Are some infinities really bigger than others?

Question

I'm not a mathematician, but I recently read a thread about how some infinities are bigger than others. The argument put forward was that of mapping pairs of numbers from reals to naturals. There is also the argument of:

How many numbers are there between and including 1 and 2? 
An infinite set.

How many different numbers are there between and including 1 and 3?
An infinite set.

Since the number 3 cannot be counted in the set between 1 and 2, 
the infinite set which includes the numbers between and including 1 and 3 
must be a greater infinity than the set of numbers between and including 1 and 2.

I don't disagree with any of this. The way this is defined with cardinality of sets and whatnot, I can see how some "infinite" sets have more items than another.

But I think this is a bit of a linguistic trick. Maybe it's because we don't have enough words to express this concept or the existing words aren't specialized enough.

Isn't area/volume a better way of thinking about this? When someone says "one infinity is larger than another", then I believe most people "larger" consider larger to be referring to size.

That is, the set of reals may have a larger cardinality than the set of naturals, but the total "size" of their infinities are the same.

For example, let's say a girl and a boy are each given two apples. The boy cuts his apple in half, such that he now has "3 pieces of apples". I do not consider his share of apples to be larger than the girl's. I'm not sure why mathematicians are using cardinality rather than total size when saying "one infinity is larger than another". I think it would be better to say "some sets have a larger cardinality than others". You could say he has more pieces of apple, sure, but they both have the same volume of total apple.

I believe this is akin to what people are doing with numbers. Just because you split the number up further doesn't mean you actually have any more numbers. The size (total area, or volume, or whatever) of your set is still the same.

Similarly, pi is not "infinite". It's between 2 and 4.

And is infinity even infinitely large? Couldn't you argue it itself has a limit to its size? If what we're concerned with is area/total volume, then perhaps even infinity has a limit? I suppose it would depend on whether or not the universe is finitely sized. If it is, then the total volume that could be expressed in reality (not on paper) would be bounded by the volume of the universe.

We could use smaller units, but eventually are we not bounded by planck length being the smallest unit we could use? So the most we would be able to express would be (volume of the universe in cubic meters) / (planck length in meters), which is 2.2×10^115 m^2 (square meters) in size?

Edit: Okay, I thought of a better way of phrasing this. There are multiple definitions of size. For example, you could be referring to cardinality, or maybe total area. This is why I believe the statement "some infinities are bigger than others" is wrong. Because if "bigger" in this context was referring to "volume", then it is not the case. The statement itself is too general and requires qualifiers (such as explicitly denoting that we're referring to cardinality) to be accurate. I think this is why it sounds more profound than it actually is (because the statement is fallaciously implying some infinities are larger than others in the general sense, and not just in the cardinal sense)

Cardinality is a subset of the concept of size. I believe this is why laypeople like myself often have difficulty accepting this statement (some infinities are larger than others), because it's assuming that we are referring to cardinality (which most people don't have that as their default interpretation of size). I would argue the more common interpretation is that of volume (as in my apple example, where no one would argue the boy has a bigger share of apple than the girl).

It is now common to hear people just throw out "some infinities are larger than others" in the general sense (just look at how many popular YouTube videos exclaim just this without any other qualifiers). Then when lay-people contest it the claimant retreat back to "oh, I just mean that some sets have a larger cardinality than others". Oh, well now it is much less profound/interesting and more-agreeable and boring. It seems like a mathematical motte-and-bailey!

Answer 1

Mathematicians use many definitions of size, and it's specious reasoning to suppose that there's only one way to measure size.

In terms of cardinality, $[1,2]$ and $[1,3]$ are exactly the same, since $f(x):=2x-1$ is a bijection from the first to second. It says that as sets, the two have equivalent cardinality. This is useful if you're asking which set has more/less singleton elements.

In terms of length, $[1,3]$ is longer than $[1,2]$ in the sense of measure. However, size and measure are not always intuitively related, as can be seen for example by studying Cantor sets.

There are also comparative versions of infinity based on finite "filtrations," for example, the ratio of lengths of the family of sets $[1,x]$ and $[1,x^2]$ as: $\lim_{x\rightarrow\infty} \frac{x^2}{x}=\infty$ despite both numerator and denominator tending to infinity.

Answer 2

No general definition of the "size" of a set, or of two sets being "equal in size", has gained much traction, except the one using cardinalities. If you can suggest an alternative equivalence relation with a nice ordering of its equivalence classes to serve the purpose, we can discuss it.

In some narrow contexts you can use something different. For example, all countable sets of real numbers, and also some uncountable ones, have zero "measure", while still others may have positive, infinite or undefined measure. But in general, there's no more benefit to characterising a set of cardinality $2^{\aleph_0}$ as being "the same size" as one of cardinality $\aleph_0$ than to try the same thing with $2^5$ vs $5$.

Maybe it's because we don't have enough words to express this concept or the existing words aren't specialized enough.

Just the opposite. When our terminology was less specialised, we just lumped together "infinite" sets. Now, however, we can be much more specific about their sizes.