Thoughts on the solution to 0 x 0, as declared to pertain over a given set, in certain conditions.

Definition of ‘nothingness’: the attribute which is being nothing; the being of nothing; the essence of nothing.

If a set is empty of nothing, then it can contain anything except nothing. In the event that there are, thus, zero lots of zero in that set, a condition must pertain which constitutes non-zeroness throughout that set.

Set-spanning non-zeroness is the result of multiplying zero, not by any non-zero number (which would create the absence of that non-zero number), but by zero, since ‘zero lots of zero’ means the ‘non-occurrence of zero’, which is any number, except zero; in other words, something, somewhere!

Considering the concept of 0 x 0 philosophically thus shows, simply, that the answer is not 0, which many wrongly assert.

If the multiplication is expanded to its additional nature, the same result is evident.

0 x n = 0 + 0 + … + 0 (n times)

For the case where n = 0, the answer is a list devoid of zeros: and therefore, also, a set which is empty of nothingness and, over the span of which, there must only be non-zero items.

So, 0 x 0 describes a situation which is empty of nothingness, and can, therefore, be anything except zero. The solution is therefore the complete set of possibilities, except zero.

Accordingly, 0 x 0 is not equal to 0, but may be anything else.