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?
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.
