this post was submitted on 02 Oct 2025
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Okay, so what’s the point of “proving” that there some “infinities” are “bigger” than others? What’s the practical application here? Because an infinite hotel with an infinite number of guests is physically impossible, so I don’t see the point.
A practical application is for example in probability theory (or anywhere that deals with measures) such as this question:
If we generate a random real number from 0 to 1, what is the probability that it is rational?
Because we know that the continuum is so much larger in a sense than the set of rationals, we can answer this confidently and say the probability is zero, even though it is theoretically possible for us to get a rational number.
Statistics deals with similar scenarios quite frequently, and without it we wouldn't have the modern scientific method.
Practical application in math tends to be like three degrees of separation and half a century removed from the math at play. In this case, all of modern mathematics is based on set theory, so it's more that this stuff allows us to do other, more practically useful math while knowing what we're talking about.
I've never been a fan of just saying "some infinities are bigger than others," to be honest. Way too easy to misunderstand and it's also kind of meaningless by itself.
The real numbers also includes the integers.
The practical consequence of this example is that the integers die regardless of what you choose.
However infinitely many people will survive if you choose the first option.
And yet there can never, and will never, be a situation where an infinite number of people are tied to a railroad track. So this thought experiment is meaningless.
It's more about set theory than the actual numbers.
Let's say you have 100 people with everyone tied up across both tracks. Heads on one track and legs on the other. Let's assume they die if the train touches any part of them, but you still need to choose between running over heads or legs.
The best choice is then legs, because there's a probability of some of them being handicapped and not having legs.