Russell's Paradox - a simple explanation of a profound problem

Introduction to Set Theory and the Russell Paradox

In this section, the speaker introduces the paradox at the heart of mathematics and science discovered by Bertrand Russell in 1901. The paradox concerns set theory, a foundational branch of mathematics. The speaker explains that he will teach set theory in eight minutes and then show how the paradox arises.

What is a Number?

• A number is not four potatoes or four tomatoes but rather the number itself.

• The number itself is something that no one has ever seen or touched, but we seem to know things about it.

• Numbers are essential to science, technology, and all of human life.

Logicism

• Logicism is a view developed by Gottlob Frege and Bertrand Russell that mathematics is a branch of logic.

• According to logicism, arithmetic could be reduced to just first-order basic logic and set theory.

Naive Set Theory

• A set is a collection of objects studied in set theory invented by Georg Cantor in the 1870s.

• Naive set theory is ordinary set theory formulated in ordinary languages like English.

The Russell Paradox

• The paradox specifically concerns a foundational branch of mathematics called set theory.

• Naive set theory runs into a terrible problem known as the Russell Paradox.

Set Theory and Cantor's Paradox

In this section, the speaker introduces set theory and explains how sets are used to group objects together. The speaker also discusses Cantor's paradox, which is a logical problem that arises in set theory.

Introduction to Set Theory

• Sets are used to group objects together using squiggly brackets .

• The notation for picking out a set in writing is "the set of all x's such that x is a cat."

• A set contains all the objects that are members of that set.

• A set can be thought of as one entity or a gathering together into a whole of definite distinct objects.

Cantor's Paradox

• Cantor invented set theory in the 1870s.

• Russell discovered a logical problem called Cantor's paradox in 1901 concerning one or more rules of set theory.

• There are five rules of set theory: unrestricted composition, set identity determined by membership, order of elements doesn't matter, repeats don't change anything, and description doesn't matter.

• Rule number two states that what makes a certain set the set that it is just what's inside it.

• Rule number four states that repeating an element does not change the identity of the original set.

Overall, this section provides an introduction to sets and their use in grouping objects together. It also introduces Cantor's paradox and briefly explains some of the rules of set theory.

Introduction to Set Theory

In this section, the speaker introduces set theory and explains the basic rules of sets.

Basic Rules of Sets

• A set is a collection of distinct objects.

• The union of any two or more sets is itself a set.

• Any subset is a set.

• A set can have just one member. This is called a singleton set.

• A set can have no members. This is called an empty or null set.

• You can have sets of sets.

Paradoxes in Set Theory

• Fragile and Russell believed that numbers were just sets, but this idea was later disproven by Russell's paradox.

• Sets can contain themselves, which leads to paradoxes.

Russell's Paradox

In this section, the speaker discusses Russell's paradox and how it led to the breakdown of set theory.

Sets that contain themselves and those that don't

• Russell discovered a paradox in 1901 and 1902 when he was developing the idea that sets can contain themselves.

• Some sets do not contain themselves, such as a singleton set with just LeBron James or the set of all cats.

• Other sets do contain themselves, such as the set of all non-singleton sets or the set of all sets mentioned in a room.

The Set of Sets That Do Not Contain Themselves

• Russell proposed collecting all sets that contain themselves into one set and making another set out of all those that do not contain themselves.

• This led to the question of whether the set of all sets that do not contain themselves contains itself.

• If it does, then it meets its own condition and doesn't contain itself. If it doesn't, then it doesn't meet its own condition and contains itself. This is a contradiction known as Russell's paradox.

Attempts to Solve the Paradox

• Changing the rules of set theory was attempted by Russel but did not work since other mathematicians have done so before him without success.

The Rules of Set Theory and Predication

In this section, the speaker argues that the rules of set theory are not made-up rules but real non-made-up objective rules that already exist. He explains the relationship between sets and objects and predicates and subjects.

Sets and Objects

• The rules of set theory are not made-up rules but real non-made-up objective rules that already exist.

• Sets contain objects, meaning objects are in the sets.

Predicates and Subjects

• Predicates are true of certain subjects.

• The relationship between predicates and subjects is that a predicate is true of a subject.

• Predicates can be true of themselves just like sets can contain themselves.

Regenerating Russell's Paradox

• Predication is just the practice of saying things about things.

• Rule number 11: predicates can be true of themselves.

• We can predicate things of predicates just like we can have sets of sets.

• Some predicates are not true of themselves, like "is a cat."

Paradox of Self-Reference

In this section, the speaker discusses the paradox of self-reference in predication. The speaker explains how a predicate can be true of itself and how this leads to a paradox.

Predicate is True of Itself

• A predicate is true of itself if it is a string of words.

• This predicate typically comes at the end of a sentence.

• Let's try to make a predicate that is true of all predicates that are true of themselves.

• The predicate "is true of itself" is an example.

Paradox

• Let's try to make a predicate that is true for all predicates that are not true for themselves.

• The predicate "is not true of itself" would be an example.

• If this predicate is true, then it isn't because it says it's not true.

• If this predicate isn't true, then it meets the condition set out by the predicate and therefore must be true.

• This creates a contradiction where the statement can both be true and false.

Rule 11

• We cannot escape this paradox by declaring that predicates cannot be true for themselves because they can.

• Once we give rule 11, which states that predicates can be used as subjects or objects in sentences, we generate the paradox.