Russel's Paradox

Russell's Paradox
Philosophical Ponderings of a Confused Discrete Student

So, who is this Russell dude anyway, and what is his paradox?

Russell was a mathematician that introduced an interesting concept sometime about the time the 19th century was rolling into the 20th. He asked us to consider a question. Something along the lines of, "Does the set of all sets that do not contain themselves contain itself?" A mouthful. How do we answer such a question? Let X be the set of all sets that do not contain themselves. The question now becomes "Is X a member of X?"

Well, let's assume X is not a member of X. Thus X is a set that does not contain itself. So by definition of X, X is a member of X. So our assumption that X is not a member of X logically leads to the statement that X is a member of X. This is a contradiction, so our assumption must be wrong.

So, let's try assuming that X is a member of X. Thus X is a set that contains itself. But X is a set that has only members that do not contain themselves, so X cannot be a member of X. So our assumption that X is a member of X logically leads to the statement that X is not a member of X. This is a contradiction, so our assumption must be wrong.

Now, if you have a statement A, A must be either true or false, which means that exactly one of the statements "A" or "not A" is true. Let A be that statement "X contains X". So, logic states that X must either be a member of X or not be a member of X. But we have just disproved, via logic, both of these statements. We have logically shown that "A" is false and "not A" is false. Thus, we have logically disproved logic. So logically, logic is illogical. This is clearly some sort of paradox.

How do we intend on resolving the paradox?

Let's look a moment into the nature of paradox. There are many different types of paradoxes, including a certain kind that Russell's Paradox can be classified under: "A paradox is a concept that, given a set of premises, logically allows you to form two or more contradictory statements."

For example, let's look at another well-known paradox, Xeno's Paradox. "In order to move some distance, you must first move half that distance. But once you've moved half that distance, you still have a distance ahead of you to move. So you must first move half the new distance before you can move the whole distance. So on and so forth, you can never reach your destination. Yet in reality you do."

One of the premises is directly stated: "In order to move some distance, you must first move half that distance." Without such a premise, there is no Xeno's Paradox. Now, scientific advances in the field of Quantum Mechanics show that underlying reality is a very discrete, quantized theory. In such a system, changes in energy can occur discretely; a system may only posses certain discrete energy states. That is, in order for particle A to move from some location to C to another location D, it does not necessarily have to move half way first. It may go directly from C to D if they are adjacent legal energy states. Thus the physical premise, "In order to move some distance, you must first move half that distance" is simply wrong. In general, an incorrect premise can lead to a paradox.

Because "a paradox is a concept that, given a set of premises, logically allows you to form two or more contradictory statements," the only thing that can lead to a paradox defined in this way is an incorrect premise. A paradox, thus, is resolved by identifying the incorrect premise(s).

How does this tie in with Russell's Paradox?

Okay, so we've established that Russell's Paradox exists because we are relying on incorrect premises. To help identify potential incorrect premises, let us examine once again Russell's infamous set, the set of all sets that do not contain themselves. To understand this set, we must understand sets in general.

What do we mean when we say set?

To simplify the world to an analyzable state, let there exist only one primitive element (that is, an element that is not a set): the letter Q. Now let us start enumerating sets (other than the null set):

{Q}
{Q, {Q}}
{{Q}, {Q, {Q}}
Etc…

As you can see, this can go on forever. But enumerating like this forever will still leave out sets that contain themselves. Notice that none of the above sets contain themselves. I dare you to write one out. Go ahead and try.

That, having failed, forces us to give our sets names. Let there exist a set that has only the members Q and itself. Let us call this set A. Thus, A = {Q, A}. We could also write this as "the set that has only the members Q and itself = {Q, the set that has only the members Q and itself}." Thus, "the set that has only the members A and itself = {Q, {Q, the set that has only the members Q and itself}}." Or A = {Q, {Q, A}}. Notice that we could do this forever. But regardless of how far we take this, we will never be able to write out a set only in terms of its primitives and structure. There will always be the elusive A, the elusive "set that has only the members Q and itself."

This is infinite recursion, a concept that human minds don't really understand, because it can't really exist in our everyday experiences. It's just a very abstract philosophical toy. So what does it really mean?

One way to look at how this concept occurs is that in reality all we can have is indefinite recursion. I can expand the above set as long as I want to (ex: A = {Q, A}, A = {Q, {Q, A}}, A = {Q, {Q, {Q, A}}}, etc…). As long as I have time and inclination, I can expand that further and further. But notice that in order to complete it, I need the "etc…" Thus, in reality, the set is indefinitely recursive, not infinitely recursively, because I am limited by time. In set theory, time is a foreign concept. So infinite recursion only exists where time does not exist, and that is why we never really encounter infinite recursion.

Now that we understand that we will never entirely understand infinite recursion, let us start examining infinite recursion in more detail, so as to allow ourselves to become even more confused. We start by going right to the definition of sets.

We define a set (any set) by saying "set A is the set of all elements that satisfy criteria B, C, etc…" This is also written {x: x satisfies B, x satisfies C, etc…}. That is, the elements of a set do not define a set, the criteria do. In the above example, we have the set A defined by two criteria. Let B be the criterion, "is the element under consideration Q?" And also let C be the criterion, "is the element under consideration the set A?" This is a perfectly legitimate set based on our definition.

To find out what exactly is in the set, we use an algorithm; for every possible element, if either B or C is true, the element is part of the set A. Or more generally, for every possible element, if the element meets any of the criteria for a set, the element is a member of that set.

So, let's enumerate the above set A using this algorithm. Remember A is defined as {x: x is Q, x is A}. Let us first consider Q and ask ourselves, "Is Q the element Q or the set A?" Q is most definitely Q. So Q is in the set A. Let's move on to consider {x: x is Q}. "Is {x: x is Q} the element Q or the set A?" The set {x: x is Q} is not the element Q, but is it the set A? We compare the two by comparing their criteria. The set {x: x is Q} is defined by the sole criterion "is the element under consideration Q?" so the set {x: x is Q} is not the set A, because A has different criteria. Let us now consider {x: x is Q, x is A}. This set and set A share exactly the same criteria, so they are equivalent sets. Thus, the set {x: x is Q, x is A} is a member of A. So on and so forth for every element we eventually find that the only elements that are members of A are Q and {x: x is Q, x is A}.

So what now if we have Russell's set, the set R, defined with just one criterion "is the element under consideration a set, call it S, that has no criterion that S satisfies?" Confusing as it is, based on our definition of a set it is a perfectly legitimate set, and can be written R = {x: x is S, such that S = {y: y satisfies B, y satisfies C, …} and S does not satisfy B and S does not satisfy C and …}.

So let's use the algorithm presented above to figure out what exactly is in R. Well let's consider Q. Q is not even a set, so Q could not possibly be the set S. Thus Q is a not member of R. Let's look at another possibility, {x: x is Q}. So, is {x: x is Q} a member of the set R? We check this by seeing if this set satisfies the criterion for R. Based on the criterion's naming conventions, S = {x: x is Q}. Now we must ask ourselves if S satisfies any of the criteria for S. So, the question is "Is the element under consideration, S, equal to Q." The answer is no. So S does not satisfy any of the criteria for S. As a result, S, which in this case is {x: x is Q} does satisfy the only criterion for R. Thus, {x: x is Q} is a member of R.

Now let us consider the set R as our next element. This is where we have to be careful. From above we know that R = {x: x is S, such that S = {y: y satisfies B, y satisfies C, …} and S does not satisfy B and S does not satisfy C and …}. So we have to ask ourselves now, "is our element under consideration, R, equal to the set S, such that S = {y: y satisfies B, y satisfies C, …} and S does not satisfy B and S does not satisfy C and …?" Or more succinctly, "Does R = {y: y satisfies B, y satisfies C, …} and R does not satisfy B and R does not satisfy C and …?" The only criterion for R, our element under consideration, is B = "Is the element under consideration equal to S, such that S = {y: y satisfies B, y satisfies C, …} and S does not satisfy B and S does not satisfy C and …?" So if R as our element under consideration satisfies B, then R is a member of R as our main set. So, does R satisfy B? Phrased differently, "is R the set S, such that S = {y: y satisfies B, y satisfies C, …} and S does not satisfy B and S does not satisfy C and …?" But wait, we just asked ourselves this question. If we tried to answer it again, we'd just end up asking ourselves again. Using this algorithm, we are never able to enumerate the elements of R. Complex as it is, if you read this paragraph enough times, you'll notice that it does make sense, and does indeed indicate an infinite loop.

How does all this resolve Russell's Paradox?

You see, the reason our algorithm failed is that we had a criterion for R that referenced criteria in a temporary set that it can't even necessarily see. That is, we had the criterion "x is S, such that S = {y: y satisfies B, y satisfies C, …} and S satisfies B or S satisfies C, or …" This criterion references B and C, which only really exist in the temporary set S, which is declared, used, and eliminated, all before B and C are referenced in the main criterion. The criteria for our set R might not even be allowed to see the criteria B and C, which exist only in set S. Yet we are operating on the premise that the criteria for R are indeed allowed to reference the criteria for the temporary set S. This may very well be a false premise.

If it is a false premise, then there is no such thing as a set that contains all sets that don't contain themselves. If there is no such set, Russell's set doesn't really exist. Poor Russell. Without such a set, he has no paradox. Thus, the paradox is resolved when we found the false premise.

So what makes you think you're right?

All right, so here's the disclaimer. I'm probably wrong. I don't necessarily think I'm right. Above, I am claiming that the ability of one set's criteria to reference criteria in temporary sets does not exist. If this is true, then that is the resolution for the paradox. But I have no right to claim this is true. I don't even know if I'm using the right definition of paradox. I don't even know if I'm tackling it from the right angle at all. After all, I am just a confused Discrete student.