Set Theory, Gödel’s Incompleteness Theorem, & Paradox in Math
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Background
As a result of A Compact History of $\infty$ by David Foster-Wallace and Gödel, Escher, Bach by Douglas Hofstadter, I’ve been thinking about set theory, infinity, and paradox in math for a bit now. In particular, Foster-Wallace’s book considers the work of Georg Cantor. Furthermore, these ideas have been coupled with the notion of consciousness as that’s the main topic for G.E.B.
Set Theory’s Basics
I’ll be cribbing liberally from Richard Hammack’s Book of Proof as I found it to be the most useful, succint description for Set Theory and the fundamentals of it. Link Here.
Hammack’s Book of Proof
“A set is a collection of things. The things are called elements of the set. We are mainly concerned with sets whose elements are mathematical entities, such as numbers, points, functions, etc.
A set is often expressed by listing its elements between commas, enclosed by braces. For example, the collection {2,4,6,8} is a set which has four elements, the numbers 2,4,6 and 8. Some sets have infinitely many elements. For example, consider the collection of all integers,
\[\mathbb{Z} = \{ ... -4, -3, -2, -1, 0, 1, 2, 3, 4 ... \}.\]Here the dots indicate a pattern of numbers that continues forever in both the positive and negative directions. A set is called an infinite set if it has infinitely many elements; otherwise it is called a finite set.
Two sets are equal if they contain exactly the same elements. Thus {2,4,6,8} = {4,2,8,6} because even though they are listed in a different order, the elements are identical; but {2,4,6,8} $\neq$ {2,4,6,7}.”
Skipping ahead a bit, mathematicians use the symbol, $\in$ to denote something is an element of a set. If
\[A = \{2,4,6,8\} \longrightarrow 2 \in A.\]A special notation called set-builder notation is used to describe sets that are too big or complex to list between braces. Consider the infinite set of even integers E = {…,−6,−4,−2,0,2,4,6,…}. In set-builder notation this set is written as
\[E = \left\{ 2n: n \in \mathbb{Z} \right\}\]We read the first brace as “the set of all things of form,” and the colon as “such that.” So the above expression reads as “E equals the set of all things of form 2n, such that n is an element of $\mathbb{Z}$.”
A general form could be written as follows:
\[\chi = \left\{ expression: rule \right\}\]Gödel
Austrian mathematician and logician Kurt Gödel is responsible for what Douglas Hofstadter would refer to as a strange loop.