for a number system, and already the objects in sets exist in such a form, to each object one digit one corresponds, at a serie of modular comparisons, it gives a serie of zeroes. A division by 1 does not decrease the digit size of a number and it does not compress the notation.
The modular operation is essentially a mechanical one. In the first step the line of elements is cutted into rows by the modulo. The last line which is incomplete (it can be empty) is the result of the modular operation.
***** mod **: **
**
Rest * = 1
The column of the complete rows is transposed and the operation is repeated till all elements are exhausted
** mod ** **
Rest 0 = 0
* mod ** 0 (the number of complete rows)
Rest * = 1.
The resulting binary notation: 5 = 101. The third modular operation is the division by the third power of 2, the rest represents the number of fours in the original set. In the notation, they are determined as such by their position. A number of a smaller modulo is simultaneously a number of a greater modulo, to fours in a binary number corresponds hundreds in a decadic number. It is one hundred and one.
Two natural numbers are equal, if they are obtained from the same set {n} and comparable, if they are determined using the same modular set {m}.
Comparing with von Neumann set model, where joined sets {{0}, {1}} produce the number 2, here the set {2} generates the numbers 0 and 1.
The advantages of the proposed definition are obvious: It connects the natural numbers with the cardinal numerals by the algorithm, how the names and notations of the natural numbers are formed from the numerals. It is logical: Numbers which are described in natural languages by combinations of the cardinal numerals are the natural numbers.
References
1. A. Kolman, Istoriya matematiki v drevnosti, Gos. Izd. Fiz. Mat. Lit., Moskva, 1961.
2. V. Pořízka, Hindí Language Course, SPN, Praha, 1972.
3. E. A. Wallis Budge, Egyptian Language, Dower Publ., New York, 1970.