At the risk of repeating some of what Alphanumeric said (please read his useful post)... It's not really a number, but we can fudge treating it like a number in some circumstances. When we do that, we're usually implicitly dealing with some kind of limit process. For example, if I ask what's 12 divided by infinity, the answer seems to be zero. But really what we're saying is that the value 12/n becomes smaller and smaller as n gets larger and larger, until it's effectively zero when n gets very large. In transfinite arithmetic there's actually a distinction between $1+\infty$ and $\infty+1$. The answer in the first case is just $\infty$, whereas in the second case we're talking about the next number greater than whichever '$\infty$' we're talking about. In our more familiar mathematical landscape, adding 1 to infinity essentially doesn't change it, because 1 is titchy compared to infinity. All of them! (Well, discounting transfinite arithmetic.) Actually, no. The set of all positive even numbers is a set with the same number of elements as the set of all even numbers. That is, the sets have the same cardinality. The reason is that we can put the two sets into one-to-one correspondence, with each element from one set matching exactly one element from the other set. Arguably, it has an end, but you can't get there from here. You can never, for example, count up to infinity, even in principle, because every finite number is less than infinite. On the other hand, maybe you could count to infinity if you have infinite time available, because you could then work through the infinite number of finite numbers to arrive at the end. Lots of functions in maths include infinities, but again we usually end up importing some kind of limit process when we talk about them. For example, take the function $y=1/x$. Let $x$ approach zero from above. Clearly, $y$ gets bigger and bigger, until, in a rough sense, it is infinite when $x=0$. On the other hand, we might choose to ignore the problem simply saying that the function is not really defined at $x=0$.
Interesting. Had to look up cardinality. "Obvious" isn't always obvious when dealing with the very large. Or the very small for that matter.
Wow, thank you James. I’ll come back later to discuss, I have a few questions. Math is not my thing, but you and others have helped me better understand.
For example, here's one way to match up the two sets: Set 1: Positive even numbers: 2,4,6,8,10,.... Set 2: All even numbers: 0,2,-2,4,-4,6,-6,... Every element in set 1 has one and only one corresponding element in set 2. For example, the number "8" in set 1 corresponds to the number "4" in set 2. Due to the one-to-one correspondence, both sets must have the same size.
Do you mean infinite in size or infinite in age? Our universe has a finite age, so far: around 13.7 billion years. As for its size, we only know about what we can see. The visible universe has a finite size.* However, it is theoretically possible that the universe as a whole is actually spatially infinite. Personally, I don't like that idea - I'm suspicious of actual infinities as opposed to mathematical ones - but I can't disprove it. --- * The furthest distance we can see is about 46.5 billion light years. You might think that it should be more like 13.7 billion light years, because light has only had time to travel through space for 13.7 billion years at the most. But the universe has expanded since the big bang, so the earliest light reaching us now has travelled from further away than 13.7 billion light years.
Well, as the universe ages, then the size of it will expand, so perhaps both, but more so regarding size.
Infinity can never be reached. If you imagine a clock that runs one-minute fast: the "real clock" can never catch it. Please Register or Log in to view the hidden image!
Ahhh. Okay. I get it! Please Register or Log in to view the hidden image! Infinity can never be reached. I've been mistaken in my thinking that it was a mathematical "end point" that we could eventually come to.
Yes. It continues "forever." +1+1+1... Of course the clock is running at the present time for us to begin with, and must be "set-fast" for infinity to exist. Anything that will remain true, forever. Please Register or Log in to view the hidden image!
Ah, but that is the question. What fundamental condition can remain true forever and be causal to our BB and the formation of the universe, or maybe even a multiverse? Infinite time doesn't mean anything without a starting point. Infinite time without a dynamic change does not really exist other than as a permittive condition and any change is simultaneously the measurable beginning of time. For our Universe this beginning was the BB and our universe is 13.7 billion years old and its size still expanding into the permittive vacuum. Now imagine a prior Universe with a BB beginning and continuously expanding until it was so far stretched that for all intents and purposes it had reached maximun entropy.......until another BB creates a new universe and spacetime....etc. etc. For an infinite number of iterations, dynamical starts and a slow entropy until stasis is reached ...waiting for the next BB. String theory is really intriguing to me. I like the fundamental aspects of fields and the wave function.
The Earth is protected by a layer of ozone, without which life on Earth could not exist. If I had a time-machine I would reverse time, and inform mankind earlier of the damage that is being done. Perhaps this is proof that I will never invent such a contraption, and that I am destined to age and wither.
"Perhaps this is proof...". What is "this"? If you had a time-machine and reversed time too much, Earth would not have an ozone layer either. Earth did not always have an ozone layer.
...But I haven't warned humanity and prevented global warming. Please Register or Log in to view the hidden image!
Actually the ozone layer is now in pretty good health. https://www.bbc.com/news/av/science...-layer-shows-signs-of-recovery-say-scientists This is as expected. https://en.wikipedia.org/wiki/Ozone_layer It takes on average about 35 - 50 years for a CFC molecule to circulate in and out of the lower stratosphere Please Register or Log in to view the hidden image! https://en.wikipedia.org/wiki/Ozone_layer
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I saw on a news broadcast that simply reducing our carbon footprint will not undo the damage to the ozone layer.
