THE STUPID NUMBERS

Introduction:

The stupid numbers, * S*, form a set which may or may not be a field. The size of the set

*has not been determined and*

**S***may be the empty set. There are two binary operations on*

**S***, addition ( + ) and multiplication ( * ), which may or may not make the set*

**S***a commutative group under addition and may or may not make the non-zero elements of*

**S***a commutative group under multiplication. ***S ***is based on the following axioms:*

**S**```
Axiom of Reflexivity: For every x, x = x.
Axiom of Symmetry: For every x and y, if x = y then y = x.
Axiom of Transitivity: For every x, y, and z, if x = y and y = z then x = z.
Axiom of Additive Commutativity: For every x and y, x + y = y + x.
Axiom of Additive Associativity: For every x, y, and z, x + (y + z) = (x + y) + z.
Axiom of the Existence of the Additive Identity: There exists an element x such that for every y, x + y = y.
Axiom of the Existence of the Additive Inverse: For every x there exists a y such that x + y = 0.
Axiom of Multiplicative Commutativity: For every x and y, x * y = y * x.
Axiom of Multiplicative Associativity: For every x, y, and z, x * (y * z) = (x * y) * z.
Axiom of the Existence of the Multiplicative Identity: There exists an element x such that for every y, x * y = y.
Axiom of the Existence of the Multiplicative Inverse: For every x there exists a y such that x * y = 1.
Axiom of Distributivity: For every x, y, and z, x * (y + z) = (x * y) + (x * z).
Axiom of Stupidity: 0.999 … ≠ 1. Note: 0.999 … = 0.9 ̅
```

Discussion # 1:

Are the axioms of the stupid numbers consistent or inconsistent? If the axioms are inconsistent, which axiom(s) could be removed, without the removal of the Axiom of Stupidity, to make the remaining axioms consistent? If the axioms are inconsistent, can more than one scheme of axiom removal without the removal of the Axiom of Stupidity be devised to make the remaining axioms consistent? If so, which scheme(s) create interesting or useful axiomatic system(s)?

One scheme of axiom removal to ensure that the axiomatic system is consistent would be to remove all of the axioms except the Axiom of Stupidity. Would this result in S = {0.999 … , 1} and the operators of addition and multiplication being undefined?

Alternatively, if the axioms of stupid numbers are inconsistent, could additional axioms be added to make the axiomatic system consistent?

Discussion # 2:

If the axioms of the stupid numbers are consistent:

What is size of * S*?

Is

*a field?*

**S**Is

*a commutative group under addition?*

**S**Are the non-zero elements of

*a commutative group under multiplication?*

**S**Is 1 > 0.999 … ?

What are some of the other properties of

*?*

**S**If axioms, other than the Axioms of Stupidity, must be removed or added to make the axiomatic system of the stupid numbers consistent, what are the answers to the above questions using the new axiomatic system(s)?

Discussion # 3:

You may note that the axioms of * S* do not include any axioms of order. What are some of the consequences of this omission? Without any axioms of order, does the relation > have any meaning? Can consistent axioms of order be added to the axiomatic system of the stupid numbers?