Gottlob Frege published a work called Grundgesetze der Arithmetik (Basic Laws of Arithmetic) which attempted to reduce arithmetic to logic. Unfortunately for him, it was not well-received. He intended it to be a multi-volume work, but Volume I was so badly reviewed that he had to publish Volume 2 at his own expense. There was at least one interested reader though: Bertrand Russell. While Volume 2 of Frege’s work was still in proof, Russell sent him a letter about one of the axioms contained in Volume I.
Basic Law 5 of the Grundgesetze allows for the creation of sets merely by describing the properties of their members. So Russell invited Frege to create the set of all sets that are not members of themselves, and then asked him whether this very set is a member of itself. If it is, then it must have the characteristic of not being a member of itself; but on the other hand, if it is not a member of itself, then it must be included in the set. Attempting to answer either way leads to a self-contradiction. Hence, the paradox.
Frege said that the letter left him ‘thunderstruck’ and later remarked that it had destroyed his entire life’s work. He hastily attempted to amend his system to account for it, and noted it in an appendix in Volume 2, but this solution proved to be unsatisfactory, even to him.[1]
Russell also attempted to find a solution, and actually took up Frege’s task (along with Alfred North Whitehead) of trying to reduce mathematics to logic, with this solution (known as the theory of types) playing a prominent role in their system. I feel that they were somewhat on the right track, but not entirely. Several other proposed solutions to the paradox (as well as modifications to set theory in order to avoid it) have been given, but none of them address the real heart of the problem.
Before giving my answer, though, let’s first make sure that we fully understand the paradox. The way that it is worded makes it somewhat difficult to follow, so to make it a little easier, let us begin by separating how we refer to the sets. Let’s call the set of all sets that do not contain themselves as members ‘N’:
The question is whether N itself is a member of the set. Let’s first assume that it is, and see where that takes us. According to how the set is defined, N could only be included if it is not a member of itself. So, if it is a member of itself, then it has to be one of the sets that is not a member of itself, an obvious self-contradiction. That cannot be correct, so let’s instead try starting from the assumption that it is not a member of the set. Well, if it is not a member of itself, then it would have to be a member of N based upon how N is defined, so it would be a member of itself. Once again we arrive at a self-contradiction. Is N a member of itself or not? No matter how you answer it seems to lead to a self-contradiction, which is why it is considered paradoxical.
Now to analysis. What exactly is it even supposed to mean for a class to be a ‘member of itself’?It may sound similar to being ‘identical with oneself’ but they are not equivalent. Being identical with oneself is tautological; that is true of everything. But a category, class, or set could never be a member of itself. That claim does not even make sense; it would be like saying that all of the cells in my body are also a cell of my body.
The member-class would have to be identical with the class in order to be considered a member of ‘itself’, yet at the same time there would also have to be some difference between them that would make them distinct. If there is not a difference then one would not be a member (or subclass) of the other, it would just be the same class, even if called by a different name, as with ‘dogs’ and ‘Canidae’. In this case, even the names would be exactly the same: it would be ‘dogs’ and ‘dogs’.
If the category ‘dogs’ was a member of itself, what would its other members be, exactly? I assume that there would have to be other members, in addition to itself, or we would just say that it is one category. So what would these other members be? Are they dogs? If they are, why would they not be included in the member-class? If they are included, and both the superclass and the member-class have exactly the same members, and exclude exactly the same members, then what is the distinction between them that would make them two separate categories?
To put this point more formally, consider the method of definition by genus and difference. If a class could be a member of itself it would have to simultaneously be both genus and species, and there would have to be some difference between them. The difference would be the attribute that distinguishes that species from other species within the genus.[2] ‘Ice’, for example, could be defined as ‘frozen water’. In this case, ‘Ice’ is a species or type of ‘water’, which is the genus, and ‘frozen’ is the difference between ice and other species of water. The specific difference narrows the genus class, restricting it in some way to only some members of the genus. If a class were a member of itself, or a subclass of itself, what could possibly be the difference between its genus form and its species form? And, if there is a difference, how could it be considered identical with the genus class? ‘Ice’ is not identical to ‘water’; ‘water’ is not a species of itself. No species is ever identical to its genus! It does not even make sense to say that it could be. If they were identical there would not be a species.
Consider classes that have only one member, such as: ‘All things identical to Socrates’. If it was a member of itself, then it would actually have two members rather than one: it would include Socrates, of course, and itself. But wouldn’t Socrates have to be a member of the member-class as well? It seems to me that he would. So then Socrates would be categorized in two places, one as a member of the original class, and another as a member of the member-class. Why the redundancy? Why would he need to be counted multiple times? Moreover, the member-class, in addition to having Socrates as a member, would also have to include itself as a member or it would not be identical to the superclass, and this would continue ad infinitum. ‘All things identical to Socrates’ would have an infinite number of members because the superclass must include all the members of its subclasses. (‘Mammals’ must include all members of ‘dogs’, ‘cats’, ‘horses’, etc.) Not only that, each one of those member-classes would also have an infinite number of members themselves. All classes that include themselves as members would have an infinite number of members because those member-classes would also have to include themselves as members. Why is it that we never count all these additional members if they really exist?
None of this makes any sense. No class is ever a member of itself. According to our definition above, that means that every class would be a member of N. The problem with asking whether N is a member of itself is that it seems to imply that it is possible for N to be a member of itself, that indeed some classes are members of themselves; but that is not really a genuine possibility for any class or set.
Russell obviously would have disagreed. He said in a book titled Introduction to Mathematical Philosophy (1919) that he originally thought of the paradox in 1901 while analyzing a mathematical proof by Georg Cantor. It was after considering a supposed ‘class of all imaginable objects’ that he thought of the potential contradiction:
The comprehensive class we are considering, which is to embrace everything, must embrace itself as one of its members. In other words, if there is such a thing as “everything,” then, “everything” is something, and is a member of the class “everything.” But normally a class is not a member of itself. Mankind, for example, is not a man. Form now the assemblage of all classes which are not members of themselves. This is a class: is it a member of itself or not? If it is, it is one of those classes that are not members of themselves, i.e., it is not a member of itself. If it is not, it is not one of those classes that are not members of themselves, i.e. it is a member of itself. Thus of the two hypotheses – that it is, and that it is not, a member of itself – each implies its contradictory. This is a contradiction.
Based upon this, one would expect that Russell would reply to me by saying something like the following: ‘If you acknowledge that it is not a member of itself, how could you possibly deny that N is a member of N? Even if most, or nearly all classes are not members of themselves, surely this one would be, because it has the characteristic of “not being a member of itself” which necessarily means membership in N.’ Well, not so fast Bertrand. In the quote above he makes it seem as though if ‘everything’ was not included in the class ‘everything’ that the class would be incomplete because some object that should have been included was left out. But the class is not an object. A class is a collection of things, not a ‘thing’ itself. He claims that ‘“everything” is something’ but actually it is not. It is not ‘nothing’, I’ll give him that, but the term ‘everything’ does not refer to a ‘thing’, it refers to the collection of all things. Similarly, N just stands for a collection or grouping of sets. The key characteristic that defines N as a set is that it is a collection of sets that are not members of themselves rather than an individual set that is not a member of itself.
No one thinks that the class ‘dogs’ would be a member of itself because obviously the class is not a dog. But when the members of the set are sets themselves it introduces ambiguity into the terminology so that one ends up referring to both the members of the set and the set in the same way. This makes it seem as though the set could be a member of itself if it had the same characteristic that the member-sets do. But in truth the set does not have the same defining characteristic that the member-sets have. For example,‘sets that contain two or more members’ may seem as though it would be a member of itself because it does have more than two members, but it is the ‘set of sets that contain two or more members’ so it would actually be a member of the ‘set of sets of sets that contain two or more members’. Likewise, N is a member of the ‘set of sets of sets that are not members of themselves’.
Another type of set that is often thought to be a member of itself is formed by what I refer to as anti-predicates, such as non-humans, non-mammals, or non-trees. (Actually, the prior example of ‘not a member of itself’ would be an anti-predicate.) The classes that are created from anti-predicates are sometimes referred to as class complements. They are formed by grouping things based upon a characteristic or property that all members of the class lack.
Because the class ‘non-humans’ includes everything except humans, the claim is that the class itself is also not a human, and therefore it would have to be included in the class. By this reasoning, just about every category formed by an anti-predicate would have to include itself, since the category would also be a ‘non-x’, whatever x happens to be. (The only exception I can think of is ‘non-sets’ or ‘things that are not a set’.) It is true, of course, that the class ‘non-humans’ is not a human, but I do not believe that means that it is a member of itself because that is not the characteristic by which this class is organized. It is a member of the ‘class of classes of non-humans’ not ‘non-humans’.
If the class ‘non-humans’ has to be included as a member of itself because it is not a human then all of its subclasses, such as ‘dogs’, ‘shoes’, and ‘tables’ would also have to be members for the same reason. But should these really be considered members of the class? I have no problem with including all of the individual things within those subclasses as members, but the subclasses themselves should not be included. Subclasses are not objects, they just represent different ways of organizing and grouping the members. Sometimes we might say that ‘chairs’ or ‘books’ are included in ‘non-humans’, but that just refers to the objects that fall under those categories, not the categories themselves. It would make for a very cluttered category if all of the subcategories were also included as members.
Perhaps one might argue that if all the numbers in the set of even numbers were added together it would result in an even number, and thus the set itself would have to be included as an element of the set. However, there is a difference in saying that if all the numbers in the set were added together it would result in an even number and saying that the set of even numbers is itself an even number. The set is a collection or grouping of numbers, not a number itself. It is not really possible to add them all together anyway, because the set is infinite, but even if it was, the set is not identical to this number. Sets are not numbers, even or odd.
But there are some other possible objections arising from how sets are used in math. Curiously, in mathematics every set is thought to be a subset of itself. If A and B are two sets, and every element (or member) of set A is also an element of B, then A is considered a subset of B. If A and B have identical members this has been fulfilled. Thus, A would be a subset of A. I cannot argue with the fact that every element of A is also an element of A, but this is a very strange and ambiguous way of using the term ‘subset’. In any context outside of math and set theory, ‘subset’ or ‘subclass’ or ‘subcategory’ is always used to mean a more specific division within the set or class or category. If you told someone outside of math that a subcategory of ‘animals’ is ‘animals’ they would think you were insane. Can a ‘subsection’ also at the same time be considered ‘the section’? Or, for that matter, can a ‘section’ also be the entire thing? This makes no sense. Something must either be a part of the whole, or the whole, it cannot be both at the same time. A set cannot be a subset or member of itself, a subset cannot be the set or a member of the set, and a member cannot be a subset or superset that it is a member of.
I would, of course, acknowledge that a set can be a subset of some other set, which is quite common, or even a member of some other set, such as when a sports team is a member of a league. But even then we would not say that the franchise, or the team, is on the team, as though it were one of its own players. The team could be treated as though it were a single unit that has properties and characteristics of its own, but it would not be exactly the same characteristics or properties that the individual players have. If some of the players on the team are fast and the team as a whole is fast, the latter is only relative to other teams, not players. In other words, it would be a unit only within some other set. I am convinced that it would be that way for all sets, classes, and categories.
If A was a subset of A, what specific difference does the subset have? If there is no difference, why the hell are you calling one a ‘subset’ of the other instead of just saying that they are identical? A equals A; just leave it at that. Or, if A equals B, then they are also the same set. If two sets have identical members then really it is the same set called by a different name, making the terms A and B synonymous.
The term ‘subset’ should be defined as a set which is fully contained within another set that is not fully contained in it. The latter could be referred to as either the superset, or just the set. For example, all dogs (subset) are mammals (set), but of course not all mammals are dogs; the superset has at least one element that is not included in the subset. If the subset contained all the elements of the set and/or some additional elements not included in the set, then it is not really a ‘subset’ at all. ‘Fast rabbits’ is a subset of ‘rabbits’; ‘rabbits’ is not a subset of ‘rabbits’. This is similar to how mathematicians define ‘proper subset’ but there should be no need to refer to them as ‘proper subsets’ and ‘proper supersets’ because those are the only genuine subsets and supersets that there are.
Another very strange claim from mathematics is that supposedly the empty set is a subset of every set. Since the empty set has no elements at all, it would be true that all of its elements (there are none) belong to a set, no matter what that set is. One could also put it negatively by saying that unless there is some element of the empty set that is not included in whatever set you are referring to, the empty set must be considered a subset of it. Because the empty set has no elements at all, it is true that none of its elements are outside of that set, or any other set. Thus, the empty set is a subset of every set, including itself. This reminds me of the reasoning used in the Modern Square of Opposition, where literally any claim that one can possibly dream up about an empty category is considered true for roughly the same reasons as given above. ‘All unicorns are non-unicorns’ is thought to be true because since there are no actual unicorns at all, it is true that there are no unicorns outside of the non-unicorns category. ‘No unicorns are unicorns’ or ‘No unicorns are one-horned creatures’ are also thought to be true because there are no members of the subject category in the predicate category. Literally any universal claim about an empty category is considered true, no matter how silly or even self-contradictory it is. But this is just a weird quirk of the system when you are referring to empty categories. It is like trying to divide by zero; there ought to simply be an ‘error’ message like there is on the calculator when you try to divide by zero.
Claims about empty categories are not true or false, they are simply undefined. These propositions have no actual truth value unless the categories have actual members. In a similar way, an empty set is not a subset or superset. Those terms do not apply to it. We should not think of the empty set (or any set) as an independent ‘thing’; if it has no members, there is nothing there at all. There can be no collection of members without members. If one were to argue the contrary then when it is a subset of itself that would mean that the empty set contains ‘something’, which would be a very odd claim. It would not have any elements, but it would contain a subset that is acknowledged to be ‘something’, so could it really be considered an ‘empty set’?
An empty set is nothing. One may be able to hypothetically conceive of it, but it does not really exist. If there are no elements there is no set. The empty set does not contain any elements or subsets, and is not a subset or element of any other set.
In sum, it is logically impossible for a set to be a member of itself. That is like saying that something is both the whole, and also simultaneously a part of the whole but not the whole. It cannot be a component of the thing and also the thing. The fact that the opposite view leads to absurdities like Russell’s Paradox only further demonstrates this.
David Johnson
2018
[1] 1 Some of the information given in the preceding paragraphs is based upon a short biography of Frege given in A Concise Introduction to Logic by Patrick Hurley, 12th edition. Other sources of information about the paradox include several entries from the Stanford Encyclopedia of Philosophy and Wikipedia. Various websites that discuss mathematical concepts were consulted for that portion, such as math-only-math.com/subset.html and mathcentral.uregina.ca, etc.
[2] In logic, the terms genus and species have a somewhat different meaning than they have in biology. The taxonomy used in biology is actually based upon Aristotle’s ideas, but he used the terms much more broadly. However, it is a relatively similar idea, other than being more abstract. In logic, genus simply refers to a relatively larger class, and species means a relatively smaller subclass of the genus. Thus, we could speak of the genus ‘animals’ and the species ‘mammal’ or the genus ‘human’ and the species ‘female’ or the genus ‘chair’ and the species ‘rocking chair’, etc.