关于无穷大的技术问询：反复减一无穷次是否等价于未定义的∞-∞？

Great question—this cuts right to the tricky core of why infinity isn’t treated like a regular number in formal math. Let’s unpack this clearly, step by step:

First: ∞ - 1 = ∞ (and why this is well-defined) #

In the extended real number system (the framework where we formally work with ∞ and -∞), subtracting 1 from ∞ is a defined operation. Think of infinity as the limit of a sequence that grows without bound—like (1, 2, 3, 4, ...). If you take each term in that sequence and subtract 1, you get (0, 1, 2, 3, ...)—which still grows without bound. So we define (∞ - 1 = ∞) because the underlying limit doesn’t change. Even subtracting any finite number from ∞ leaves you with ∞.

Now: "Repeating subtraction 1 infinitely many times" #

This is where ambiguity creeps in—your phrase "infinite times" can be interpreted in two very different ways, and only one ties to the undefined (∞ - ∞):

Interpretation 1: Iterating ∞ - 1 over and over #

If you start with ∞, subtract 1 to get ∞, subtract 1 again to get ∞, and keep repeating this "infinitely many times"—every single step still leaves you with ∞. This isn’t the same as (∞ - ∞), because you’re never actually accumulating an infinite total of subtractions. Each individual subtraction only removes a finite value, and no number of finite subtractions can "reduce" infinity.

Interpretation 2: Subtracting an infinite total of 1s #

If you mean "subtract 1, infinite times" as in subtracting the total sum (1 + 1 + 1 + ...) (which equals ∞) from ∞—that’s exactly the undefined (∞ - ∞) operation. The problem here is that infinity isn’t a fixed value; when two infinities interact, their relative "speed" or "size" determines the result. For example:

• If you take (n - n) where (n → ∞), the result is 0.
• If you take (2n - n) where (n → ∞), the result is ∞.
• If you take (n - 2n) where (n → ∞), the result is -∞.
• If you take (n - (n + 5)) where (n → ∞), the result is -5.

Since there’s no single consistent outcome, (∞ - ∞) is left undefined in standard mathematics.

The key takeaway #

Your initial scenario of repeating (∞ - 1) infinitely (interpretation 1) doesn’t count as (∞ - ∞), because you’re never subtracting an infinite quantity—just repeating a finite subtraction that doesn’t alter infinity. But if you frame it as subtracting an infinite total from infinity (interpretation 2), that’s the undefined operation you’re asking about.

内容的提问来源于stack exchange，提问作者Basil Hallaq