I’m reading Peter Hallward’s book on Badiou, _A Subject to Truth_. I find the Appendix helpful as an overview of the math stuff. Here’s one thing I don’t understand. Hallward lists the axioms basic to Zermelo- Fraenkel set theory, the first two of which are

“1. Axiom of extensionality: “If two sets have the same elements, then they are identical; in other words, a set’s identity is determined entirely by its elements.

2. Null-set axiom: “There is a set that contains no elements, written Ø.” This is “the only set whose existence is directly asserted. Every other set is constructed in some way or other from this set.”

Fine and good. Here’s where I don’t get it. Hallward adds parenthetically under axiom 2 “Given the axiom of extensionality, the uniqueness of the empty set follows as an obvious theorem.” Why is this obvious?

It seems to be something like this: for any set X, X contains Y elements. If a set N contains the same Y elements as X then N=X. If set X is the set “cats who live in my apartment,” set N is the set “living cats belonging to my wife,” and set M is “cats I have met which I have known to be older than 19 years of age” then in all three cases Y=1 and that 1 is the same cat. Therefore M=N=X where the equal sign signifies identity. Unless I’ve made an error, this seems to be how things are working, and to be an unproblematic use of axiom 1.

This is distinct from something like the following where set X is “cats who live in my apartment,” set N is “my living grandmothers” and set M is “the apartments I live in.” In this case for all three sets Y=1, which means that these are quantitatively the same but M≠N≠X because the single element contained in each is not the same single element: my cat is not my grandmother is not my apartment. The sets in this second case could be said to be equal but are not identical.

Turning to axiom 2, axiom 2 says there is a set with no elements, the empty set or null set. If X is the empty set then X contains Y elements such that Y=0. If set X is “empty set”, set N is “null set”, and set M is “set with no elements” then Y=0. Fair enough. But why is this a case of these sets being identical instead of equal? In all three Y=0, but is that 0 is the same no elements in all three? How is a non-element comparable such that it can held as either the same as or different from another non-element? Put differently, on what basis are there non-elements vs a single non-element? The latter seems to be implied in this case if M=N=X where Y=0 and that 0 is the same. This implies a single zero or single emptiness. Why not many zeroes or many emptinesses (which is to say, many empty sets rather than one empty set)? That (many) seems no less (and no more) coherent to me than one.

Is there a difference here if we keep X as empty set but make N “the set of round squares” and M “my currently living grandfathers”? As far as I can tell, according to the view that there’s one empty set, these would be the same. In all three cases Y=0 and it’s the same zero. But while I admit this is a bit silly, I don’t see why my no currently living grandfathers should be the same as my no round squares. That is, I don’t see why their sameness is any more obvious than their difference. It may end up that neither is coherent, such that 0 is neither the same as nor different from 0.

Maybe that’s the question I guess. In terms not of equality but in terms of identity, does 0=0? Does 0≠0? Neither really makes sense to me (it seems like a bad pair of questions). Hallwards claim that the uniqueness of empty set seems to involve a slip from “=” as equality to “=” as identity.

Of course, I may be totally off base, this is all new to me and I feel clumsy at it. Can anyone sort me out?

This is a really interesting set of questions, Nate. And I will also confess a certain perverse desire for conversations full of logical operators…

You’re right that the question about the multiplicity of empty sets hangs on the difference between equality and identity. Let’s say there’s empty set (a) and empty set (b), such that (a=b). Here, both empty sets would be equal. One might be able to formally distinguish between (a) and (b), such that (a) is the empty set on the left-hand side, and (b) is the empty set on the right-hand side, or something of that order. That would still allow you to state (a=b), but, in that case, the sets would be equal with respect to their members (both are empty), but unequal with respect to their position (left- versus right-).

Because of the inequality of their positions, one would have grounds for saying ~(a=b) in some cases, or in answer to some questions. There would be grounds for distinguishing them and differentiating them, based on their relative positions. Where sets are solely defined by their members, there can be no such principle of distinction or differentiation. Because there is nothing to distinguish or differentiate empty sets on the basis of their members, every empty set would be indiscernible, and axiom 1 implies that things which are indiscernible are identical.

I would be inclined to agree with you that the living grandfathers of people with no living grandfathers and round squares are not identical contents. But I’m inclined to say that because there are cases where (a)=f and (b)=f but ~(a=b). Here, there are still grounds for distinguishing (a) and (b). The fact that the sole criterion for the identity of the empty set is its membership takes away that principle.

I may well have no idea what I’m talking about, and might not have responded to the question you’re getting at. But, maybe it’ll be a start.

Colin–

I think it’s precisely the axiom of extensionality that allows Hallward to, as you put it, slip from equality to identity; the axiom basically asserts that equality and identity are the same thing. It’s an axiom, i.e., a decision – set theorists have decided, for whatever reason, to treat non-existing grandmothers and non-existing grandfathers as the same. They could have decided differently, and indeed there are intensional logics that do make that kind of distinction.

Hmm, just looking at your post again, I think I’ve misunderstood how you were using the term “equal”; I hadn’t realized you meant it in the sense of “having the same number of members.” But I don’t think the number of members is important here; “having the same members” is the important relation.

Maybe you’re concentrating on the number of members because the empty set is the set with 0 as the number of it’s members? But that’s not quite how the empty set is defined – rather, Axiom 2 is (formally): (∃x)¬(∃y)(y∈x) , i.e., there is a set such there are no things which are a member of that set. It’s just a coincidence that the number of elements in the empty set is 0 (well, it’s not really a coincidence: rather, 0 is defined as the number of elements in the empty set). So the question isn’t, how do we know that all these 0s (the number of elements in each empty set) are the same; rather it is, how do we know that one set with no members is the same as another set with no members. And the answer is, how would we tell them apart (clearly, there are no objects that are in one empty set, but not in another)?

hi Colin, V,

Nice to hear from you. I think the question I’ve got here is one of the application of the axiom of extensionality.

Let’s take the following:

1.

X=cats who live in my apartment

N=living cats belonging to my wife

M=living cats that I know to be over 19 years old

In each case there’s a quantity and a unit. This is clumsy but let’s say –

X=yz, N=op, M=kl. y=1, o=1, k=1 therefore y=o=k.

z, p, and l, all =an actually existing cat which is the same cat therefore z=p=l. So, X=N=M. The axiom of extensionality applies unproblematically, because the quantity and the units are the same.

Now,

2.

X=times I have fired a gun

N=round squares

M=my currently living grandfathers

Again in each case there’s a quantity and a unit.

X=yz, N=op, M=kl.

y=0, o=0, k=0 therefore y=o=k. Thus far, no problem to the application of the axiom of extensionality here.

But what about z, p, and l in this case, the “units”? Hallward’s saying “the uniqueness of the empty set is obvious” seems to suggest that the three are the same. This would seem to imply that my firing a gun, round squares, and my grandfathers are the same. To my mind this is cleary not the case with quantities greater than 0. They could be said to be all the same of course, depending on the framework – say if one is looking for any object whatsoever which can be plugged into a variable then they are all the same, irrespective of quantity – but Hallward’s point isn’t irrespective of quantity, rather it’s specific to the quantity 0.

The reason I ran through this again is because I think it’s quantity that Hallward’s point turns around. X, N, and M are different sets if they involve the same quantity of different object provided said quantity is > 0. But, for Hallward, they are the same set if they involve the same quantity of different objects if the quantity is = 0. I don’t see why that should be. Given “1 round square”, “1 time firing a gun” and “1 living grandfather” are not the same from at least one point of view, I don’t see why “0 round square”, “0 times firing a gun” and “0 living grandfathers” should not similarly be not the same from at least one point of view.

V’s question, “how would we tell them apart” when there are 0 of each is fair. But in that case, any two objects or units which can not be told apart can count as the same for the purposes of the axiom of extensionality. That would mean that the axiom isn’t necessarily one of elements being the same but rather of elements being judged as the same, treated as is the same. Also, while “we can’t tell them apart” may authorize treating-as-same. The principle here seems to be “same until proven differentiated.” That’s reasonable. But it doesn’t seem to me that treating-as-same is the only treatment one can do in response to “we can’t tell them apart.” One could just as well treat objects which can’t be told apart as different, a principle of “differentiated until proven same.”

I’m not particularly troubled by the treatment of empty sets as all being the same set, but I don’t find it obvious nor do I understand why this is being done. I would find it no less (and no more) obvious to say that there as many empty sets as there objects about which we can say there are none of those objects. (Just are there are as many sets containing one element as there are objects about which we can say there is one of those objects.)

take care,

Nate

Hey Nate–

I think the question here is more about quality than quantity. An empty set isn’t empty because it has a certain number of members, bur rather because it has the quality of being member-less.

The non-equivalence of sets has to follow from its members, so if I have a set of round squares or a set of present kings of France, both of those sets have a defined membership. It may be that the only member of each set is a square we say is also round, or a man we call the present king of France. But we could also call every man the present king of France, and include him in the set of present kings of France. Likewise, we might suppose an infinite number of rounded squares, and include them in that set. Whether the quantity of members is one or infinitely large makes no difference, because we’ve still defined a particular membership for that set.

It may be that a round square is impossible to construct or that the present king of France is just a name we call somebody, but I think that’s enough for set-membership, and for establishing that a set isn’t empty. A set that lacks even that kind of membership would have to be totally empty, such that there would be no possible inclusion, and consequently no way of distinguishing them. Sets of round squares and present kings of France is distinguishable, precisely because those sets have members. Take away even that membership, and they’re identical, because there’s simply no membership.

Again, no idea if this makes any sense.

Colin–

hi Colin,

That’s interesting and I hadn’t thought of it. Do I understand correctly that you’re saying the set of my living grandfathers and the set of times I’ve fired a gun etc do have members? The same for the set of round squares and present kings of france? If that’s what you’re saying then I think that would mean that there would be members of these sets but members with 0 quantity. (As opposed to the set of cats in my apartments, which has members with a quantity of 1.) Is that right? That’s interesting and a bit odd I think. You’re saying a set containing quantity 0 of some member/unit (round square etc) is not the same as a set with no members. So, 0≠Ø, right? That’s interesting. What I’d been arguing was basically that 0x≠0y if x and y are different objects. I’d just been presuming that 0=Ø (such that one could have Øx and Øy where x and y are different objects). It seems like here with sets containing 0 members of some type there’s something about the “members of some type” being specified – I’m saying something different when I talk about 0 times firing a gun vs 0 living grandfathers etc – whereas with Ø there’s no specification of any particular members of which there is 0 quantity. Hmm. I’ll have to think more about this.

take care,

Nate

Hey Nate–

I think I was actually confused in my last post. I wanted to make two points, I think…

First, that it isn’t clear to me why there couldn’t be a set of round squares or present kings of france or the living grandparents of someone without living grandparents or number of times a person who’s never fired fired a gun has fired a gun. At a formal level, I can imagine writing something like Ex(Sx & Rx), which I think says there is something (x) which is both square (S) and round (R). The contradiction of circularity and squareness wouldn’t necessarily enter into the constitution of the member of the set. There could be a set whose members were all contradictions, for examples. If that’s possible, then I imagine one could also say Ey(Sy & Ry), and so on, to infinitey. In that case, every (x) or (y) could be a member of a set of round squares. Someone else should slap me around and tell me why this is wrong.

Second, I think the point of the empty-set identity is that every other set, no matter how many members it has, is equivalent (equal) in a certain sense, simply by virtue of having members. We can say set (a) is equal to set (b) because both (a) and (b) have members. That doesn’t imply that they’re identical, because there could very well be reasons to say ~(a=b). If sets are defined by their members, and (a) and (b) have different members, or different numbers of members, then they’re different sets. But if it’s true that a set is defined by it’s members, then there’s nothing which would serve to distinguish empty sets. They’d be equal, indiscernible, and therefore identical, in the same way that the products of 5×0 and 100,000,000×0 are both equal and identical. They’re both 0, there’s no way of discriminating between 0’s, and so there’s no way of distinguishing one product from another. They’re effectively identical, because indistinguishable-indiscernible.

I think thinking in terms of quantities of units is confusing. Why not just say, directly, that X1, N1 and M1 all contain just this particular cat, and therefore have the same members, and that X2, N2 and M2 all contain nothing, and therefore, likewise, have the same members? Describing things in terms of quantities of units gives the impression that “no grandfathers” is, as it were, a peculiar type of grandfather, and therefore different from “no round squares.”

And it’s not really a case of “the same until proven different,” because we can prove that two empty sets are identical – we take all the members of each set, and we can see that all of them have an indentical counterpart in the other set. The fact that “all x are y” is taken to be true if there are no x is a more-or-less chance feature of modern logic (if memory serves, Aristotle didn’t think that was the case); but it turns out to be quite convenient.

hi Colin,

Interesting. I’m not sure I get something about your first point. When you say “every (x) or (y) could be a member of a set of round squares” do you mean that the set of round squares contains everything?

On the second point… I get this: every set with members is equal, in the sense of equivalent, qua set-with-members. I get also that every set with no members is equivalent, qua set-with-no-members. I think where I’m unclear is on what counts as being a member of a set or counts as having a member. A set with 1 cat (like the set of cats who live in my apartments) is a set with members. What about a set with 0 grandfathers (like the set of my living grandfathers)? Is that a set with members? Or it a set of no members? If the latter, then, according to Hallward it’s not _a_ set of no members, it’s _the_ set of no members, because the empty set is unique.

I’m still thinking this out…

Let’s take the following –

B = the set of bachelors who are married.

P = the present kings of france.

R = round squares.

G = my living grandfathers.

T = the times I’ve fired a gun.

B and R are sets of contradictions (subsets of C, the set of contradictions).

P, G, and T are sets of things don’t exist but which it not be contradictory if they did exist (subsets of N, things which don’t exist but which are not logical contradictions).

How many elements do these sets have?

P contains 0 present kings of france, G contains 0 living grandfathers of mine, and T contains 0 times I have fired a gun. It’s not clear to me that B and R have contain 0 bachelors who are married and 0 round squares respectively. If we stipulate that B and R are the sets of _existing_ bachelors who are married and _existing_ round squares, respectively, then clearly they contain 0 bachelors who are married and 0 round squares. Let’s say we do that for now. In that case, all of these sets – B,P,R,G,T – have 0 members. Are they sets with no members? Or are they sets with members, members of 0 quantity? In either case, clearly they’re equal. In the former case, they’re the same. In the latter case they’re not the same, only equal.

take care,

Nate

hey Voyou,

We were writing comments at the same time. I think it’s fair to say that adding terms like “my living grandfathers” etc may not be a productive way to get at the empty set. But that doesn’t get over this confusion I’m having. I think the following hypothetical situation that demonstrates the confusion. This hypothetical is odd but I think it’s reasonable, and I think it’s similar to some things that do occasionally happen in the world.

Say there’s a bar wherein two groups of people are discussing some topic. The first group is discussing having fired a gun at a shooting range. The second group is discussing their living grandfathers. One person in each group hasn’t yet contributed to the discussion, we’ll call them the apparently shy person. Coincidentally at exactly 7:30pm in each group, one person (we’ll call them the facilitator) turns to the apparently shy person, and asks them a question in the attempt to include them in the conversation. In group 1, the facilitator asks “and how many time have you fired a gun?” and the apparently shy person answers, “Actually, I haven’t fired a gun.” In the second group, the facilitator asks “and what about your living grandfathers?” and the apparently shy person answers, “Actually I don’t have any living grandfathers.”

One paraphrase of shy person #1’s answer is “I have been skiing 0 times.” We could say that answer names the set of times that person has fired a gun, which has a membership of 0 times having fired a gun. We can similarly paraphrase shy person #2’s answer as naming the number of living grandfathers that person has, which has a membership of 0 living grandfathers. (You might say each set has no members.)

Is the set indicated by each shy person a different set? If each set indicated names a different set, are those sets empty? If so, then there is more than one empty rather than one such that there is not one empty set contra Hallward.

If each set indicated by each is empty and there is only one empty set, then each set indicated is a synonym for the empty set. That means each is also a synonym for the other. And each is also a synonym for every other synonym for the empty set (such as the set of round squares). That seems weird to me, and here’s why, returning to the hypothetical:

Let’s say a member of each group (let’s call this one the well known person in each group) looks up after the exchange between facilitators and shy persons (say, at 7:31pm) and looks around the bar. Each well known person realizes they know the well known person in the other group. They greet each other. Well known person two says “what were you people just talking about?” Well known person one says “we were just talking about how shy person one has never fired a gun.” Well know person two says “we were just talking about how shy person two has no living grandfathers.”

Here’s where I’m confused and I think things are weird re: the uniqueness of the empty set —

Next comes what I will call the synonymy utterances:

Well known person #1 says “what a coincidence, both of our groups were talking about the same thing!” Facilitator #1 adds “we were also talking about round squares!” to which facilitator #2 adds “so were we!”

If the sets named by each shy person are synonyms for the empty set, and thus synonyms for each other, then there’s no mistake here in the synonymy utterances, the synonymy utterances are true. Each group was in fact talking about the same thing for a moment – the empty set – which means they were also talking about every synonym for the empty set.

If the sets named by each shy person is not the same, then the synonymy utterances are a mistake and are false, because they render things which are not synonyms into being synonyms.

My intuition is that the synonym utterances would be mistaken. In the respective context of each group the fact that shy person #1 is talking about the set of their 0 times having fired a gun is an important difference from the fact that shy person #2 is talking about the set of their 0 living grandfathers, such that they are not the same as each other and neither is the same as the set of round squares.

I’m not invested in there being one empty set vs more than one. I just don’t get it. I just mentioned this to a friend, and in our brief discussion here’s what I thought of. Is having-quantity a quality? If so, it seems strange to me to say that nothingness has a quantity, because that means nothingness has a quality – the quality of having quantity. It seems that to say “there is one empty set” is to say there is ascribe a quantity to nothingness – the quantity of one – which is to ascribe a quality to it, the quality of having a quantity.

There are three basic questions for me here. Is Ø a quantity (and/or, does it have a quantity)? Is 0 a quantity? Or not? And, is 0 the same as Ø?

Leaving aside quantity, it also seems to me that “sameness” is a quality just as is “difference”. (A quality, or a relation.) So, saying “0 is the same as 0” or “Ø is the same as Ø” or “the empty set is the same as the empty set” or “this empty set is the same as that empty set” – that is, the application of the axiom of extensionality to the empty set as Hallward does when he asserts the uniqueness of the empty set – gives nothingness a quality (and/or a relation). But nothingness has no qualities (and/or no relations).

I may be making a mistake here, conflating “nothingness”, “no members”, and “empty set(s)”, but I don’t know how to think of this otherwise.

take care,

Nate

If the issue is one of synonymy, though, I don’t think it just applies to the empty set. Say a group of people are talking about the most evil person who ever lived, and another group are talking about the first female prime minister of the UK. Now, maybe in each group there is a disagreement – the first group is debating whether or not George Bush is the most evil person, and in the second group there is one person who insists, all evidence to the contrary, that Winston Churchill was secretly a woman.

Then, someone comes in and sets both groups straight – they’re both talking about Margret Thatcher. But that doesn’t mean they were talking about “the same thing,” just that the object that answered to the description they were debating happens be the same in both cases. The empty set case doesn’t strike me as being any wierder. I don’t know if this is a useful example, but imagine some property that identifies Margret Thatcher, but completely by chance – being the however-many-millionth person born in the UK, or being made up of exactly such-and-such a number of molecules, or something. Completely coincidentally, there could be a third group in the bar discussing this description; this seems to be analogous to the case of the empty set, where it’s completely a matter of coincidence that the object that answers to the two descriptions is the same.

It’s maybe significant that your example involves what people are “talking about.” “Talking about” seems like a context where, not just the object referred to, but also the description by which its referred to, seems important. And, as I said in my earlier comment, it’s a matter of decision in set theory to only consider identity in terms of objects, not in terms of the properties with which we describe them. This is a (self-imposed) limitation of set theory.

As for whether the empty set is nothingness: my inclination is to think that the empty set is a thing (in particular, a set), and so is something different from nothingness. You can, for example, have a set containing the empty set (which is different from the empty set itself), which seems like it would be odd if the empty set were nothingness.

hey there V,

Thanks for that. I think you’re right that the synonymy issue isn’t unique to the empty set. In some ways this is just replaying Frege on the morning star and evening star both being Venus. In the conversations unknowingly about Thatcher in your hypothetical it would indeed be weird for people to know that she was the Xth person born in the UK or the person with Y quantity of molecules. If that were known, though, it wouldn’t be a mistake for someone to say “The Xth person born in the UK? You mean Thatcher?” anymore than it would be weird for someone to say “The Prime Minister in the 80s? You mean Thatcher?” Someone else could then say “She’s also the person with Y quantity molecules in her body” and for people to use those terms as synonyms. In that case nothing would be lost in using the terms synonymously provided everyone knew the terms were synonyms. It seems to me that it would be a mistake, though, for me to say “The number of my currently living grandfathers? you mean the number of times I’ve fired a gun? you mean the empty set?” Those are quantitatively equivalent (and I’ve become more confused about the empty set and quantity actually), but it’s not clear to me that they’re identical, just as it’s no longer clear to me anymore whether the set “my living grandfathers” (or the times I’ve fired a gun etc) is the empty set or a set with members the quantity of which happens to be zero.

I understand the point about set theory focusing on objects themselves sans contexts, properties, and descriptions, but I’m not sure what’s left of “object” if we bracket those, beyond a whatever-object which is simply a variable. I actually like the abstraction here, but I’m not sure the discussion of sameness makes sense without contexts, properties, and descriptions.

This reminds me of another thing I found odd in the Hallward. He says at one point that part of the set theory stuff is to get away from the world as foundation of mathematics, to make math self-founding or resting on self-founding axioms that don’t rely on reference to the world to make sense. But the “same as” as involved in the axiom of extensionality doesn’t seem to have that quality – at least when talking about sets with members other than abstractions, logical constructs, etc. This isn’t an objection to the axiom of extensionality, just a question about the range it applies to unproblematically – there are normal cases where the axiom applies unproblematically. Hallward says the empty set is one such case, which I still don’t get.

That’s interesting about the empty set being a thing while nothing(ness) isn’t. I hadn’t thought of that. That makes sense. The set of no members is a thing. That doesn’t say the content must be a thing (loosely anologous to the set of televisions being a set and not a television while its members are televisions and not sets). Are the empty set’s no-members a thing, or nothing? That is, is the content of the empty set different from nothing(ness)?

cheers,

Nate

Hey Nate–

I’m probably way out of my depth…

But…

In the first case, I was wondering why exactly sets whose members would be something whose truth-value was false, or which contained a manifest contradiction would be empty. Formally, a contradiction can still be a member of a set. A round square, for instance, could very easily belong to a set of things which are impossible to construct. In that case, round square (x), round square (y), and an infinite number of other round squares would be impossible to construct, and would be members of the set of things which are impossible to construct.

Similarly, we can just stipulate that there is something, some (x) or (y) or (nth) thing, that is both square and round. In that case, it could be a member of the set of round squares, and the set of round squares would not be empty. That’s not to say that everything is a round square, but rather that there is an infinite number of things which could be defined as both square and round. A square which is not round would not belong to the set of round squares, and an infinite number of squares which are not round would not eliminate the possibility of an infinite number of round squares.

With my second point, I think I was trying to say that as sets, B, P, R, G, and T are identical because they have no members. Let’s say that B, P, R, G, and T are all different numbers…

B=the square root of 2.

P=2

R=4

G=6

T=8

Now multiply every number by 0. The product is both equal and identical. In every case, it’s 0. Do you want to say that one 0 is different from another because it’s the product of multiplying the square root of 2 and 0, rather than 2 and 0? I don’t think so. I don’t know how that would make sense. It makes a lot more sense to say there’s nothing which could possibly distinguish them, and that they’re therefore identical.

The uniqueness of the empty set seems to me to say that there’s no other number that you could multiply every other number by, and get the same (equal and identical) product. That only occurs when you multiply a number by 0. No other number has that quality, so nullity is unique. The set which contain no members are unique, because if you take away all of the members of any set, you’re left with exactly the same thing: A set without any members, without anything to differentiate it from anything else.

Also, when I was taught set-theory, we were told that there was no such thing as an empty set. I’m not sure exactly what version of set theory the textbook (Nolte, Logics) used, but by definition a set had to have members. Every set had to have at least one member.

The affirmation of the empty set seems to me to be doing exactly what you (Nate) say it does: giving nothingness a quality, saying something which is “nothing” “is,” in a certain way. Nothingness is a set which is not a set, because it’s a set with no members.

Colin–

hi Colin,

I get it now re: the first thing. That’s interesting. I’ll accept that set B, P, and R (married bachelors, present kings of france, and round squares) are not empty. I do think that sets B’, P’, and R’, where the ‘ adds “actually existing” to each set would be empty. That’s what I had in mind initially when I first raised sets whose members are logical contradictions as empty sets.

Re: the second point, I’m still not clear. You’re right that zero is a unique number in that X times 0 is always 0 for any X. But I’m not sure that means “unique” in the sense of “all zeros are one” [insert joke here about binary, or better yet, pomo anti-binary-distinction talk]. I think means unique in the sense of being distinctive or special. I think I contest your example where you give numbers for sets B-T. They have numbers, quantity, but the sets are not _only_ quantities. They’re quantities of different members. Each set could = 1 in quantity but would not be the same set (unless considered only quantitatively). It’s not clear to me why they become the same set if each set = 0 in quantity. To be clear, I have an intuition that this is so, that’s my first impulse actually, but if someone else had a different intuition (akin to the positions I’ve been trying to sketch in this discussion) I don’t know that I have a solid argument as to why their intuition is wrong. I think in part I just think there are contexts where 0 of one thing is not the same as 0 of another thing, such that there might be reasons to make a distinction between two sets with the same number (zero) of members of two different types. I’m also still not clear if a set is an empty set if it contains members of some type X but with a quantity 0.

Perhaps it’s a rule that a set can only be said to “contain members” if that set has some quantity greater than 0 of those members. In that case, my hypothetical bar conversation in the comment to Voyou is mistaken as is my positing sets G and T (my living grandfathers and times I’ve fired a gun), because there is no such set possible. (I guess I could accept that, though it seems a bit odd given possible – and I think common – speech situations where people say things like “I have no X.”) But if that’s so, then what’s the empty set and how is it a set, since it has no members? If member quantity must be > 0 then for the empty set to exist would seem to require a substantialization of 0 or nothingness, such that it has a quantity, as we’ve talked about a bit. Hmm. Maybe I’m just further demonstrating my own confusion.

Two more things I just thought of, a little tangential perhaps. How many members would there be in the set of round squares? Is that identifiable? Is it quantifiable – such that there could be said to be some quantity – or is it no quantity (but not 0, rather, a non-quantity, neither one nor multiple). Ditto for any other logical contradiction.

Second, somewhat related to the first, you and Voyou have both mentioned undifferentiability, effectively “how would we tell them apart?” I find that fairly convincing. But, despite certain anti-realist intuitions, I still want to say “we can’t tell X from Y” is not the same as “X and Y are the same thing”, at least in some contexts. So, on that, could we have a set of items which are indistinguishable from each other? If so, how many members does it have? One? More than one? Some indeterminate quantity? (I think this relates to “how many members would there be in the set of round squares” question.)

take care,

Nate

hey guys,

I think you’re complicating things:

Nate is totally right in the post when he says that equivalence does not equal identity. In the basic case that Hallward is refering to, he is refering to identity and not directly equality.

Here is why the ax. of null set and the ax. of extenstionality means that there is one and only one null set(uniqueness):

Ask yourself a question, is there another identity avaliable for the null set. No, that is why nate, your example of cats living in your apartment and the cats living in your grandma’s apartment both “equal” to one, but is not unique. One is not unique, there are many sets which count to one. Only the null set is unique because there is no other set-theoretical identity that can express the lack of belonging that the null-set “captures.”

So, to be clear, “king of france” “number of MxPx albums I own” “numbers of cakes baked today” all provide the same identity. Since there are NO, count it, 0, referents to any of these predicates, they share an identity.

by the way… Hallward’s appendix is almost entirely gotten from Mary Tile’s book “Philosophy of Set theory” she is extremely clear (though there are parts i don’t necessary agree with) and very nice read.

Simple.

hi Tzuchien,

Nice to hear from you. When I first started on this I thought “Tzuchien could probly straighten me out here, maybe I’m making things too complicated.” I’ve got that Tiles book out the library (on your recommendation actually) but I haven’t read it yet. I expect that will clear up a lot for me. Question: Is the set of my living grandfathers and the set of your MxPx albums the same as the null set?

I think in general the disconnect I’m having here is the move from “indistinguishable”/”undifferentiable” to “identical.” Maybe I have an intuitive want for a substantive postive identity which is naive or misplaced. I think this relates to my confusions in the discussion with Colin — here:

http://whatinthehell.blogsome.com/2007/03/10/is-the-big-deal-about-sufficient-reason/ —

over the equivalence of A and ~(~A).

As I said, I’m confused. Let me start again, trying to be clearer.

1. There is a set with no members, the null set. So far so good.

2. There is one and only one set with no members. This is what I don’t get.

3. There is a set with exactly one member. There are many sets like this, as many as there are objects we can specify one of (like “the set of that guy there”).

And, there is a set which is the set of sets with exactly one member. This has more than one member.

4. Say there is a set of sets with no members. This set, according to #2, has a membership of one. This means this set is a member of the set in #3, the set of sets with exactly one member.

5. The axiom of extensionality states that two sets having the same elements are identical. That makes sense to me. Let’s take three sets named in my post, X, N, and M, where X is the set “cats who live in my apartment,” N is the set “living cats belonging to my wife,” and M is “cats I have met which I have known to be older than 19 years of age.” Each of these sets has exactly one member, and that member is the same – the cat you and Colin have met. According to the axiom of extensionality, these sets are the same. No problem so far. Let’s say we’re naming and counting some members of the set named in point #3, the set of sets with one member (let’s call it “*1”). Sets X, N, and M, are all members of *1. Are they three members thereof, or one member?

take care,

Nate

The advantage some see in set theory is that we can move away from predicates and thus away from universal.

So, the man named “nate” and the dashing young man in the corner with the beer can both refer to the same object, but has nothing to do with set belonging.

A set is nothing but its members, that is what the ax. of extensionality says, effectively, a set’s extension are what, explicitly and precisely speaking, make it it. A set with my cat, your cat and your neighbor’s dog is nothing but that, my cat, your cat and your neighbor’s dog.

So, there are if i may be so bold, an infinite number of sets which count to one. By conseqence, all the PREDICATES which refer to no objects “pick-out” and form the null set since there is nothing in them. Now, we might “mean” differenting things when we are attemping to pick them out, but in the end we get the same thing: nothing.

To answer your question specifically, If sets X,N and M are members of the set *1, then they are exactly three members of *1 and there are the only three. But most sets are not like this, the interesting ones anyway.

THink about using predicates as a “loose” way of speaking, that connects normal language with set theory. The theory of reference helps us by thinking that a predicate “MxPx albums that Tzuchien owns” picks out items in the world referenced by it. Like a library search, it can land you with nothing, search “secret sex life autobiography of George WH Bush” – there are NO object that corresponds to this search.

So, you have picked out nothing. The question is whether this “nothing” differs from all the other “nothings” in the world. The very fundamental point of set theory will be to try to fix principles that can say “no.” There is nothing in nothing that allows us to distinguish these different “0” results.

Hope that’s clear enough. Remember one of the central intentions of set theory was its clarity that there is nothing that organizes the sets other than the members of the sets themselves, this means that the member can be anything and the sets, thereby, also as diverse. This necessitates the finding of basic laws to determine singular and unique points.

Tzuchien

hi Tzuchien,

That’s (or rather, I’m) somewhat clearer now, thanks. I want to make sure I understand something which I think is pretty basic, I hope it doesn’t seem like I’m being purposefully obtuse.

Take three of the sets I made up above, sets X, N, and M. Sets X, N, and M each have exactly one member, item x, n, and m respectively. Items x, n, and m are the same item, therefore sets X, N, and M are the same. That’s right so far, right? Here’s (one thing) what I’m not clear on. Given the above in this paragraph, how many sets does the list “X, N, M” contain/name? Three? Or one?

take care,

Nate

ok, i see the problem. THe “set” X contains x… So, by the axiom of extensionality, it’s identity is wholly constituted by its member x. As such, the set “N” is also constituted by n, and M by n.

It is extremely simply, if x=n=m, then the sets X=M=N. They are THREE sets that are identical, by the axiom of extensionality. simple.

Thanks for clarifying. Let’s make this make this about cats, the three sets I named all of which include my cat, where x, n, and m are all my one and only cat. Since x=n=m then X=N=M, where “=” means “is the same as” or “is identical to.” X, N, and M = each other but they are still three sets. Their being the same as or identical to each other does not make them one set. I get that now, thanks again for clarifying.

Let’s change the example now, though. Let’s make x= times I’ve fired a gun. I have never fired a gun. So X has a membership of 0 times I’ve fired a gun. It has no other members either. In this case, X is an empty set, a set with no members. Right? Or am I making a mistake here by saying the set “times I have fired a gun” is an empty set?

If I’m not making a mistake and X is an empty set, then let’s continue revising the example. Let n = number of MxPx CDs you own, and m= my living grandfathers. In this case, sets X, N, and M each have a membership of 0. In one sense then, X=N=M because each has 0 members. If so, in this case, is the list “X, N, M” a list of three empty sets or one empty set?

take care,

Nate

ok, in this case, we would have three sets that are identical with the null set. Uniqueness of the null set means that all those sets which contain nothing are identical, meaning that they all constituted by the same element (as a facon de parler): nothing.

The uniqueness of the null set means that there is no other identity avaliable for the null set.

does this clear things up at all?, what is the problem, really, i have this feeling that there is somewhere a real problem… would like to hear it of course.

Tzu.

Actually, you’ve just now cleared it all up bigstyle.

The problem was I misunderstood “the empty set is unique” to mean “there is one and only one empty set”, which would mean there could not be more than one empty set. Rather, “the empty set is unique” means that every empty set is identical, but does not mean that there is only one empty set. Just like there are many sets which have my cat as their sole member, there are many sets that have no members. That was my intuition (though I didn’t manage to formulate it previously) which my misinterpretation of Hallward conflicted with. Much clearer now. Thanks! I should be able to proceed further without this hang up (when I get the time). Also clearer now is the meaning of “identical” – the stuff on “unique” had me confused about how identical sets should be counted, as one or as however many sets one is holding as identical. Now I get that it’s the latter.

take care,

Nate

One other thing… new question, on this:

“The uniqueness of the null set means that there is no other identity avaliable for the null set.”

Can you unpack “other identity available”, a case where something has more than one? I’m not sure I understand the point.

ok, yea, let me just add that for set theory, you need to seperate what can be loosely called “reference” and “sense” in Frege’s exposition. Extention is, by the axiom, meant to secure the basis for what a “set” “is.” So, the identity of a set is what it “is” … your cat, the number zero, the set of blue crayons in the world. etc. So uniqueness means that there is only one “identity” of the set… indeed, the set is nothing but that identity, you can represented it in various ways but it remains what it is -its composition.

You can call it anything you like. As a formal language, set theory calls a duck a duck… if you will, the whole calculus is devised around the composition of sets and not the name… this is one of the thing that distinguishes set theory from other formal languages.

so in this way, there are many empty sets all which are nothing but the empty set… because in set theory we disgard the notion of “sense”, we end up really with one set for the empty set, it is unique. We are not so troubled by the multiple ways of calling it such.

The trouble i see is that you rely too much on predicates to intrepret what is going on in set theory… no doubt it uses predicates at times, but when the identity of the set is concerned, it really doesn’t matter what we call it. In this sense, we get the colloquial sense of “uniqueness” back -the null set only has one identity.

Now, when numbers are concerned, are there multiple ways to get at the concept “3”…. of course there are. In different number systems, you will get different representations of “3” with different members and thus different extensions. Not all “3”s are alike… this posed a big problem. I would cf. Benacerraf’s famous piece on “What numbers cannot be” where he argues that numbers cannot be set theoretical. Its a good read and highlights exactly the multiplicity of ways to get at number structure.

So, to be clear, I think your confusion comes from thinking about “referential sets” as sets. Sets can be thought independently from predicates, “number of times i’ve fired a gun” and so forth. While we get many predicates that refer to the same thing, and thus, different designations or names for the set “null”, in the end, from the perspective of set theory, they are the same thing because we are not so much concerned with the “sense” of the thing.

clearer?

Tzu.

That helps, thanks again. I think this stuff is going to require some work from me in a sort of moral sense, fighting to bracket or suspend some of my intuitions and prejudices (which is hard because, as you know because I’ve told you, I’m always right). Suspending one’s intuitions and prejudices is always healthy though, so I welcome the challenge.

One more question – is my cat also unique in the colloquial sense in which the empty set unique? If so then I don’t have an objection at this point. What I was objecting to before in an unclear (to myself) way was what I took to be an implication that “these three sets with membership ‘the same cat’ are three sets and those three sets with membership ‘no members’ are one set.” That didn’t make any sense to me at all.

gotta run

take care,

Nate

to consider your cat might take a few more axioms, perhaps we should add the axiom of foundation.

The cat is made up for organs, parts, molecules, atoms, etc. So there are many ways to sum up the identity of your cat. So that collection of atoms there ={x,y,z….} can be your cat just as well as the summation of its parts ={a,b,c….}. There are many distinct elements that can consitute the identity of your cat. Here, I think you see that when we speak of material objects, since there is no smallest unit to compose them, there will be an infinite collection of objects that we can speak of. Now, referentially then, there are many different sets that can comprise your object=the cat.

The uniqueness of the null set means that there is only one identity that can comprise that idenity, so it is unique.

ok, i think that might be clear enough.

That makes sense. Is this right:

My cat is divisible, has subsets so to speak, members or elements. Any member or element itself has members or elements, subsets. The no members of the empty set is/are not divisble and has/have no subsets or members. Is that it?

this is probably a month and a half too late, but i thought i would add something else for your enjoyment.

Finally, we can say that there are also many different interpretation of the nature of set theory.

When speaking about cats and other objects with identities independent of their belonging to the set -this is called naive set theory.

It has its own proper field, and naive is not meant to be pejorative here, but indicate simply its limits.

Proper ZF set theory, has neither definitions for the “object” set, nor any identity of the objects “within” the set outside of their pure logical relations. So while we can say that this set “a” contain powerset with the cardinality 2^n, where n=number of elements, we are cannot thereby say that it is the number of cats in my neighbors house.

we might be mixing up the two “kinds” of discussing sets here.

IT is of course easier to introduce sets with naive set theory, its more immediate and intuitive, but it has its limitations, like the difficulty of seperating the identity of “elements” from what they are “actually” (cats, oranges…etc.)

xoxo

tzuchien