If you subtract a finite number from an infinity, does the infinity become smaller?

Question

I'm working with 10th grade math here, so I hope I can get an explanation that I understand. I was reading The Kalam Cosmological Argument, by William Lane Craig, and it sort of opens on this section about a Medieval Muslim philosopher who came up with several premises that together allegedly show that actual infinities do NOT exist, and therefore the Universe must have had a beginning. All of it is rather tedious; the above link directs to the page if you are interested.

The bit that I am interested in is where it says:

For if one has an infinite body and removes from it a body of finite magnitude, then the remainder will be either a finite or infinite magnitude. If it is finite, then when the finite body that was taken from it is added back to it again, the result would have to be a finite magnitude (principle five), which is self-contradictory, since before the finite body was removed, it was infinite. On the other hand, if it remains infinite when the finite body is removed, then when the finite body is added back again, the result will be either greater than or equal to what it was before the addition.

Now, I was wondering how adding and subtracting from infinities work. My intuition is that

∞ - x = ∞ + x

so long as x is a finite number. Meaning, adding or subtracting a finite number to an infinity does not change its value, but I vaguely remember a YouTube video that talked about different kinds of infinities, such as ∞! but it was all well above my head.

So the question is, does subtracting finite numbers from an infinity make it smaller? Is this even a problem I can understand with rudimentary math skills? Thanks.

Answer 1

So we can't really treat infinity like a number because certain rules would break. It may be meaningful to say something like $1 +\infty=\infty$ but it becomes less clear when we write something like $\infty - \infty$. Sometimes a token for $\pm \infty$ will be used to represent sequences that are eventually unbounded in one direction or the othe but they should be used with caution as treating them as numbers can lead to nonsensical results.

With that in mind the author has essentially captured what it means to be an infinite set by discussing removing some elements and still having infinitely many left. Said another way a set $A$ is infinite if and only if there exists a proper subset $B \subset A$ and a bijection $f:B \rightarrow A$. Such a set is said to be Dedekind-infinite. This allows us to definite infinite sets without appealing to finite ones.

Answer 2

In the traditional Real number system, $\infty$ is not a number. But there are contexts where $\infty$ is a specific number that can be arithmetically manipulated.

The extended Reals, for example, introduce $\infty$ and $-\infty$ as distinct numbers, with the following rules:

• $\infty + x = \infty$ for all finite $x$
• $x - \infty = -\infty$ for all finite $x$

There are some other rules which exist to handle things like multiplication and division, but you asked specifically about subtraction. The takeaway here is that your intuition is correct; $\infty$ remains $\infty$ even after subtracting a finite number from it.

I'd say in any context I've seen in which there is a notion of an "infinitely large number," your intuition holds true. But all of this depends on how we've defined things. A lot of Mathematics starts first with some definitions, and then we discover the consequences of those definitions.