Is there some other definition of cardinality that captures the notion that they are twice #integers than #oddintegers.
Something like:
Even numbers can be defined as the map x->2x. Odd numbers like x->2x+1. All numbers x->x is the baseline. Square numbers, the map x->x*x
There is some way to "count"(quantify/measure) this sets based on these distributions over the line?
It seems I would like to take derivative over x when dealing with linear forms. I would like to consider iterative derivatives when dealing with multiplicative forms like x*x.
Maybe the "count" could be encoded/expressed like:
- oddnumbers: 2
- evennumbers: 2
- quadrating numbers: e^2
- (x->x^3+2x+3): e^3+2
Or another way of looking at that. I'm taking derivatives over x until there is no x, while exponentiating at each step to encode the process.
But then, how can I "count" 2^x... The "derivative until there is no x" would run forever... 2^x grows too fast to be "counted" by this silly method...
Maybe I should first take the log and then iterative derivative over x (*).
So in general I would take iterative log mixed with iterative derivative until I hit something that grows too fast.
Is there any hiperlog function out there?
It is like taking derivative over the line, then over the powerset of the line, the powerset of the powerset of the line, etc... efectively counting at the rigth (ordinal?) level...
(*) Taking a log and then derivate over x, it is like a derivative over 2^x? Not sure the right name or if derivative over 2^x makes sense. The idea is to modify the underlying measure and choose one more suitable for exponential growth...
Well I should stop here... What I say has any sense? There is some paper, book or mathematical concept that captures these "intuitions"? I think it is the very definition of countable ordinals (w, 2w, E0, etc...)