Assessing Set Theory and Russell’s Paradox

In the past four years, I have gone head-first into the world of analytical philosophy to get a deeper understanding of the craft. When I think I have a basic understanding of the concepts, I observe another theory that humbles and excites me at the same time – Set Theory with the introduction of Russell’s Paradox is one of those times. Set Theory, devised by Georg Cantor in the late 19th century, provides a formal framework for understanding collections of objects and their properties. Throughout its development, set theory has played a pivotal role in shaping various branches of mathematics and has been instrumental in establishing rigorous logical proofs. However, within the seemingly impeccable structure of set theory lies Russell’s Paradox which continues to intrigue and challenge mathematicians today. This piece attempts to explain Set Theory, how Russell’s Paradox impacts this theory, and its implications from it.

What is Set Theory?

First, we have to make sense of Set Theory and its relation to the world of mathematics. Set Theory is a branch of mathematics that deals with the study of, well, sets – which are collections of distinct objects, elements, and properties. Furthermore, it is understanding the relationship between these objects, elements, and properties. In my book, Pedagogy of the Depressed, I develop concepts of analytical epistemology using the tradition of First Order Logic – which according to Gottlob Frege and Bertrand Russell – is simply one of two twigs on the mathematical branch of a logic tree. Set Theory is the other twig, and according to Russell and Frege, is considered the precise understanding of mathematics.

Let’s take a basket of apples, this basket has 5 apples in it. Say we also have a sports stadium with 20,000 people in it. Both items are sets; although they are unrelated, they share properties of sets; in addition, their level of infinity differs. For example, an infinite set of baskets of 5 apples with an infinite set of sports stadiums with 20,000 people in it, solely measuring apples to people within the set one infinity will always be greater than the other (∞ < ∞). We can also say everything that is baskets of apples and sports stadiums with 20,000 people in it is a set, and everything that is not baskets of apples and sports stadiums with 20,000 people in it is a set.

I have thought about how to go through this clearly, I think it is best to define key concepts – based loosely on Cantor’s rules – of Set Theory.

1. The Axiom of Unrestricted Comprehension: This essentially means that everything and anything that you can think of, or imagine, or plan to imagine even if you have not imagined it yet can be a set. It can be defined as {x:x is a set} or {all of x so long as x is a set}. It states that for any well-defined property or condition, there exists a set that contains precisely those objects that satisfy that property. It can be expressed as follows: For any property P(x), there exists a set A such that for every object x, x is an element of A if and only if x satisfies property P(x).
   
2. Elements and Membership: A central notion in Set Theory is the concept of membership. For example, if A = {1, 2, 3}, then 2 is an element of A or (2 ∈ A), while 4 is not an element of A, or (4 ∉ A). Membership establishes the connection between elements and sets, indicating whether an object is included or excluded from a particular set based on its presence or absence as a constituent part.
   
3. Cardinality: The cardinality of a set refers to the number of elements it contains. For a finite set, the cardinality is simply the count of its elements. For example, the set A = {1, 2, 3} has a cardinality of 3. In the case of infinite sets, such as the set of all natural numbers, the concept of cardinality becomes more nuanced.
   
4. Subsets and Supersets: A set B is said to be a subset of another set A if every element of B is also an element of A. If there exists at least one element in A that is not in B, then B is a proper subset of A, denoted as B ⊂ A. A set A is considered a superset of B if B is a subset of A. Simply put, any subset is a set.
   
5. Operations on Sets: Set theory provides several fundamental operations to manipulate and analyze sets. These include union, intersection, and complement. Operations on sets involve manipulating and combining sets to create new sets, including operations such as union, intersection, and complement. So, A is in union with B is a set that all elements are contained in either A or B (A ∪ B). A intersecting with B is a set of elements common – but not contained – to A and B (A ∩ B). Lastly, the complement of a set A with respect to a universal set U, denoted as A’, is the set of elements in U that are not in A.
   
6. Singleton Sets and Null Sets are Sets: A singleton set is a set with one item in it {apple} and a null set is written as { } Ø. So, a set with one item in it, or a set with nothing in it is also a set. This is understood by the first point of The Axiom of Unrestricted Comprehension. To simplify this further: one thing = set, all things = set, nothing = set, one thing + all things + nothing = set.

What is Russell’s Paradox?

Where does Bertrand Russell come into play here? Well, in 1901, he developed and tested a distinct problem with Cantor’s rules of Set Theory. We start by understanding that you can have sets of sets or sets being able to contain themselves – this has been established throughout the first section of understanding Set Theory – so where does that leave us? {apple} is a Singleton Set inside of a set {basket:5 apples}, and {x:x is a Singleton Set}; also, {x:x is a set} with { } Ø are all fair play in this theory.

The basis of this paradox comes from the first loosely based rule from Cantor: The Axiom of Unrestricted Comprehension. Russell’s Paradox raises a question of logical consistency. The paradox points out the inherent contradiction that forms when considering the set of all sets {x:x is a set} that do not contain themselves as elements. If such a set exists, it leads to a logical contradiction, as it both contains itself and does not contain itself. This paradox leads to two possibilities:

1. If A does not contain itself as an element, then it satisfies the property of not being an element of itself. However, this means that according to the definition of A, it should be an element of A, leading to a contradiction.
   
2. On the other hand, if A contains itself as an element, it does not satisfy the property of not being an element of itself. Again, this contradicts the definition of A.

No matter how you look at it, the fundamental rule based on The Axiom of Unrestricted Comprehension leads to a logical contradiction; thus, a logical contradiction on the whole of Set Theory. Simply put, if nothing is not an element it is not a set, but null set is a set, then nothing is both not a set and a set at the same time. Even “simpler”: if it is then it isn’t and if it isn’t then it is???

Is this an Effective Understanding?

Russell’s Paradox highlighted the limitations and inconsistencies of the Axiom of Unrestricted Comprehension, demonstrating that not all properties can be used to define sets without leading to contradictions. As a result, set theories like Zermelo-Fraenkel Set Theory (ZF) introduced more restricted comprehension axioms, such as the Axiom of Separation, to avoid these paradoxes and ensure a consistent and well-defined foundation for Set Theory.

But alas, there seems to be an issue with not having unrestricted comprehension, as it is the building block of Set Theory. Remember we discussed (∞ < ∞), in this instance, not all infinities are the same, and they can just be the infinity of sets, but once again we run into the same problem with an infinity of sets and get to the conundrum of “infinity is not infinity, but also not infinity is infinity”. In a video by Jeffrey Kaplan he excellently describes the concept of predication avoiding this paradox in the future, but eventually becoming another paradox. I’ll add that a paradox can be escaped, by escaping Set Theory altogether, and ensuring that axiomatic principles a defined as worthy of being true – this idea comes from Roderick Chisholm.

• If all implies all is true A → A = T,
• And none implies none is true N → N = T,
• Then all implies not none is true A → ¬ N = T, or N → ¬ A = T
• And all implies none is false A → N = F, likewise none implies all is false N → A = F

Some in this field may think that this is a cop-out, but it encompasses all the things and none of the things that are clearly defined. Even though my statement has predicates, the predicates are reliant on the proper axiomatic principles defined by propositions. Meaning predicates are not just predicates on their own, but predicates proceeding axiomatic principles.