J.E.FREUND’S SYSTEM OF NATURAL NUMB

J.E.FREUND’S SYSTEM OF NATURAL NUMBERS POSTULATES
Modern mathematicians are accustomed to derive properties of natural numbers from a set of axioms or postulates (i.e., undefined and unproven statements that disclose the meaning of the abstract concepts).
The well known system of 5 axioms of the Italian mathematician, Peano provides the description of natural numbers. These axioms are:
First: 1 is a natural number.
Second: Any number which is a successor (follower) of a natural number is itself a natural number.
Third: No two natural numbers have the same follower.
Fourth: The natural number 1 is not the follower of any other natural number.
Fifth: If a series of natural numbers includes both the number 1 and the follower of every natural number, then the series contains all natural numbers. The fifth axiom is the principle (law) of math induction.
From the axioms it follows that there must be infinitely many natural numbers since the series cannot stop. It cannot circle back to its starting point either because 1 is not the immediate follower of any natural number. In essence, Peano’s theory states that the series of natural numbers is well ordered and presents a general problem of quantification. It places the natural numbers in an ordinal relation and the commonest example of ordination is the counting of things. The domain of applications of Peano’s theory is much wider than the series of natural numbers alone e.g., the relational fractions 1 , 1, 1, 1 2 3 4 and so on, satisfy the axioms similarly. From Peano’s five rules we can state and enumerate all the familiar characteristics and properties of natural numbers. Other mathematicians define these properties in terms of 8 or even 12 axioms (J.E.Freund) and these systems characterize properties of natural numbers much more comprehensively and they specify the notion of operations both arithmetical and logical.
Note that sums and products of natural numbers are written as a + b and a . b or ab, respectively.
Postulate No.1: For every pair of natural numbers, a and b, in that order, there is a unique (one and only one) natural number called the sum of a and b.
Postulate No.2: If a and b are natural numbers, then a + b = b + a
Postulate No.3: If a, b and c are natural numbers, then (a+b)+c=a+(b+c)
Postulate No.4: For every pair of natural numbers, a and b, in that order, there is a unique (one and only one) natural number called the product.
Postulate No.5: If a and b are natural numbers, then ab = ba
Postulate No.6: If a, b and c are natural numbers, then (ab)c = a(bc)
Postulate No.7: If a, b and c are natural numbers, then a( b + c ) = ab + ac
Postulate No.8: There is a natural number called “one” and written 1 so that if a is an arbitrary natural number, then a.1 = a
Postulate No.9: If a, b and c are natural numbers and if ac = bc then a = b
Postulate No.10: If a, b and c are natural numbers and if a + c = b + c then a = b
Postulate No.11: Any set of natural numbers which (1) includes the number 1 and which (2) includes a + 1 whenever it includes the natural number a, includes every natural number.
Postulate No.12: For any pair of natural numbers, a and b, one and only one of the following alternatives must hold: either a = b, or there is a natural number x such that a + x = b, or there is a natural number y such that b + y = a .
Freund’s system of 12 postulates provides the possibility to characterize natural numbers when we explain how they behave and what math rules they must obey. To conclude the definition of “natural numbers” we can say that they must be interpreted either as standing for the whole number or else for math objects which share all their math properties. Using these postulates mathematicians are able to prove all other rules about natural numbers with which people have long been familiar.

Результаты (русский) 1:

J.E.FREUND'S СИСТЕМА ПОСТУЛАТА НАТУРАЛЬНЫХ ЧИСЕЛСовременные математики привыкли к наследовать свойства натуральных чисел из набора аксиом или постулаты (то есть, неопределенные и недоказанные заявления, которые раскрывают смысл абстрактных понятий). Хорошо известная система 5 аксиом итальянский математик, Пеано предоставляет описание натуральных чисел. Эти аксиомы являются: Первый: 1 — натуральное число. Второе: Любое число, которое является преемником (последователь) натуральное число является само натуральное число. Третий: Нет двух натуральных чисел имеют же последователь. Четвертое: Натуральное число 1 не является последователем любое натуральное число. Пятое: Если серия натуральных чисел включает номер 1 и последователь каждое натуральное число, то серия содержит все натуральные числа. Пятая аксиома является принцип (Закон) математики индукции. From the axioms it follows that there must be infinitely many natural numbers since the series cannot stop. It cannot circle back to its starting point either because 1 is not the immediate follower of any natural number. In essence, Peano’s theory states that the series of natural numbers is well ordered and presents a general problem of quantification. It places the natural numbers in an ordinal relation and the commonest example of ordination is the counting of things. The domain of applications of Peano’s theory is much wider than the series of natural numbers alone e.g., the relational fractions 1 , 1, 1, 1 2 3 4 and so on, satisfy the axioms similarly. From Peano’s five rules we can state and enumerate all the familiar characteristics and properties of natural numbers. Other mathematicians define these properties in terms of 8 or even 12 axioms (J.E.Freund) and these systems characterize properties of natural numbers much more comprehensively and they specify the notion of operations both arithmetical and logical. Note that sums and products of natural numbers are written as a + b and a . b or ab, respectively. Postulate No.1: For every pair of natural numbers, a and b, in that order, there is a unique (one and only one) natural number called the sum of a and b. Postulate No.2: If a and b are natural numbers, then a + b = b + a Postulate No.3: If a, b and c are natural numbers, then (a+b)+c=a+(b+c) Postulate No.4: For every pair of natural numbers, a and b, in that order, there is a unique (one and only one) natural number called the product. Postulate No.5: If a and b are natural numbers, then ab = baPostulate No.6: If a, b and c are natural numbers, then (ab)c = a(bc) Postulate No.7: If a, b and c are natural numbers, then a( b + c ) = ab + ac Postulate No.8: There is a natural number called “one” and written 1 so that if a is an arbitrary natural number, then a.1 = a Postulate No.9: If a, b and c are natural numbers and if ac = bc then a = b Postulate No.10: If a, b and c are natural numbers and if a + c = b + c then a = b Postulate No.11: Any set of natural numbers which (1) includes the number 1 and which (2) includes a + 1 whenever it includes the natural number a, includes every natural number. Postulate No.12: For any pair of natural numbers, a and b, one and only one of the following alternatives must hold: either a = b, or there is a natural number x such that a + x = b, or there is a natural number y such that b + y = a . Freund’s system of 12 postulates provides the possibility to characterize natural numbers when we explain how they behave and what math rules they must obey. To conclude the definition of “natural numbers” we can say that they must be interpreted either as standing for the whole number or else for math objects which share all their math properties. Using these postulates mathematicians are able to prove all other rules about natural numbers with which people have long been familiar.