Comment by thaumasiotes 4 days ago
> but you need that you can take a Cartesian product and turn it into a set of its elements as well
Huh? In your model, what is a Cartesian product? How can it have elements without being a set?
> Even ignoring that, a set containing n elements is different from a set containing n sets of one element each.
Different in what way? A set containing n sets of one element each is a set containing n elements.
In ZF, everything that's an element of a set is itself a set, so unless what's bothering you is the idea that "a set containing n elements" might contain the empty set, or a set with two elements, I'm not seeing it.
> you’d need some model of index, for one. And I’m not sure how you’d construct that with uncountably many elements
Why would the number of elements matter? Set elements aren't indexed. What are you using your model of index for?
I really feel I should repeat the question I asked you to begin with: You say you need to convert a Cartesian product into a set containing "its elements". In your mind, what is a Cartesian product, before that conversion takes place?
No. We don't.
Let's note first of all that (a, b, c, a) is a tuple, not a Cartesian product.
Where are you getting your ideas from? What do you think a Cartesian product is? Give me a definition.
Well, you’d need some model of index, for one. And I’m not sure how you’d construct that with uncountably many elements. Even ignoring that, a set containing n elements is different from a set containing n sets of one element each.
Not saying your formulation is wrong, just that there’s a fair amount of non-obvious work to get to the classic formulation.