Russell's Paradox

                                                               Herr Cantor in a set of one

In this post, I'm taking a break from my latest obsession with 'Roman History' and interlarding my general nonsense with something very sensible and philosophical   

Bertrand Russell was an English mathematician, logician, philosopher, and undoubted intellectual powerhouse of the 20th century. I consider him the last 'Great Polymath' as his interests and abilities were diverse and multitudinous. His book: 'A History of Western Philosophy' is a wonder to behold. A great book by a great man. In this book, Russell not only encapsulates philosophic development spanning 2,500 years he also manages to place individual philosophers within historical and intellectual contexts. He cogently and eloquently represents philosophers within their intellectual milieu. He goes to great effort to consider the influence of prior philosophers upon man (philosophers are always, men- except in modern times when they are not), and the subsequent furtherance of intellectual development on those to come. In addition, his style is compact, elegant and without unnecessary embellishment. He comes from a time when folk of genius seemed to burst forth like ripe fruit in the summer sun and their abundant cornucopia spillethed (not a real word) upon a florid landscape (steady Flaxen-Arse). Alas, those times are no more.

Anyway, I've waxed enough- tis time to get to the point. Today's fare is a little on the dry side and intrudes upon the esoteric. It concerns, 'Set Theory'. Set theory was initiated in the 1870s by the brilliant German mathematician, Georg Cantor. Simply stated it concerns stuffing stuff into boxes, of different hues, or the same hue,  just because we can. As you will note, my style for the following is vastly different from my usual grandiloquent style. Tis more in keeping with my professional stance, in times past and not a single 'Arse' shall impinge, unless I get bored.  This post is not for all as it is, as a consequence of the subject matter a tad dry. But, gentle reader, it is difficult to present the problem in a more 'user-friendly' manner.  

Russell's paradox is a classic paradox in set theory that is named after the English philosopher and logician Bertrand Russell. The paradox arises when we consider the set of all sets that do not contain themselves. This set, known as the Russell set, is defined as follows:

R = {X | X is a set that does not contain itself}

The paradox arises when we ask the question: Does R contain itself? If R contains itself, then it must satisfy the condition of being a set that does not contain itself, which is a contradiction. On the other hand, if R does not contain itself, then it must satisfy the condition of being a set that does contain itself, which is also a contradiction. Thus, the paradox shows that there cannot be a set of all sets that do not contain themselves.

The paradox was first discovered by Russell in 1901 when he was attempting to find a way to avoid the logical paradoxes that had been discovered by the German mathematician Georg Cantor. Cantor had shown that there are different sizes of infinity and that the set of all sets is a larger infinity than any other infinity. This led to paradoxes like the set of all sets that do not contain themselves, which seemed to defy logic.

Russell's paradox is significant because it shows that there are limits to what we can define using set theory. It reveals a fundamental inconsistency in the way we think about sets and collections. It demonstrates that some assumptions we make about sets can lead to contradictions and inconsistencies.

To understand the paradox in more detail, let's consider the two cases that arise when we ask whether R contains itself or not.

Case 1: R contains itself

Suppose that R is a set that contains itself as an element. This means that R satisfies the condition of being a set that does not contain itself because R is a set that contains itself as an element. But this leads to a contradiction because R cannot both contain itself and not contain itself at the same time.

To see why, suppose that R contains itself as an element. Then R satisfies the condition of being a set that does not contain itself because R is a set that contains itself as an element. But this means that R does not belong to the set R, because the set R consists only of sets that do not contain themselves. This leads to a contradiction because R must belong to the set R since we assumed that R contains itself as an element.

Case 2: R does not contain itself

Suppose that R is a set that does not contain itself as an element. This means that R satisfies the condition of being a set that does not contain itself. But this leads to another contradiction because R must be an element of the set of all sets that do not contain themselves. But R is itself a set that does not contain itself, so it must be an element of this set. This contradiction arises because we cannot consistently define the set of all sets that do not contain themselves.

The paradox shows that there is something fundamentally wrong with the way we think about sets and collections. We assume that any collection of objects can be made into a set, but this assumption leads to paradoxes like Russell's paradox. We cannot define a set of all sets that do not contain themselves, because such a set leads to a contradiction.

Russell's paradox has significant implications for the foundations of mathematics and logic. It shows that some of our most basic assumptions about sets and collections are flawed and that we need to be careful when defining sets and collections. It also shows that there are limits to what we can prove using set theory and that we need to be aware of the limitations of our theories.

A select few great thinkers have proposed ways of circumventing the above contradiction. However, I deem that if I continue in this vein I will lose what little readership I already have.

Tis enough for today. I will endeavour to pen a less boring post on the morrow, but only if I remember to take my medication.