How to understand mathematic concepts like cardinal numbers as opposed to only remembering set of rules and formulas for someone thinking spatially?

Question

I have had a lot of math courses in my life, some of which i had to repeat due to... lifestyle choices.

I observed that a lot of topics such as linear algebra are vastly easier to understand perhaps thanks to geometry analogies or outright geometry based definitions. Perhaps i have a mind oriented for spatial reasoning.

Concepts such as cardinal numbers however while i can learn by heart all the rules and memorize it i can't seem to figure some kind of 3d analogy that would let me draw my own conclusions. How is it possible for two infinite sets like natural and integer numbers to have the same amount of numbers seems beyond my imagination and even when shown ordering 0 -1 1 2 -2 etc my brain remains very against the idea. There must be more of those numbers here they are double the amount! How is it possible? It seems to not make any sense

Also do skilled mathematics understand every concept or do they just use mathematic proofs and treat the resulting concepts as truth without trying to somehow grasp it imaginatively? Navigating the labyrinth of mathematics proof by proof?

Answer 1

The question is what does it mean for two sets to have "the same size"? Naively, this is very intuitive for finite sets, but not so much for infinite sets. Perhaps it seems obvious that the even natural numbers are a smaller set than the natural numbers since one is strictly included in the other, but how do we compare the size of the set of natural numbers to the set of all finite strings of letters? Or the size of the set of real numbers to the set of all increasing functions from $\mathbb N\to \mathbb R$? Or the size the set of all topologies on the three-sphere to the set of all finite multigraphs?

The point of those random examples is that using strict inclusion to compare the size of infinite sets is a pretty limited notion, since we won't be able to compare the size of mathematical objects whose elements have nothing to do with one another. Also note that it doesn't really even work for finite sets if we take it literally. How do we know $\{1,3\}$ is smaller than $\{10,11,12\}$?

Regardless, if we're going to have a notion of size of any sort, we need a precise definition. One robust, intuitive, but not very useful approach would be to simply say any two infinite sets have the same size ("infinity"). However, the decision that the mathematical community made was to instead say that two sets have the same size if there is a bijection between them. The first important thing to realize is that this is a definition. We can't prove in any meaningful sense that it corresponds to exactly to the intuitive notion of "having the same size" because this intuitive notion is not precise. We make it precise by defining it precisely, and this is how we have chosen to do that.

Perhaps it might be best to think of it this way: we have a sharp idea of what "size" is for finite sets... we are generalizing that notion to the infinite by using the idea of bijections.

So, why was this choice made, and is it any good? First off, hopefully it's clear that the definition works exactly as in our intuition tells us it should in the case of finite sets. On the other hand, there are many rules about size of finite sets that don't generalize to infinite sets, like your example that a strict proper subset can still have the same size. There is a reason why this (commonly know as Hilbert's Hotel) is often called a "paradox"... it is a feature of the definition of equinumerousness as applied to infinite sets that doesn't conform to our intuitions from finite sets.

In practice, mathematicians who are trying to learn an established field generally just build their intuitions off of the definitions rather than trying to make a definition fit whatever preconception they have about what the term should mean (which usually isn't an issue cause usually you don't have a strong opinion about what something should mean). When I think about two sets being the same size, I literally just think about there being a bijection between them. The way I visualize this is simply to imagine the two sets being paired off against each other. So visualizing your example is quite simple from this perspective: just list off the naturals 0,1,2,3..., then list off your enumeration for the integers 0,-1,1,-2,2,... below that, and draw a vertical line connecting 0 to 0, 1 to -1, 2, to 1, etc. That picture is what it means for them to have the same size.

I haven't done much in the way of convincing that the defintion is good so I'll wrap it up with a couple thoughts to that end:

1. The idea of pairing two sets off is intuitively compelling on its own. If you thought about comparing the size of the set of natural numbers to the size of the set of all strings of text you might come up with this yourself: just imagine any string of letters as a number in base 26 (or whatever size of alphabet). This encoding is just a bijection between the sets, and it seems natural to consider these sets to be the same size since one is a code for the other. In this sense it's maybe not so hard to reconcile Hilbert's Hotel either... one can simply think of an even number as a code for that number divided by two. On the other hand one might imagine a silly world where we need to convince an extreme skeptic that $\{1,2\}$ is smaller than $\{10,11,12\}$... one way you might try is convincing them that no matter how you try to pair them off, one of the element of the second set is left over.
2. This definition has been extremely fruitful, to say the least. The idea of cardinality is used frequently in almost every field of mathematics. A couple examples: it's easy to prove there are transcendental numbers, and Lebesgue-measurable sets that aren't Borel just using cardinality arguments (e.g. that the real numbers are a strictly larger set than the algebraic numbers). It's also worth mentioning that an entire fascinating (imo) field of mathematics grew out of this notion.