Wednesday, June 24, 2009

The Axioms of Arithmetic

During the 20th Century, mathematicians decided that they had to properly define the foundations of maths. Work by philosophers such as Bertrand Russell showed that there were inherent contradictions in the subject that might put many important assumptions in doubt. As several theorems proved to be false, there were concerns that many branches of maths were built on houses of cards that may be found to tumble.

To try to resolve this, a list was produced of axioms which can be taken as self-evident. This means they can be assumed as first principles upon which all other theories can be built. However, even these axioms have not been immune to questioning - and, if you need axioms upon which you build the axioms, doesn´t that also mean you need axioms to build the axioms of the axoms....(ad infinitum)? Anyway, these are the seven Axioms of Arithmetic:

  1. For any numbers mn: m + n = n + m
  2. For any numbers m, n, k: (m + n) + k = m + (n + k)
  3. For any numbers m, n, k: m(n + k) = mn + mk
  4. There is a number 0 which has the property that, for any number n: n + 0 = n
  5. There is a number 1 which has the property that, for any number n: n x 1 = n
  6. For every number n, there is another number k such that: n + k = 0
  7. For any numbers m, n, k, if k ≠ 0 and kn = km, then m = n

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