Here’s a link to a clear and well-written explanation that’s probably more layman-friendly than the Wikipedia pages.
(I just now found it; if I had known about it, I would have linked to it earlier. It’s a pretty good explanation at the popular level.)
Thank you very much. I’m still trying to wrap my brain around it, and that will take quite some time.
For the benefit of others who (like me) couldn’t figure out your perspective, I am now going to attempt a very dumbed-down version of what you’ve been saying. I hope I’m not far off the mark…
The main idea is that it is possible to have two different things which are both infinitely large, yet somehow it is clear that one of those infinities is larger than the other. (I am tempted to give an example, but it would probably be wrong. Those who want/need examples are encouraged to read Thudlow’s links.)
Okay, if one is willing to accept that, then here’s the next step.
The number of things which can happen is infinitely large. The number of instants in eternity is also infinitely large. Let’s say that the first infinity is larger than the second infinity. (If you don’t accept that, okay, but you should have a better reason than “Infinity is infinity!”, because we have already accepted the idea that one infinity can be larger than another infinity.)
Now, if one accepts both of the above admittedly-whacko ideas (but then, isn’t everything about infinity quite whacko?), then we are forced to conclude that the OP can be answered in the negative: There are some things which CAN happen, but even after infinite time passes, they still will not have happened.
The logic is actually simple. It’s not that “they couldn’t happen”. After all, we’ve already defined them as things which CAN happen. But, unfortunately, as is the way of the world so often, we simply ran out of time.
Get it? If the infinite number of things which can happen is larger than the infinite number of instants in eternity, then even if eternity passes, there will still be some things that have not happened YET, simply because not enough time has passed. And waiting another eternity - or infinite eternities - is not going to change that.
I don’t necessarily agree with the above – especially the part about there being more events than instants - but I admit that it has a certain logic to it. And in math, that’s all that matters.
The number of possible events and the number of instants are both countable infinities of the same order. You can’t casually declare that one is bigger than the other.
You also can’t say that mathematical things have a certain logic to them by using words. You can only do so by using math.You don’t yet understand the math.
I appreciate and sympathize that you’re trying to understand infinities but this isn’t the way to go about it.
This is false if you’re considering countable sequences of reals in the interval [0, 1].
Yeah, I thought Keeve’s last post was alright. It also exactly captures what we were saying about the dartboard: if you think of the moments at which things can happen as the countably many crack-of-dawns from here on out to eternity, and the possible events as the continuum of points on the board to be hit, then there’s too many of the latter to be entirely covered by the former, even though each of the latter is, by stipulation, possible.
Also, Exapno Mapcase’s line “The number of possible events and the number of instants are both countable infinities of the same order” is falsified even without going to countable sequences of reals; even if events are just single reals, they aren’t countably infinite.
ignore this post
OK, then I apologize. But how are these sets not capable of being put one-to-one with the rationals rather than the reals? I don’t see possible events as being a continuum constituting a higher infinity. There have been several posts already doubting this as well. You seem to be making a definitional connection between events and the total number space of the dartboard and I’m saying that this is not true. I understand the dartboard difference; I don’t understand why you’re extending it to possible events.
Well, for each point on the dartboard, one possible event, we could say, is the event of the dart hitting that point.
I mean, it’s up to you to stipulate what you want to consider to be the possible events. But that’s certainly one setup you could analyze.