Could a "fuzzy", or non-symmetric, relationship change the size of a power set?

Question

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.

Answer 1

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.

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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.

Answer 2

I know this is a zombie response to a two-year-old question, but it seems to me that the OP here reinvented/rediscovered the concepts of cardinality and equivalence classes, restricted to the set of natural numbers $\Bbb{N}_0 := \{0, 1, 2, 3, ...\}$. From OP's comments:

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.

In other words, OP is really suggesting that 2 subsets of the "Unnatural Numbers" are to be considered "the same" if they can be paired off in a 1-1 correspondence. This is functionally equivalent to taking the subsets of the regular natural numbers $\Bbb{N}_0$ and forgetting the (labels/values/numerical properties) on the elements of any subset.

If we do this, then the only way to tell two subsets of $\Bbb{N}_0$ apart is if they have different numbers of elements. Contrapositively, this is the same as saying two subsets of $\Bbb{N}_0$ are "the same" if they have the same number of elements (i.e. there's a bijection between them). "Indistinguishability", in this sense, turns out to define what is usually referred to as an equivalence relation $\sim$ on the subsets of $\Bbb{N}_0$. Although OP specifies "fuzziness" to mean non-symmetric in their post, their example of "fuzziness" is more like the "forgetfulness" one gets from an equivalence relation--we can indeed check that "indistinguishability" is an equivalence relation on subsets of $\Bbb{N}_0$, which actually means it's symmetric (as well as reflexive and transitive). And the names of these equivalence classes are exactly the finite and countable cardinalities, because "cardinal numbers" are exactly what one gets by "forgetting" the labels, or interpretations, or relationships, between the elements of a set, and just keeping its "size".

(The process of "forgetting" some or all irrelevant aspects of the structure on a highly structured set is very common in math and plays a role in foundations: see, for instance, forgetful functors. I suppose this could be considered another topic the OP 'discovered' for themselves, in a particular context.)