ZFC considered harmful

Revision en1, by adamant, 2023-04-17 15:30:26

Hi everyone!

As it is widely known, Zermelo–Fraenkel set theory with the axiom of choice, also widely known as ZFC, has several fatal flaws:

  • Nobody remembers the axioms accurately;
  • in ZFC, it is always valid to ask of a set ‘what are the elements of its elements?’, and in ordinary mathematical practice, it is not.

From now on, please use the Lawvere's Elementary Theory of the Category of Sets instead of ZFC:

  1. Composition of functions is associative and has identities
  2. There is a set with exactly one element
  3. There is a set with no elements
  4. A function is determined by its effect on elements
  5. Given sets $$$X$$$ and $$$Y$$$, one can form their cartesian product $$$X \times Y$$$
  6. Given sets $$$X$$$ and $$$Y$$$, one can form the set of functions from $$$X$$$ to $$$Y$$$
  7. Given $$$f : X \to Y$$$ and $$$y \in Y$$$, one can form the inverse image $$$f^{-1}(y)$$$
  8. The subsets of a set $$$X$$$ correspond to the functions from $$$X$$$ to $$$\{0, 1\}$$$
  9. The natural numbers form a set
  10. Every surjection has a right inverse

Only by switching to a superior set of set theory axioms we can save mathematics. Thank you for your attention.

P.S. On a more serious note, I think that their approach is quite interesting and it is useful to revisit the fundamentals once in a while. Overall, as highlighted in An Infinitely Large Napkin, we understand things much better when we think about them as "sets and structure-preserving maps between them" rather than just "sets and their elements", as suggested by ZFC.

Tags shitpost, set theory

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en1 English adamant 2023-04-17 15:30:26 1778 Initial revision (published)