A (possible) puzzle regarding John Lane Bell's "Abstract Sets"

Question

John Lane Bell, is his paper "Abstract and Variable Sets in Category Theory" (go to Bell's Homepage to download it), defines an abstract set as follows:

"An abstract set is then an image of pure discreteness, an embodiment of raw plurality; in short, it is an assemblage of featureless but nevertheless distinct 'dots' or 'motes' [footnote 3: "Perhaps also as 'marks' or 'strokes' in Hilbert's sense.]. The sole intrinsic attribute of an abstract set is the number of its elements." (pg. 10)

With this in mind let us consider a countably infinite set of such strokes

{|,|,|,...} = S

and attempt to define the power set P(S) of S. If one allows the existence of the empty set { } one can define P(S) as follows:

{ { }, {|}, {|,|}, {|,|,|},...}

which is fine as long as the subsets are finite. However, things seem to break down when one tries to define the countably infinite subsets of S because by the definition of S, any countably infinite subset S' of S is simply S, which seems to make the cardinal number|P(S)| of P(S) Aleph-null.

Although Bell states two paragraphs down that "an abstract set cannot be regarded as the extension of an attribute...", it seems that one needs the notion of an 'attribute' to be able to distinguish countably infinite subsets S' of S from S (and from each other) in order to make |P(S)| greater than Aleph-null. But this seems to equate the cardinality of |P(S)| with the number of attributes which allows one to distinguish the countably infinite subsets S' of S from one another (such attributes may, for lack of a better term, may be designated as 'extensional attributes'). Herein lies (at least for me) the puzzle, if in fact it IS a puzzle.

Answer 1

You have a number of incongruities in your question, I think:

1. If S is made up of infinite strokes, as you presented your question, its cardinality is ℵ₀
2. So, the set S is countably infinite
3. This also means that the set ℘(S) is not countable since |℘(S)| = 2^ℵ₀ which, assuming continuum hypothesis, is equivalent to ℵ₁
4. Ergo, all possible subsets of S (otherwise known as ℘(S)) are not countable; keep in mind that 2^ℵ₀ > ℵ₀ which is the same thing as ℵ₁ > ℵ₀

Also, when looking at a power set ℘(S) of a set S, there is only one element in ℘(S) that's simply S.

Consider L = {a, b, c}
Then ℘(L) = {Ø, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}

L is only equivalent to the last discrete element in ℘(L), so I guess one could axiomize α ∈ ℘(α) since the same holds for countably infinite sets (and I think uncountably infinite sets, as well but I could be wrong).

I don't think there needs to be any distinguishing between L and (in our case) the last element of ℘(L) as they represent the same "thing" - in one case L is a set made up of discrete "things" and in the second, L itself is a "thing" inside a larger set. I don't think this breaks Bell's discreteness (or Set Theory, for that matter).

If your question was about breaking set identity (which Bell argues is simply the number of elements - also known as cardinality) when using the power set, worry not! Directly from Wikipedia:

Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (informally the power set must be larger than the original set).

So there is never any ambiguity. Or maybe I missed the entire point of the question.

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@ThomasBenjamin, to answer your comment:

• I have the set of all natural numbers N
• I have the set of all even natural numbers, lets call this N_e
• I have the set of all odd natural numbers, lets call this N_o

Set theory says that |N| = |N_e| = |N_o| - that is, the cardinality of the set of natural evens is equivalent to that of the set of natural odds and to that of the set of all naturals. This can be confusing because N_e and N_o are subsets of N.

However, that doesn't really seem to be a problem for Bell. In a sense, you're right to worry about |N| = |N_e| = |N_o|. How can we say that the even numbers are the same thing as the odd numbers?

Well, it looks like Bell argues that all sets of cardinality ℵ₀ or greater are abstract sets (as opposed to concrete sets like L = {a, b, c}). So you could say L possesses the attribute a, but you could not say N_e possesses the attribute 2. So then how can we know that 2 is in N_e? Well, Bell, at the bottom of pg. 10, shows us how to create a relation between two abstract sets. In our case, it would maybe be f: n → 2n.

So, going back to your question (and comment), there is no way to distinguish between two sets if they have the same cardinality unless we map them. So until we specify that the set of all even naturals maps to n → 2n and the set of all odd naturals maps to n → 2n + 1, the two are identical.

Answer 2

I'm not sure what "intrinsic" means, but if the sole intrinsic property of sets is the number of elements, you can't have power sets in the normal sense.

2^{|,|} = {x|x is one of ({}, {|}, {|}, {|,|})} = { {}, {|}, {|,|} }

If you work through this, you may end up defining a power set operator that is identical to the successor operator. And that seems to me to be what you are doing (I have not checked to see if Bell makes the same mistake; I hope not).

Calling successor "power set" strikes me as unhelpful. But, anyway, yes, if you only have successor, you get stuck at ℵ₀.

Incidentally, this is also why the natural numbers are usually defined as {}, {{}}, {{},{{}}}, ...--so you have a way to distinguish the elements.

Alternatively, a countably infinite subset of a countably infinite set has the same cardinality as the original (by definition), but it is not the same set. For example, N = {0, 1, 2, ...} but Z = {..., -2, -1, 0, 1, 2, ...}, and they are not the same because -1∈Z but -1∉N. So if there is more to a set than its mere cardinality, then we're okay.

Answer 3

Perhaps the most important thing about a set S in itself is its cardinality, as the labels of the elements are not important to the set-in-itself (compare to the notion of a thing-in-itself, which is to metaphysics as a set-in-itself is to the philosophy of mathematics). As soon as you entertain the power-set, you are no longer considering the set in itself: although the elements of the set may as well be anonymous (albeit distinguishable) if you are only considering the set in itself, the distinguishability of the elements is significant to the distinguishability of the subsets, in which case the cardinality no longer characterizes the subsets.

If every subset of order k is indistinguishable from every other subet of order k, perhaps because we have adopted a dogma that all sets of order k are "essentially the same", then indeed one cannot obtain a power-set ℘(S) which has a cardinality larger than S for infinite sets S. However, the question one should ask is why all sets of a given cardinality should be regarded as being for all intents and purposes as equivalent. I certainly do not regard a set of two ten dollar bills as being equivalent to a set of two twenty dollar bills, nor to a set consisting of one ten dollar bill and one twenty dollar bill. The reasons for this may be uninteresting to set theorists, but this is only to say that the subject matter of set theory cannot be neatly protected from application to non-set-theoretic problem domains. In this respect, I have a motivation to care, if I have a set of bills consisting of two twenty dollar bills and two ten dollar bills, which subset of two elements I consider. Thus the structure of the set, which may fade away if I consider the elements to be interchangeable tokens due to some detachment I have from the contents of the set, reasserts itself through the significance I attach to the elements as a consequence of the significance of the elements to a wider context.

It is difficult to find a better example than the integers, if one wishes to consider an infinite set. Perhaps the difference in the significance of a googol compared to a googol-plus-one to me is pitiful, but I recognize that there is in principle a difference, and to the extent that I should care at all about the number googol, I should care about the difference between googol and googol-plus-one. Thus I should care about the difference between any two positive integers in principle; and so the proliferation of subsets of the integers of any given cardinality is of interest. If I grant this, it is difficult to avoid the fact of the uncountability of ℘(ℕ).

The fact that we interpret ℘(ℕ) as a set with an extension larger than any possible notation system is an interesting one; one might suppose perhaps not that there are different degrees of infinity, but that there are systems of concepts which exceed our ability to reason non-constructively, by virtue of the fact that we can always construct a subset of the naturals about which any enumeration of proofs says nothing. However, distinguishability of labels is in essence the foundation on which mathematics is established, and to suppose that all sets of objects of a given cardinality are equivalent, despite that their elements may be given distinct labels, is to suppose that we can abolish all of the structure which mathematics is meant to describe. It is perhaps a self-consistent viewpoint, but I would maintain that it is an unproductive one, to say nothing about whether it is a necessary one.