Let's say we have a set of numbers called the Unnatural Numbers (UN). These numbers have the following properties.
- They're equinumerous with the natural numbers
- They have no defined position
- They have no defined value
- The well ordering principle doesn't apply
- They have just enough "distinctness" that we can add n UNs to a set and have the set contain n members
I think of them like a countably infinite pile of absolutely identical grains of sand.
The power set of the UNs seems to be countably infinite, which is the same size as the set of UN. This seems to be because the relationships between the UNs are low. My questions are:
- What am I getting wrong?
- Can power sets be used as a measure of how many relationships there are or is there a better way to do it?
- Can relationships be made "fuzzy", or non-symmetrical, and what does that do to power sets?
Regarding (3), there seems to be no issue numerically, but I can't really work out what it means. Does a "fuzzy" relationship only carry certain properties, would this mean that subsets we think of as distinct would now be the same, would the power set only change under certain conditions, etc.
I may be talking complete nonsense with this, but it's been fun thinking about what "fuzziness" could mean.