Set theory with multiple countable infinities

Question

In set theory, all sets that are countably infinite are generally considered to have the same size since there is a bijection between them. Has anyone tried formalising set theory in a way which rejects this hypothesis, that is so that adding an element to a countably infinite set increases its size?

Answer 1

This is not an unreasonable question, naively.

However, if you really think about it, it starts to become a very unclear question. And for a good reason too.

First, what does it mean for a set to have cardinality? Cardinality is our way of measuring size of sets. So you need to decide what sort of thing you want to measure, and how to measure it. Using bijection means that we strip the entire structure from the set. This means that $\Bbb N$ and $\Bbb Z$ are no longer ordered, they do not have addition and multiplication. They are just two sets. We are allowed to use this additional structure to define functions, but these functions need not preserve this structure. Which is why we can define a bijection between $\Bbb N$ and $\Bbb Z$, but not an order preserving bijection.

It turns out that infinite sets are weird. This is a fact of nature. If you want to consider infinite sets that adding an element increases their size, then you are looking at models of the failure of the axiom of choice in which infinite Dedekind-finite sets exist. Those sets can be, and cause, far more counterintuitive results than "adding an element to an infinite set does not change the size".

For example, it is possible there is a vector space over $\Bbb F_2$ which is not finitely generated, by every proper subspace is finitely generated. It is not hard to show that such vector space is in fact a Dedekind-finite set, and that it cannot be split into two disjoint infinite sets. Other examples include a finitely splitting tree without maximal elements, but no infinite branches. And this is just the tip of the iceberg. The infinitely weird iceberg.

The second problem, is that countable by definition means that the set is in bijection with $\Bbb N$. Or at least "have the same cardinality as $\Bbb N$". In the latter case, you are forfeiting the fact that cardinality induces an equivalence relation on the universe ($|A|=|\Bbb N|$ and $|B|=|\Bbb N|$ need not imply $|A|=|B|$) which is bad; and in the former case, well... it means that you are going to measure the cardinality of sets using some structure.

But not all infinite sets have natural structure. Not only that, you can endow an "unnatural" structure on sets just as well. Given a bijection $f\colon\Bbb N\to\Bbb Q$, it defines, via a transport of structure, a dense order on $\Bbb Q$. So if we measure the cardinality of $\Bbb N$, and we chose this structure, is the cardinality of $\Bbb Q$ equal to that of $\Bbb N$? What about $\Bbb Z$?

This is why cardinality, as we have it defined now, works so well. It does not discriminate between one structure or another that we can endow our set. It just ignores all of them. And then, when there are no favorites, it is easier to get a smooth sailing theory.

So, how do you handle the fact that cardinality becomes counterintuitive with infinite sets? Well. Infinite sets are weird. With a capital E. wEird. And that is a weird way to write "weird". That's how weird they are, and even more.