Re:Is infinity an odd or even number? (2011)
by crazygringo on Tuesday May 02, 2023 @05:38PM
(#35794583)
Thanks, but you may have misunderstood the definition I have for defining odd numbers, because that corresponds equally to the natural numbers as well.
So there is no better parity rule as you say, it is entirely arbitrary. It's not extending the same pattern, it's seeing that there are two ways of extending it and picking one arbitrarily.
In that case, if it were me, I'd call it something else. Not parity or even/odd. Because it's not the natural obvious extension. It's arbitrary.
Modern mathematics is all about coming up with definitions and rules that give rise to interesting (to a mathematician!) properties when further investigated.
The definition given naturally lets the ordinal numbers continue the odd/even/odd... pattern. Choosing the alternative definition would not.
In one sense that's 'arbitrary' because we decided on one definition over another. But another sense, we picked the parity rule that lets us extend the same pattern from the natural numbers, so it's a 'better' parity rule. And the fact that one rule gives this pattern while the other does not, did not come from humans, but is a 'metamathematical fact' from the universe of possible ways to define things.
So I would say this definition is not fully arbitrary, it's an interaction between what mathematicians find interesting and the Platonic realm of possible mathematical constructs.
Anyway, I'm not a mathematician but it seems this is how the game of math is played: to continually discover new rules that give rise to more interesting math.