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 Originally Posted by wufwugy
it's not reasonable to think that something with infinite speed and 3d travel facilities can't reach every point in space at any given time?
I don't see how any macroscopic thing can be in all the places at one time. Even with particles, we're talking about their wave function being non-zero over any volume, but it's so damn near to zero over most of the universe. "Most" being an understatement of universal proportions. The probability of finding an electron which is bound to an atom further than 1 nm from that nucleus is nearly infinitesimal. It drops off exponentially. While it is non-zero everywhere, it is ridiculously small even a few atomic radii in distance from the parent nucleus.
Originally Posted by wufwugy
so for something to be infinite, it has to be infinite at any one point in time?
I'm saying the balloon is finite in surface area at the beginning of our little thought experiment. Its rate of expansion is always finite. Therefore, it can only be of finite surface area after any finite amount of time passes.
I would say that the only way this balloon can have infinite surface area is if an infinite amount of time passes. When I use infinite here, I mean the non-number version if infinity which is a concept of unendingness.
I'm basically calling BS on your notion that the balloon is infinite at any "real" time, but not on your notion that the idea of infinity is not completely discarded from this balloon's state in some arbitrarily distant future.
Originally Posted by wufwugy
it's intuitive. what i think i dont understand is this: there are an infinite many numbers between 0 and 1. this suggests to me that something that travels to each of those numbers can never reach them all. from his perspective, he is traveling infinitely and reaching new places. if so, why can't we say that the distance he is traveling is infinite?
... but the distance between the numbers is infinitesimal.
So, sure, you can say he's traveled infinite distance, but does it hold significant meaning?
There are many uses of infinity. Even mathematically, we acknowledge that not all infinities are equal. Sometimes we treat infinity like a number, especially when doing integrals, but there are so many other examples.
The various uses of infinities are appropriate in their own contexts.
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