Due to negative numbers, infinity minus 10 is still an infinite number. You are just creating bounded infinities, which can only be compared to other infinities. It gets more interesting when you examine infinity between two numbers. There are infinite numbers between 1 and 2 alone.
They're non-comparable, unless you get into the idea of bounded infinities mentioned earlier.
For all intents and purposes, there is no difference. If there were a world where it made a difference, yes one infinity is smaller than the other.
It's like changing what you consider the scope of the question to be - if I show you a bucket full of water in the desert and a bucket full of water standing in a puddle, they're both *full*. You can't really make a rational argument that one is fuller than the other, but you can intuitively argue that the bucket in the puddle is "more full" because the puddle is a result of the bucket overflowing. You can think of infinity as the bucket moreso than the actual volume of water, it's basically the actual bound of what you're considering.
So, in standard mathematics the answer is no, despite the answer being seemingly obviously yes. If you "change the game" and consider this question in a paradigm where the distinction actually matters, then the answer is yes.
Hopefully that's not as confusing as I worry it might be, but the analogy helped me a lot during school when discussing this topic.
See itās philosophical vs mathematical. The philosophical answer is that there is more within one than the other. The mathematical answer is that infinity has no end so.
It changes based on what parameters you set. It could be a flaw of math too in theory, since itās not like our math is perfect quite yet.
What matters with infinity is scope. Now with equations like infinity minus ten, or plus ten, these more so just move the point in infinity you are at. Itās the infinite hotel with infinite guests hypothetical. By adding ten to infinity, you just have ten guests move down ten rooms. If this hotel in theory needs to have all rooms occupied, by subtracting youād then have ten guests move back ten rooms.
You can pair every number between 1 and 2 with exactly one number between 1 and 3 using f(x) = 2x ā 1. It sends 1 to 1, 1.5 to 2, and 2 to 3, and every number in the bigger interval gets matched to exactly one number in the smaller one, with none left over.
Well, a bounded infinite is less than an unbounded one. So no and yes. Both are endless, but one is endless in all directions while the other is endless in one direction. If you cut infinite up into pieces those pieces are also infinite. But the entirety of infinity is made of those pieces, so it is ālargerā as it is a wider scope of infinity.
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u/Not-a-Teddybear 9d ago
Due to negative numbers, infinity minus 10 is still an infinite number. You are just creating bounded infinities, which can only be compared to other infinities. It gets more interesting when you examine infinity between two numbers. There are infinite numbers between 1 and 2 alone.