What Is Russell's Paradox in Set Theory?

TL;DR

Russell's Paradox, discovered by Bertrand Russell in 1901, is a logical problem sitting at the heart of set theory, the foundational branch of mathematics invented by Georg Cantor in the 1870s. It arises from set theory's own rules, and Jeffrey Kaplan argues that although Russell and other mathematicians believed they had solved it, they never actually did.

Transcript

in 1901 the english philosopher and mathematician bertrand russell discovered a problem a paradox at the heart of mathematics and all of science the paradox specifically concerns a foundational branch of mathematics called set theory so in this lecture i teach you all of set theory in like eight minutes and then i show how the paradox arises russel... Read More

Key Insights

• Russell's Paradox was discovered in 1901 by Bertrand Russell, the English philosopher and mathematician, and it concerns set theory, a foundational branch of mathematics. Because set theory underlies mathematics, the problem reaches into all of science, not just one specialized corner of math.
• Logicism is the view developed by Gottlob Frege and Bertrand Russell that mathematics is a branch of logic and that arithmetic, the most basic type of math, can be reduced to first order basic logic plus set theory. Its payoff would be answering what a number is: numbers are sets.
• Immanuel Kant, a Prussian philosopher living in the 1700s, thought mathematics was a construction of the human mind, which makes mathematical truths subjective in some sense. Frege and Russell disliked this because they held that mathematics had to be objective, and logicism was their counteraction.
• Set theory was invented in the 1870s by Georg Cantor, the Russian German mathematician, and not as a mathematical game for its own sake. He needed it to prove that some infinities are larger than other infinities, meaning you can have two infinite collections with more in one than the other.
• A set is a collection of objects whose members need not be gathered together in space or time and need not be related to each other in any meaningful way. A set can contain LeBron James and the top half of the Eiffel Tower, or a real person and the fictional Harry Potter.
• Set identity is determined by membership alone, which is the axiom of extensionality. From this follow the rules that order does not matter, that repeated members change nothing, and that how you describe a member does not matter, so the set of LeBron James equals the set of the NBA all-time scoring leader.
• Unrestricted composition, called the axiom of unrestricted comprehension in formal set theory, says any set you can think of is a set. From it follow the rules that the union of two or more sets is itself a set and that any subset of a set is also a set.
• A singleton set contains exactly one member and is not identical to that member. The singleton set containing LeBron James is a zero-time NBA champion even though LeBron James is a four-time NBA champion, showing that a set and its lone member have different properties.

Questions & Answers

Q: What is Russell's Paradox and when was it discovered?

Russell's Paradox is a logical problem that the English philosopher and mathematician Bertrand Russell discovered in 1901. It sits at the heart of mathematics and all of science, and it specifically concerns set theory, a foundational branch of mathematics. The paradox arises out of one or more of set theory's own rules, which is what makes it so troubling: it is not an outside attack but a problem generated internally. Russell himself and many other mathematicians thought they could solve the paradox, but Professor Jeffrey Kaplan argues in this lecture that they cannot and they do not, and that the paradox goes even deeper than Russell realized.

Q: What is a set in set theory?

A set is a collection of objects. Georg Cantor, who invented set theory, described a set as a gathering together into a whole of definite distinct objects of our perception or of our thought, which are called elements of the set. Crucially, the objects in a set do not need to be collected together in space or time. You can have the set of three markers sitting on a desk, but equally the set of all people watching a video who are spread all over the world and possibly across time. The members also need not be related in any meaningful way, and anything you can refer to or imagine can be a member.

Q: Why did Frege and Russell develop logicism?

Immanuel Kant, a Prussian philosopher living in the 1700s, thought that mathematics was a construction of the human mind. If that is right, then mathematical truths are in some sense subjective. The German philosopher Gottlob Frege and the English philosopher Bertrand Russell did not like this conclusion, because they thought mathematics had to be objective. To refute and counteract Kant's view they developed logicism, the position that mathematics is a branch of logic and that arithmetic, the most basic type of math, could be reduced to just first order basic logic and set theory. Success would let them answer what a number is, with the answer being that numbers are sets.

Q: Who invented set theory and why?

Set theory was invented by Georg Cantor, the Russian German mathematician, in the 1870s. He was not making up a mathematical game for the sake of it. Cantor proved that some infinities were larger than other infinities, the idea being that you could have an infinite number of one thing and an infinite number of something else and still have more of one than the other. In order to do this he had to invent set theory. Kaplan also notes that we deal with sets every day: saying a pile of potatoes is enormous is a claim about the pile, not about the individual potatoes, and the same holds for saying the world population of cats is enormous.

Q: What is the difference between naive set theory and axiomatic set theory?

Naive set theory is the ordinary set theory that can be formulated in ordinary languages like English. It is called naive simply for that reason, not because it is foolish. It contrasts with formal or axiomatic set theory, which is formulated in an artificial logical language. The rules are the same ideas under different names: unrestricted composition in the naive version is called the axiom of unrestricted comprehension in the formal version, and the rule that set identity is determined by membership is called the axiom of extensionality. The lecture uses the naive version because the paradox can be shown without the formal machinery.

Q: What are the main rules of naive set theory?

The lecture lists several rules. First, unrestricted composition: any set you can think of is a set. Second, set identity is determined by membership, so only what is inside the set matters, not the labels. Third, the order of elements does not matter, so the set of one and two is the same as the set of two and one. Fourth, repeats change nothing. Fifth, the description of the items does not matter. Sixth, the union of two or more sets is itself a set. Seventh, any subset is a set. Eighth, a set can have just one member, called a singleton set.

Q: How do you write a set that is too large to list?

Set theory uses squiggly brackets to pick out a set, with everything inside the brackets being the members. Listing every member works for small sets, such as the set containing LeBron James and the number four, but it becomes too clumsy for a big set like the set of all cats, since there are too many cats to list and we do not even know all their names. Instead, set theory uses set builder or intensional notation, read as the set of all x's such that x is a cat. The word contains is used to say that a set contains all of the objects that are members of it.

Q: Why is a singleton set not the same as the object it contains?

A singleton set is a set with just one member. The lecture uses LeBron James to make the point: LeBron James is a four-time NBA champion, but the singleton set containing LeBron James is a zero-time NBA champion, because the set itself has never won anything. The set and its sole member therefore have different properties, which means they cannot be the same thing. This distinction matters for set theory generally, since sets are abstract objects in their own right, separate from the things they collect, and it is part of what allows sets to serve as candidates for what numbers are.

Summary & Key Takeaways

• Bertrand Russell discovered a paradox in 1901 at the heart of mathematics and all of science, specifically concerning set theory. The lecture teaches set theory in roughly eight minutes, shows how the paradox arises from the rules, and argues that Russell and other mathematicians did not really solve it despite thinking they had.

• The motivation is the question of what a number is. Immanuel Kant, a Prussian philosopher of the 1700s, held that mathematics is a construction of the human mind, which makes mathematical truths subjective. Gottlob Frege and Bertrand Russell rejected this and developed logicism, the view that mathematics is a branch of logic.

• A set is a collection of objects, and its members need not be together in space or time, nor related in any meaningful way. Sets can contain a basketball player and part of a tower, real and fictional things together, and even the set of all objects that cannot be imagined.

• Naive set theory runs on rules including unrestricted composition (any set you can think of is a set), set identity determined by membership, order not mattering, repeats not mattering, descriptions not mattering, unions being sets, subsets being sets, and singleton sets containing exactly one member.