Properties of Non-Diophantine Arithmetics
Properties of Non-Diophantine Arithmetics
In some non-Diophantine arithmetics, even the most evident truth (such as 2 + 2 = 4 or two times two is equal to four) may be discarded. Some of them (projective arithmetics) possess similar properties to those of transfinite numbers arithmetics built by the great German mathematician Georg Cantor (1845-1918). For example, a non-Diophantine arithmetic may have a sequence of numbers a1 , a2 , …, an , … such that for any number b that is less than some an the equality an + b = an is valid. This is an important property of
infinity, which is formalized by transfinite (cardinal and ordinal numbers). The equality
is another interesting property of some transfinite numbers. Such equality may be also true in some non-Diophantine arithmetics. Thus, non-Diophantine arithmetics provide mathematical models in which finite objects – natural numbers – acquire features of infinite objects – transfinite numbers. In such a way, it is possible to model and to describe behavior of infinite entities in finite domains.
Also Read : Discovery of non diophantine arithmetic
In non-Diophantine arithmetics, it is also possible that two times two is not equal to four. We can add different numbers to the same number and get the same result. There are such non-Diophantine arithmetics that have the largest number. All this contrasts what is possible in the Diophantine arithmetic.
Let us consider some other features of non-Diophantine arithmetics and compare them with the properties of the Diophantine arithmetic.
We know from school that the main laws of the Diophantine arithmetic are:
1. Commutativity of addition: a + b = b + a ;
2. Associativity of addition: (a + b) + c = a + (b + c);
3. Commutativity of multiplication: a ⋅ b = b ⋅ a ;
4. Associativity of multiplication: (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c);
5. Distributivity of multiplication with respect to addition: a ⋅ (b + c) = a ⋅ b + a ⋅ c ;
6. Zero is a neutral element with respect to addition: a + 0 = 0 + a = a ;
7. One is a neutral element with respect to multiplication: a ⋅ 1 = 1 ⋅ a = a .
Now we may ask whether these laws are valid for non-Diophantine arithmetics. First, addition and multiplication are always commutative, zero is a neutral element with respect to addition, and one is a neutral element with respect to multiplication in all non-Diophantine arithmetics. At the same time, the laws of associativity and distributivity fail in the majority of non-Diophantine arithmetics. Only special conditions on the functional parameter of the non-Diophantine arithmetic in question provide validity of these laws. These conditions are obtained in (Burgin, 1997).
Second, the Diophantine arithmetic possesses the, so-called, Archimedean property, which is important for proofs of many results in arithmetic and number theory. It states that if we take any two natural numbers m and n, in spite that n may be enormously larger than m, it is always possible to add m enough times to itself, i.e., to take a sum m + m + … + m, so that the result will be larger than n. This property is also invalid in the majority of non- Diophantine arithmetics. As we have seen above (cf. the example “A Woman and a Pay Phone”), it is possible to add 1 to itself as many times as you can but never get 5. The Archimedean property is important for proving that the set of all natural numbers is infinite as well as the set of all prime numbers. Thus, having in general no Archimedean property in non-Diophantine arithmetics, we encounter such arithmetics that have only finite number of elements or such infinite arithmetics that have only finite set of prime numbers (Burgin,
Another example of a non- Archimedean arithmetic in the nonstandard arithmetic (Robinson, 1966). However, this arithmetic includes infinitely big and infinitely small numbers, going thus, beyond natural numbers.
Third, the Diophantine arithmetic always contains infinitely many numbers. Some of non-Diophantine arithmetics contain the largest number and thus, are finite. However, this is possible only for perspective arithmetics. All dual arithmetics are infinite. So, we can see that many properties of non-Diophantine arithmetics contradict our intuition and contrast to what we know from our experience.