As part of my Cosmology, Consciousness, and Computation course I have lately been thinking a lot about Zeno’s paradoxes and the ‘standard solution’ to them that we get from calculus and thinking about motion as a ‘completed infinity’, limits, etc (next week we start quantum mechanics and I am starting to wonder about how Zeno’s paradoxes might relate or not to that, but one thing at a time!). Take the function g(x)=1/x, as x approaches zero the function trends towards infinity (both positive and negative infinity depending on the direction one is going in). We put this by saying that the function’s limit is ∞ (in both directions), but ‘infinity’ is not a real number. So a different way of putting this point would be to say that there is no real number that is the limit of this function.
This got me to thinking. There may be no real number but does that mean that there is no number at all? Suppose that we introduced them on the lines of imaginary numbers. In homage to this let us call them ‘pretend numbers’. 1/0 will then be equal to some pretend number. I will use ‘Þ’ for pretend numbers, so ‘pretend one’ will be ‘Þ1‘. What is Þ1? It is the pretend number that when multiplied by real 0 will give you real 1. We can then define n/0 as follows:
These numbers are not on the real number line, or on the imaginary extension of that line. They are on a further pretend extension. Admittedly 0/0 is still tricky and I am not sure what to say about it but putting that to one side, do pretend numbers exist?