Isn't it absurd to suppose that sets can be empty or can contain other sets?

Question

I'm reading about a logician I'm just starting to learn about, Geldsetzer, and some of the commentary rings a bell. A set should be just a collection of objects. Therefore, it seems absurd to say that a set could be empty, for then the set wouldn't exist. Similarly, that a set can contain other sets also sounds absurd, for when we talk about a set of things, we're really talking about the things in the set, and not the things as well as the set. Even if we were talking about the things as well as the set, then either the set that we're talking about would be empty (and wouldn't exist), or it would contain all the things in the set (in which case, all sets would include themselves and be infinite since a set should the things in it).

Can anyone tell me where I can find out more about this sort of argument, and what is the response to it? I doubt that Geldsetzer is the only one who holds this view (who I'm having a hard time finding more about), and it's interesting in that Russel's Paradox goes away because you can't have sets containing sets.

Also any online references for Geldsetzer's work or commentary would be welcome.

Answer 1

Check your assumptions!

A set should be just a collection of objects.

Says who? Frege, at least, thought that things that fall under a concept build a set. For example, the things on my desk form the set of things that are on my desk. Note that "thing" does not necessarily mean physical things. Hence we can speak about the set of numbers that are divisible by 2.

Therefore, it seems absurd to say that a set could be empty, for then the set wouldn't exist.

This is the wrong way to think. I can speak of the set of 1000 EUR bank notes in your possession, and I can even say true sentences like the following:

The cardinality of the set of 1000 EUR bank notes in your possession is not lower than the cardinality of the set of 1000 EUR bank notes in my possession. The proof goes like this: The latter set is empty (empirical observation), and since no set can have less elements than the empty set (by definition), the former set can have no smaller cardinality. Which is just another way to say that you don't have fewer 1000 EUR bank notes than I have.

How would a true sentence about a relation between two sets be possible if the mentioned sets didn't exist? Note that you can't say

My unicorn is not heavier than yours.

because you can't relate non-existing things. Or, to put it differently, when you can say a true sentence that compares two things, those things must exist.

Similarly, that a set can contain other sets also sounds absurd, for when we talk about a set of things, we're really talking about the things in the set,

This is not so. As in the example above, the sentence speaks about a relation between two sets. It does not say anything about any concrete 1000 EUR bank notes, only about the number of 1000 EUR bank notes that are in your or mine posession. All we could presently say about 1000 EUR bank notes is that they don't exist. And yet, sets of 1000 EUR bank notes can exist, as shown.

Answer 2

This argument is about as sensible as saying that -1 is absurd because a negative apple doesn't and can't exist. Sets are abstractions. As such, "the empty set" as well as infinite sets are perfectly well defined in a wide variety of formalizations of set theory including ZF and Morse-Kelley.

That physical objects can be grouped in correspondence with the ideas of "set" and "number" should not cause one to reject the non-physically-instantiable ideas that go along with sets and numbers. (Unless, of course, you wish to physically instantiate them.)

Answer 3

Let me take up your issue about sets containing other sets. This is not as strange as it seems. To see this let us consider the fact that I have several different sets of shoes. Each pair of shoes may be considered a set, a left shoe and a right shoe. Now if I have a red pair of shoes, a blue pair of shoes and a green pairs of shoes, I may consider the fact that I have a set of pairs of shoes, consisting of {red, blue green}. Now again each pair of shoes is a set,

Red={LeftRed, RightRed},

Green={LeftGreen,RightGreen}, and

Blue={LeftBlue, RightBlue}. This is in contrast to the set

{LeftRed, RightRed,LeftGreen,RightGreen,LeftBlue, RightBlue}.

The distinction here is actually important, since otherwise I might wear a green shoe on my left foot and a red shoe on my right (and I would look like a clown). The point is that sometimes you want to keep track of the fact that certain things are bunched together, while bunching together other bunches.

Another place that you might see this is in linguistics. One may define an adjective as a collection of all nouns that satisfy that additive. For instance the adjective "clownlike" may be defined as a set

{myself, Bozo, etc...}.

One may then form sets of adjectives, which will be a sets of sets.

Now in standard ZF(C) set theory, all sets are either the empty sets or a set whose elements are themselves sets. However their are other theories where one has urelements, so while being able to form sets that contain sets is useful,it is not an absolute necessity. Also see Mozibur Ullah's answer.

Now for your query about the empty set, this is a bit more difficult. The reason one may want an empty set is similar to the reason one may want the number zero. Let us say that I do not have any shoes (just to consider the example from before). Then the set of shoes that I have is empty. Now this may seem to be silly, but it is nice to know that we can form a set without having to figure out how to inhabit it. Moreover knowing that that a set is empty is a useful notion. In the subject of probability theory, if a set is empty, then the probability of selecting an element from that set is zero.

Answer 4

You ask if it's "absurd" for sets to have two features: (a) a capability to be empty and (b) a capability to contain other sets. With that said, I'm not really sure what you mean by absurd, but you could mean two distinct things: logically incoherent (or inconsistent) OR non-intuitive.

If your question is about logical coherence or consistency, the answer is simple: all well-thought-out set theories (like ZFC) are logically coherent. There are many proofs that take care of this. So, as far as we know, there are no logical inconsistencies as far as empty sets or sets of sets are concerned.

If your question is about intuition, then it becomes semantically-loaded -- and NOT a question about logic (but maybe about meta-logic). I would argue that the number zero has about as much intuitive force as an empty set: that is, almost none. Negative numbers have even less intuitive force behind them and let's not even bring up imaginary numbers. I'm not familiar with Geldsetzer, but there is a lot of extra work that needs to be done if one wants to show that empty sets and sets-of-sets are not helpful. One couldn't simply say "well it doesn't make sense!".

If that were the case, Mary could, just the same, claim that "imaginary numbers make no sense" and there would go an entire field of mathematics (namely, complex analysis).

Answer 5

Interesting question, and possible resolution of the Russell Paradox. This is really in the nature of a comment than an answer.

One can argue that an object is never found without a context; following this thought through gets you Type Theory.

One can argue that objects in themselves are unknowable and thus it is the relationship between that is knowable. This gives Category Theory.

One can argue that inconsistency is rather the rule than the exception; and any reasonable mathematics should not 'explode' on being acquainted with it. This gives inconsistent set theory. It also has another solution to the Russell Paradox: The universal set (the set of everything) exists and is fact the union of all the members of the Russell Set.

This suggests, as you're undoubtably aware, that a good formalisation of Geldsetzers ideas on a different kind of Set Theory may give some very interesting results.

Edit

I'm no set theory expert - but the only place I can see in traditional ZFC where sets of sets are introduced is through the power set axiom. There is a paper by Gitman, Hamkins & Johnstone which specifically addresses this - what is the theory ZFC without the power set axiom. Their abstract states:

We show that the theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed-specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered-is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context...Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory ZFC_, obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach.

Further, in the perspective of category theory, you have the category of inhabited sets Set\0; this is obtained from Set by forgetting the structure of initialness. They say:

but the positive word ‘inhabited’ reminds us that this is the simpler notion, which emptiness is defined as the negation of.

Which I suppose is part of Geldsatzers point.

But also there is the category of Pointed Sets where a specific element of any set is distinguished. This can only happen when this category does not contain the empty set. One might think of this as 'compactifying' sets from the purely mathematical sense, where the compacting element is the additional distinguished point.

Finally, traditional Set Theory is built up from sets & membership, and then builds up the idea of a function. Category theory reverses this development - by starting with sets & function and then building up a notion of membership, that is membership is a secondary consideration and not a primary one. Its a structuralist point of view.

Since membership is not primary, a set is ontologically pure - it is indivisible. One cannot ask what members it contains. Similarly one cannot ask what sets, or sets of sets it contains.

For example the set of primary colours is not included within the set of colours, but one can identify some correspondance (by a function). Its a subtle distinction.

It also allows a ramification of ontology: In ZFC, one has only one type - the set; in NFU, New Foundations with ur-elements one has two types - sets & ur-elements (ur-elements are not sets but are contained in sets). In Category Theory there can be many,many more different types.

In Topos Theory, which is a generalisation of Set Theory one actually may have a set with no elements but is not empty. This is because its logic is intuitionistic. So the idea of emptiness ramifies.

Finally, Parmenides said No-Being is Not - which could be interpreted that in actual fact there is no such thing as the emptiness. Everything is always full of something. From this perspective the empty set is a theoretical construct of certain utility in a formal theory but has nothing of the Real about it. In Platos Heaven is there a Form of the Empty that all emptiness in the world is a model of?

Answer 6

A set is a primitive notion, is not defined! Some authors define the concept of set, for example Patrick Suppes in Definition 1. Also you can use the ur-elements ;)

Answer 7

Sets are (useful) abstractions used in mathematics (like numbers).

Outside mathematics is not so easy to find "examples" fitting all the abstract features of sets.

(a) About the empty (or null) set.

You can think of boxes : you can heve an empty box.

But you must think to the mathematical idea of set more as the content of the box than the box itself. So, the empty set is the content of an empty box : all empty boxes have the same content (i.e.nothing at all) : this is why the empty set is unique.

(b) About sets that contain other sets.

Think about army : an army is a set of brigades; a brigade is a set of companies; a company is a set of platoons; a platoon is a set of soldiers (they are no more sets, i.e. they are urelements).

Answer 8

A set is an artificial mental construct, defined to follow certain rules. Like many other mathematical or logical objects, it is related to concepts that have arisen more organically, but cannot be expected to conform to our naive intuitions.

However, it is true that the ability of sets to contain sets can lead to paradoxes. There are many variations on set theory, and some of them solve the problem by distinguishing between sets and classes, which are collections of sets.

You might find these wikipedia articles helpful: http://en.wikipedia.org/wiki/List_of_first-order_theories and http://en.wikipedia.org/wiki/Class_%28set_theory%29

Answer 9

You make a distinction between things and sets, but I believe that the notion of set is precisely supposed to be a model for the notion of thing that is both complete and consistent. It is not that there are sets of things and no sets of sets, but rather that there are no sets of things and only sets of sets, for things are sets.

It is clean and consistent to forget about a distinction between objects and objects that are collections of objects, and rather to consider them as being one and the same: a notion of set. In this perspective we define numbers in terms of sets in several natural ways.

Have a look at the construction of the natural numbers from sets and a set-theoretic construction of integers from the natural numbers.