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.
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?