Axioms of Arithmetic 2

Here is a list of the axioms of Arithmetic as stated by Gottlob Frege.

1. Background Information

   1. There are objects called Real Numbers. All Real numbers form an algebraic field.

   2. Fundamental Operations - there are only 2 fundamental operations with which 2 real numbers can be manipulated to produce a 3rd real number.

   3. These 2 operations are assumed to satisfy certain axioms, called axioms for a field.

      1. Addition – resulting in a “sum”

      2. Multiplication – resulting in a “product”

2. Axioms for 0 & 1

   1. Existence of 0: There is a number “0,” called the additive identity, that satisfies a + 0 = a for all real numbers a.

   2. Existence of 1: There is a number “1,” called the multiplicative identity, that satisfies a1 = a for all real numbers a.

   3. 0<>1

3. Axioms of Addition

   1. Commutative Law of addition: a + b = b + a

   2. Associative Law of Addition : a + (b + c) = (a + b) + c

   3. Closure Property: a + b is a Real number.

   4. Axioms Derived from Axioms of Addition

      1. Additive Identity Property: a + 0 = a

      2. Additive Inverse Property: For every “a” there is a “-a” such that “a + (-a) = 0”

4. Axioms of Multiplication

   1. Commutative Law of Multiplication: a X b = b X a

   2. Associative Law of Multiplication: (a X b) X c = a X (b X c)

   3. Closure Property: a X b is a Real number

   4. Axioms Derived from Axioms of Multiplication

      1. Multiplicative Identity Property: a X 1 = a

      2. Multiplicative Inverse Property: For every “a” there is a “1/a” such that “a X 1/a = 1”

   5. Distributive Property combining Addition & Multiplication

   6. (a + b) X c = a X c + b X c

5. Four Additional Axioms – Real numbers are Ordered

   1. Given a & b only one of the conditions is true: “a < b” or “a = b” or “a > b”

   2. If a > b & b > c, then a > c

   3. Monotonic Property of Addition: If a > b then a + c > b + c

   4. Monotonic Property of Multiplication: If a > b & c#0 then a X c > b X c

Axioms - Imagined Mathematical Realities

In his book Sapiens, social scientist Yuval Harari proposes a theory for the ascendance of humans on this Earth. He primarily attributes it to the human ability to imagine "worlds" whose ideas help humans to cooperate and collaborate.

According to him our ideas of religion, nation, patriotism, human rights etc are examples of such worlds.

We can think of "axioms" as "imagined worlds" created by mathematicians which have the rules which control the development of math.

The axioms cannot be proven: They are a short list of properties that we intuitively expect numbers to satisfy. However, the axioms are useful in that, from them, important additional facts about numbers can be proven. In this sense mathematics is purely a creation of the human mind.

An important theorem that can be proven directly from the axioms is that multiplying any real number by 0 produces 0 as the result.