Is a POWERSET really larger than its SET

[Antone]

Traditionally, set theory says that the [cardinal number of a power set] is larger than the [cardinal number of the set it is derived from], but I disagree.

First, I'll briefly define [powerset] and [cardinal number], for those who aren't familiar with these terms:

Traditionally, a powerset is the set of all the possible subsets of a given set. So, for example, the set {1, 2} would have the powerset {{...}, {1}, {2}, {1, 2}} where {...} is the empty set.
Personally, I don't agree that {...} should be considered a set, but that's a topic for a different thread.

And the cardinal number of a set is the number of elements that the set contains.

Now, with these definitions, it may appear that the [cardinal number] of {1, 2} would have to be larger than the cardinal number of {{...}, {1}, {2}, {1, 2}} because the first set contains [2 elements] while the second set contains [4 elements].

The reason I disagree is because I would argue that a [cardinal number] is numerically meaningless unless it has a unit of measure attached. In a practical sense, does [1] refer to something smaller than [100]? It depends. For example, [1 solar system] is larger than [100 grains of sand]. Divorced of a unit of measure, there's no way to know which number represents more.

Consider a bucket of sand: Suppose the bucket contains 100,000 grains of sand. What number does the bucket of sand represent. There are a great many ways to determine what is the appropriate number to be assigned to this state-of-affairs.
If our unit of measure is a [bucket of sand] then the appropriate number would be [1];
If the unit of measure is [grains of sand] then the appropriate number would be [100,000];
If the unit of measure is [objects] then the appropriate number is indefinable because the [unit of measure] is vague. For instance, Is the bucket a [single object] or is it two objects--after all it has a handle that is loose. If two grains of sand are stuck together by mud, is that one object, two objects, or is it three objects [two grains of sand and a bit of mud]. And so forth.

The only way to give the numbers any meaning at all is to assign to them a [unit of measure].

When we do this, there are two necessary results: First, only sets that contain [conceptually similar elements] can have a [conceptually meaningful cardinal number]; and second, the [power set of any conceptually meaningful set] is necessarily the same size as the [set] itself.

Consider the set of all the apples sitting on my desk: in the case where there are three apples sitting on my desk.
We can enumerate this set as:

(A) {A1, A2, A3},

where each element of (A) is an [apple on my desk]. Now, the powerset of (A) is:

(pA) {{...}, {A1}, {A2}, {A3}, {A1, A2}, {A1, A3}, {A2, A3}, {A1, A2, A3}},

But the unit of measure for (pA) is obviously not an [apple on my desk] anymore. Instead, each element in (pA) is a possible grouping of only those elements in (A).

We can abstract these sets as follows:

(A) {X:X is an apple on my desk}
(pA) {X:X is a possible grouping of the elements in A}

These are clearly very different [units of measure], and Groupings is not even a physical thing, so in a strictly physical sense, the abstracted version of (pA) is actually smaller than the abstracted version of (A). It's elements have no physical presence at all. In a purely conceptual sense, however, the abstracted versions of (A) and (pA) are the same size because they both contain one element: [X].

Yes, that's an nontraditional interpretation of abstract sets--but it is the common sense interpretation, if you ask me.

If we want to convert (pA) back to a physically meaningful set, we can start by introducing a new set theory principle:

The Principle of Elemental Redundancy: a set con only contain one instance of an absolutely unique element.

Now, to finish converting (pA) into (A) we can remove the idea of groupings from the power set, by removing the [set brackets] from the subsets in (pA).
The resulting is as follows:

(A) {A1, A2, A3, A1, A2, A1, A3, A2, A3, A1, A2, A3}

And when we remove all the duplicate (and thus meaningless) elements from the set, we get:

(A) {A1, A2, A3}

So once again, we see that (A) and (pA) are physically equivalent. Because they each involve [3 physical elements].

Those who are familiar with set theory may realize that this [way of understanding powersets] goes a long way towards resolving a number of paradoxes in set theory. It also puts a slightly different spin on the question:

Is the [powerset of an infinite set] is larger than the [infinite set].