A Gentle Introduction To Mathematics - Russel's Paradox
Disclaimer: this is a summary of section 4.5 from the book A Gentle Introduction to the Art of Mathematics: by Joe Fields, the content apart from rephrasing is identical, most of the equations are screenshots of the book and the same examples are treated.
Who is Russel?
![[source]](https://en.wikipedia.org/wiki/Bertrand_Russell#/media/File:Bertrand_Russell_in_1876.jpg){kind=link}
One interesting fact about Nobel Prize is that it doesn’t include the category of Mathematics. Why? There’s a lot of circulating controversy as to why it was not included, the most acceptable is the idea that Nobel believed only in utilitarian ethos. Nobel simply didn’t view mathematics as a field which provides benefits for mankind – at least not directly.
The broadest division within mathematics is between the “pure” and “applied” branches. To one side, one may call an applied mathematician a physicist (or chemist, or biologist, or economist). One of the few mathematicians to win a Nobel prize was Bertrand Russell,
" in recognition of his varied and significant writing in which he champions humanitarian ideals and freedom of thought"
It was a Nobel Prize in Literature though, not in mathematics. He was the one who helped revolutionize the foundations of mathematics, but he was better known as a philosopher. Russel’s mathematical work was of a very abstruse foundational sort. His main goal was to reduce all mathematical thought to logic and set theory.
Set of All Set
In set theory, the idea of a “set of all sets” leads to something paradoxical – this is known as Russel’s paradox.
We’ve encountered sets that contain other sets, but would it be acceptable for a set to contain itself?
Think of the following:
We could then rewrite this as
(it would be better to present this idea in a table)
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Often paradoxes seem to be caused by self-reference of this sort. (The most famous example is this sentence: “This sentence is false.”) In words, Russel’s paradox says,
The set of all sets is a set, therefore it is a member of itself.
In symbolic representation consider a universal set S with the following property, that is, we single out a subset that doesn’t contain itself:
Case 1: if we assume that S is an element of S then, it must be the case that S satisfies the membership criterion for S. Hence, S is not an element of S.
Case 2: if we assume S is not an element of S, then S does satisfy the membership criterion for S. Hence, S is an element of S.
Within the logical we’ve been developing all along, we know that within the universal set we’re working there are only two possibilities – either a set is in S or in its complement.
Note that case one and two are both true. This assumption that all statements are either true or false is limiting the space of logic hence resulting in paradoxes. One must consider other possibilities like being true and false at the same time or being not true or not false.
Is there another workaround this kind of thinking?
Type Theory
Whitehead and Russel, in a 3 volume work, developed a workaround for this kind of paradox, using a system known as type theory. They’ve introduced principles for avoiding Russel’s paradox, in which a set and its element are of different “types” and so the notion of a set being contained in itself, as an element, is not allowed.
For me, Bertrand Russel is more known for his teapot which is said to be orbiting the sun somewhere between our orbit and Mars.
While it ought to be obvious for any rational-thinking person that such a teapot does not exist, it is impossible to prove the claim false (and it will remain impossible for the foreseeable future).
Now, there are many other claims put forward that are impossible to prove - like, the existence of a god. For Russell, the existence of a Christian god was no more likely than the existence of such a teapot.
Thanks for this additional information regarding Russel. I best remember Bertrand Russel from his work, "The Conquest of Happiness". One very transformative sentence I've encountered, from his book, is about vanity, he wrote
I think this article lacks some few words, I had to look up the wikipedia article to get wiser. The paradox is not about "the set of all sets", but "the set of all sets that are not members of themselves".
I loved Russell's Paradox! And I thought Cantor's Diagonal argument was actually amazingly clever. Not sure my philosophy students appreciated me making them study them though.
Paradoxes are hard to grasp naturally. It takes hours to exit this mental dissonance before one realizes what's wrong.
There is a common misunderstanding that there exists a Nobel prize of Economics; this was not included in Nobel's will, this prize was introduced by the Swedish National Bank, and the official name of the prize is Swedish National Bank's Prize in Economic Sciences in Memory of Alfred Nobel. In the similar way one could have made up new Nobel prizes, like one for mathematics and one for environmentalism. The Peace Prize has sometimes been awarded to environmentalists, but I believe it's an abuse of the institution (unfortunately, the Nobel Peace Prize is awarded by Norwegian politicians, hence the prize has become very much political - there are many people who should never have been awarded the prize).
For Mathematics there is the Abel Prize
In addition, there is also a Fields Medal awarded by the IMU.
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