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:

  1. What am I getting wrong?
  2. Can power sets be used as a measure of how many relationships there are or is there a better way to do it?
  3. 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.

  • $\begingroup$ "The power set of the UNs seems to be countably infinite" -- could you please expound here? $\endgroup$ – BallBoy May 17 '18 at 16:38
  • $\begingroup$ How does one size $n$-"set" of UN's compare to another size $n$-"set" of UN's? Are these "equal"? If they might not be "equal", in what way can we actually say so and how can we not use that information to actually define values on them? If they are equal, then wouldn't the "power set" just consist of "sets" of the form $\{0\bullet\},\{1\bullet\},\{2\bullet\},\{3\bullet\},\dots,\{n\bullet\},\dots$ where $\{n\bullet\}$ is the "set" with $n$ UN's. $\endgroup$ – JMoravitz May 17 '18 at 16:40
  • $\begingroup$ Y. Forman, as I see it a subset that contains 3 members is exactly the same as every other member that contains 3 members, which means there's only 1 subset for each number of members, and only 1 infinite subset. $\endgroup$ – Ten O'Four May 17 '18 at 16:45
  • $\begingroup$ JMoravitz, what I was trying to do is remove as much "distinctness" as possible, but perhaps a set of n elements, is different in some way to another set of n elements, I just don't see how without adding something $\endgroup$ – Ten O'Four May 17 '18 at 16:48
  • $\begingroup$ Well., if you do, then you'll likely arrive at a set of $1$ element is different in some way to another set of $1$ element, thus allowing us to form a bijection with $\Bbb N$. It will not be an order-preserving bijection, but we don't care about keeping track of an order on the UN's, and without an order the "position" of them as well as the wellordering principle can't be talked about for them. It sounds as though you are talking about multisets and counting how many sub-multisets exist. For finite sets this is a common problem in combinatorics. $\endgroup$ – JMoravitz May 17 '18 at 16:52

You're really ignoring the following thing:

If $f\colon A\to B$ is a bijection, then $F(X)=\{f(x)\mid x\in X\}$ defines a bijection between $\mathcal P(A)$ and $\mathcal P(B)$. In other words, equinumerous sets have equinumerous power sets.

What you sort of alleging here is that because your set lacks structure, it only has "a few subsets". But just because it lacks structure doesn't mean that it can be endowed with structure.

For example $\Bbb N$ lacks the structure of a field, but it can be endowed with one by pulling it from a different countable set. For example the rational numbers.

Of course, this ignores a more serious problem, that not all subsets are necessarily definable in a fixed structure. Even in the case of the natural numbers, not all subsets are definable. Most subsets are not definable.

What you might be looking for is the collection of definable subsets of a given structure. Then on a fixed countable set there can be many ways to structure it, and ask what sort of sets are definable in a fixed structure. This acts a bit like a power set, but not quite exactly that.

Let's leave, for a second, the land of countable sets. And leave the axiom of choice behind. Without the axiom of choice you get sets which can have actual limitations on what kind of structures can be put on them. You can find sets that cannot be endowed with a structure of a field, or a linear order, and so on and so forth.

In some cases this in fact leads to this sort of "intuitive" thought that the only subsets you can have are finite or complement of finite sets. Such sets are called amorphous sets and they are a rich source for counterexamples in failures related to choice.

  • $\begingroup$ I see no issue applying another structure to it, but unlike the natural numbers that structure isn't contained in its relationships. The bijection implies capacity, certainly, but if you can't tell one set of 3 apart from another set of 3, how are they inherently distinct? Can they be inherently distinct or do we have to apply an artificial structure to them? What I'm interested in is more what we can build based on the relationships rather than what can be built if we apply artificial relationships. Hopefully, that made some sense $\endgroup$ – Ten O'Four May 17 '18 at 17:12
  • $\begingroup$ Yes. You've hit on a well-known truth. Questions of cardinality are invariant under changing the elements, and their answer depend only on the cardinality. All singletons are of size $1$. $\endgroup$ – Asaf Karagila May 17 '18 at 17:14
  • $\begingroup$ That gives me something to look up, thank you. I'm pretty sure I'm not conveying what I want to actually know lol. So is there any way to make the inherent structure "fuzzy", and in so doing, change the number of distinct subsets you can get? $\endgroup$ – Ten O'Four May 17 '18 at 18:44
  • $\begingroup$ You seem to mean that you want to remove some of the structure, or replace it with a different structure. $\endgroup$ – Asaf Karagila May 17 '18 at 18:44
  • $\begingroup$ Possibly yes. Possibly just softening the ability to know if subsets are distinct from each other could work. Honestly, I'm struggling to articulate this well, so anything vaguely related I could read would be awesome. $\endgroup$ – Ten O'Four May 17 '18 at 18:49

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