20 ms·

Why should one be bothered by it?

• hackandthink 3y ago
  It makes a simple thing (pair) complicated. This may be true for Category Theory as well (Product of two objects): https://en.wikipedia.org/wiki/Product_(category_theory) https://en.wikipedia.org/wiki/Product_(category_theory) But Category Theory captures the essence of Products/Pairs, so I am more inclined to accept it.
  • reuben364 3y ago
    Yes it captures the essence, but if you have a category of ZF sets you still need to demonstrate that a product exists. This is just a possible product (along with its projections).
  • Sharlin 3y ago
    > It makes a simple thing (pair) complicated. Well, yeah, but {x, y} is also a pair. How is (x, y) different? It's ordered, you say? All right, an ordering is a relation that's antisymmetric and so on, but in this case let's say we have a function that maps x to 0 and y to 1… But what is a function? Okay, a function is a type of relation, so let's define a relation first: a set of ordered pairs… oops. The real problem with defining (x, y) = {{x}, {x, y}} is that elements of a set must also be elements of some universal set 𝓤, {x, y} ⊂ 𝓤, but as we know, there is infamously not such a thing as a "set of all things". Sets have to be typed. But in a pair, x and y can be of entirely different kinds of entities.
• VirusNewbie 3y ago
  It seems to be ignoring β-reduction for one. But an ordered set is fundamentally different than an unordered set as well!