# Is infinity truly infinite if it has something else in it?

• B
Is infinity truly infinite if it has something else in it?
Put differently, say there's an infinite volume of water that has some rocks in it, is the volume of water truly infinite? Though there's a place where there's no water?

Orodruin
Staff Emeritus
Homework Helper
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Yes, a subset of an infinite set can also be infinite. For example, there is an infinite number of odd numbers even though you have to remove another infinite set (the even numbers) from the natural numbers to get the odd numbers. This leads to things such as Hilbert's hotel.

• Ibix
Ibix
What does "truly infinite" mean?

Certainly you can have infinite regions with boundaries - an infinite volume minus a finite volume is still infinite. I presume, since you're posting this in cosmology, that you're thinking of space or spacetime. This doesn't have any holes in it anyway, so far as we are aware.

Then when I use the term "infinite volume of water", this shouldn't mean there's water everywhere. Got it.

fresh_42
Mentor
Is infinity truly infinite if it has something else in it?
Put differently, say there's an infinite volume of water that has some rocks in it, is the volume of water truly infinite? Though there's a place where there's no water?
Infinity is a mathematical concept. If you can put rocks in it, you speak about something which exists within the universe, and then - in case it should be infinite - all rocks are already in it.

• Math_QED