Skipping a lot of posts from a lot of people who are way the fuck smarter than me, especially in this subject area, but as someone who argues theology a lot, I can say that understanding that inf != inf has a lot of philosophical applications.
The most basic explanation is that infinity isn't a number; it literally just applies to any fucking thing that isn't finite, which is an entire class things.
The best way for me to think of it is to say that if a "God" is eternal, then it is necessarily infinite. It exists for an infinite length of time, so it's impossible to count every single instance of God. But if this God's just an eternal speck of dirt on a rock on Mars, then it's very easy to imagine how you can increase God's instances in time and space. If God is all of Mars, or all of the solar system, etc, then God is WAAAAY larger than the speck of dirt, even though neither has a finite existence.
In fact, in Christian beliefs, any soul that makes it into heaven is infinite (even though they have a beginning point, they supposedly have no end point, so like a ray, it is also not finite), but clearly this soul has a shorter existence than God himself who always was and always will be.
A misunderstanding of this leads to a lot of logical shit shows. Even a supposed genius like St Augustine really cunted this up good in the Summa Theologica by making conclusions about how since God is infinite, he has to be everything everywhere all the time, which means that {insert some crappy conclusion that I forget} and BINGO this infinite thing must exist!
Anyway, didn't mean to make this about religion, but I thought that it might be interesting to bring a non-mathematic perspective to the importance of this concept.
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