Subtracting two infinities

Question

I am Curious if the following is mathematically correct:

Let $a$ be the infinite set of all nonnegative integers $0,1,2,3...$.

I take from $a$ some of its elements, say integers $10$, $11$, and $12$ only.

So now we have a new set $a'$ that is the infinite set of all nonnegative integers $0,1,2,3...$ except for $10$, $11$, and $12$.

If I subtract: $a-a'$ the result is a set comprised of $10$, $11$, $12$ only. Is this correct? Can one subtract infinities like this?

If yes, does this mean that $a \gt a'$(despite that both are infinite)?

Answer 1

The set of positive integers is denoted $\mathbb{Z}^+=\{1,2,3,\dots\}$ and has cardinality $\aleph_0$. The set of positive integers excluding $10,11,12$ also has cardinality $\aleph_0$. The difference between the sets is indeed $\{10,11,12\}$ and its cardinality is $3$. To answer your question, you should read more about cardinal number arithmetic.

Answer 2

Subtraction between infinite cardinals cannot be well-defined. This is a good example why.

We know that $|\Bbb N|$ and $|\Bbb Z|$ are both of the same cardinality, but so is $|\Bbb{Z\setminus N}|$. On the other hand, $|\Bbb N\setminus\{k\in\Bbb N\mid k>2\}|=3$ (zero is a natural number here).

So we have $\aleph_0-\aleph_0=\aleph_0$ but at the same time $\aleph_0-\aleph_0=3$ and we can easily engineer this to be any other result between $0$ and $\aleph_0$ as well. This is indeterminate much like $\infty-\infty$ is an indeterminate form in calculus, although $\infty$ and $\aleph_0$ are not directly related (these are two different, and mostly incompatible, notions of infinity).

The reason is that for infinite sets being a proper subset does not imply having strictly smaller cardinality like it does with finite sets. Our intuition from finite mathematics usually does not apply to infinite objects.

On the other hand, subtracting two concrete sets is perfectly doable and legitimate mathematically. But we cannot conclude anything on the cardinality of the result, unless we know what are the specific sets (or at least more than just "two sets", which is the most general notion).

Read more about this:

1. Why do the rationals, integers and naturals all have the same cardinality?
2. Is there a way to define the "size" of an infinite set that takes into account "intuitive" differences between sets?
3. Why the principle of counting does not match with our common sense
4. Cardinality of subtraction of sets
5. Cardinal number subtraction (See my answer, as it is related to the comment below by user14111.)