The standard answer `0*0 = 0`

is fine for most people. But the odd person who found this page will probably have second thoughts about it. As a philosoper you may feel that ‘zero times nothing’ is even less than plain empty: a deeper void.

Shouldn’t we express this as a power `0^2`

in mathematics? Or will all positive powers of zero necessarily collapse to `0`

under the usual assumptions in arithmetic? Can’t we make an exception so that zero powers make sense?

You have to ask yourself: what is zero? There are two simple answers.

First answer is the **+0** axiom that `a+0 = a`

. The function of adding zero to a number is not to change that number. This is called the identity element of addition in field theory.

Second answer is that zero is defined as one, minus one. The **1-** axiom substitutes a sign for a counted down entry `1- := 0`

in my natural array system, where a number increments by unit `1`

and the unit `-`

decrements it.

Equality `X=Y`

implies `Y=X`

(a symmetry relation), but in substitution the right side replaces the left (not symmetric).

Under the usual field rules we have:

`0*0 = 0*0 + 0 = 0*0 + 1*0 = (0+1)*0 = 1*0 = 0`

.

Using the **+0** axiom, symmetry of equality, the identity element of multiplication `1*a = a`

and the distributive law `(a+b)*c = a*c + b*c`

.

By induction this reduces `0^n = 0`

for all `n>0`

in standard algebra.

If we’d like to maintain that `0*0`

is somehow different from plain `0`

, some exceptions must be introduced to the axioms of field theory. Simplest is not to let our **+0** axiom apply to powers of zero. In turn we can discard the exception that there is no multiplicative inverse `1/0`

and allow for negative powers of zero too.

Try on this zero division ring. If `0^-1 = 1.. = ω`

continues to make sense, your arithmetical system just became rich!

With my **1-** substitution axiom an expression with just zero values like `0*0`

defies reduction altogether.

`1- + -*0 = 1 + -*1 + -*0 = 1 + -*(1+0) = 1 + -*1 = 1- := 0`

.

`1- + -*0 := 0 + -*0 = 1*0 + -*0 = 1-*0 := 0*0`

.

One way substitution to define inverses, how can we cope?

I can’t prove that `0+0 := 0`

either, should I leave it, or make it a new rule?

`1- + 0 := 0 + 0 = 0*1 + 0*1 = 0*(1+1) = 0*2`

.

`1- + 0 = 1 + -*1 + 0 = 1 + 0 + -*1 = 1 + -*1 = 1- := 0`

.

To hold on to zero powers I maintain that `-*0 ≠ 0`

and take `-*-`

to be a double negated variation of `1`

that also may raise special powers.

Some 10 years ago on my xs4all blog I wrote a series of articles, where I derived theorems from such a contrived substitution axiom. But I am sceptical now. To drive the rule of additive inverses into a one way street raises a serious obstruction in the centre of mathematics.

What do you think?