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Infinity
Infinity
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Just shuffle them all along and then when the last one is kicked out, put him back in the first one.
In both scenarios, someone is always in transit and not really in a room |
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That's good. But Hilbert makes them all occupy all the rooms successfully and no one is in transit. I wrongly said the same thing as you in one of my posts.
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https://www.youtube.com/watch?v=Uj3_KqkI9Zo
It gets crazier. He even successfully occupies an infinite amount of guests by moving each guest to twice his room number. He then does the same with an infinite number of buses containing infinite number of guests and gets them all a room too. You could abuse infinity the way you want. You could debate whether it exists or not. But I don't think you can prove what Hilbert does is wrong. That is also probably why its called a paradox. |
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It’s fun maths but it’s a terrible example if you ask me.
You can move someone before, after or at the same time as the next room is vacated. It’s never explicit but it seemed he was moving people before the next room is vacated? So guests are nearly always in transit or sharing. It’s a con, no better than a ponzi scheme. |
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But can you prove it wrong? I don't think so.
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But Hilbert makes them all occupy all the rooms successfully and no one is in transit.
Hmm, what did I miss here? Are people all moving at the same time or moving into an occupied room or moving into an empty room? |
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Not sure why you're bringing time into this.
Everyone in room n moves to room n+1. This is possible because there are infinite number of rooms. So if room n exists, then room n+1 also has to exist. |
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The method raises the question of time. That’s why I said the maths is fun but the example is terrible.
If you move people into other peoples rooms then they are sharing/ in transit so it’s a con. If you move people into empty rooms, then you don’t need all the shuffling crap and you can just move your new guests into the empty rooms. It’s either flawed or over-engineered. |
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if a hotel has an infinite number of rooms then they cannot all be full.
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Hilbert presented is just as a mathematical idea. He wasn't out to build a hotel like that. It is the mathematics he wanted to argue not the practicality or engineering of building such a hotel. Of course something like this can never exist.
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Fair enough. I have no issue with the practically, my issue is that the maths and the example don’t work together. Not even slightly.
It would have been more educational to forget the hotel nonsense and just show the maths. |
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det is right, the paradox is just highlighting that infinity+1 = infinity.
infinity is a strange concept and produces practical difficulties for mathematicians in tricky areas such as understanding black holes where you have an infinitely small singularity and infinitely dense matter and yet more mass is being continuously added. I like the poem "Auguries of innocence" by William Blake which begins: To see a world in a grain of sand, And a heaven in a wild flower, Hold infinity in the palm of your hand, And eternity in an hour. |
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Nice poem.
To put this another way (and here you will see how bad my maths is...). Hilbert was trying to show that infinity +1 = a bigger infinity But his example with all the shuffling crap seemed to prove that 1+(1-1)^infinity=1 Or.....if you have an empty room and you add an occupant and remove an occupant an infinite number of times, you will have one empty room. But that has nothing to do with what he’s trying to prove, ie that infinityx2 is bigger than, infinity+1 is bigger than infinity etc |
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Hilbert was trying to show that infinity +1 = a bigger infinity
I don't think so. it is still infinity. |
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And I am not sure what he was trying to prove with this. There's no original text for this. Some english guy presented it as Hilbert's idea.
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Quite possible the hotel crap was added for fun by someone.
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You`ll need an infinite number of chamber maids too, what happens when one is off sick?
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You`ll need an infinite number of chamber maids too, what happens when one is off sick?
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“Two things are infinite: the universe and human stupidity; and I'm not sure about the universe.”
― Albert Einstein |
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I think the motivation behind the hotel example was to explain the paradox of the fact that our universe is apparently expanding. The question arises, what are we expanding into?
The hotel example tries to explain this by arguing that, since there is no edge to the infinite [open] space, there is always room (and time, I suppose) to expand. |
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Perhaps in part, but on its own, that doesnt justify the answer to the question imo. If the hotel is expanding infinitely then put your new guests in the empty rooms. No need to wake all the current guests up and move them around.
Q. How can you accommodate an extra guest in an infinite hotel that is full? A. Put them in one of the empty rooms. Or if you want, move all your guests around for no obvious to achieve the same thing |
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The hotel example is not trying to prove that the universe is infinitely expanding. An examination of the night sky via red shift and Cepeid variable stars is enough to demonstrate that. Instead, it's trying to explain how what we are seeing can be logically possible.
You can do it neatly just by maths and numbers. But that can be hard to grasp, so the numbers are converted into the physical example of an endless number of hotel rooms. Note that this is NUMBERS of hotel rooms, not an infinite lump of hotel room material, which already had vacant individual hotel rooms. It's an illustration of the maths and numbers which underpin calculations dealing with expansion of an infinite set. I do sympathize with what you're saying. Quite frankly, anyone who says they accept all these concepts and understand the logic without any problems hasn't thought about it hard enough. |
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If you properly ( not think about infinity , or indeed it is even better to think about nothiness and what that means ( not everyone has the facility to do this )
Then you will be a funny feeling in your stomach that’s all you need to know How small life is and how big nothiness |
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* if you properly think about infinity , and better still nothiness
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If you don’t get that funny feeling in your stomach ?
then you are not capable of understanding , not everyone is |
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You see you have a future , only when you have a present
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From the comfort of my armchair, having given it 30 seconds of thought, it strikes me that these professors and award winning mathematicians would do well to keep the real universe of matter and things away from the fun but fantastical world of numbers, so they don’t get their silly little heads all confused, bless them. If one accepts that..
1. Numbers are abstract and only possible because we have tangible things in the universe to count and measure. Anything beyond countable or measurable is purely abstract and hypothetical and has very little use in the real world. 2. The only thing that is close to being tangible and infinite is nothing. It is defined as infinite because it is uncountable, immeasurably boundless, not that it is endlessly measurable or countable. Everything else in the universe is countable and measurable. ..then using hotels to explain fantastical (albeit fun) made up numbers is about as useful as having a mechanic give you a tarot card reading. |
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There are plenty of uses "in the real world" for purely abstract and hypothetical numbers. The square root of minus one is the obvious example. It doesn't exist. Yet without it you wouldn't be posting on here. The whole field of electronics would simply never have arisen without the use of imaginary numbers to calculate circuit theories or alternating currents.
The uses of infinity as a mathematical symbol? Without it Einstein would not have been able to calculate that the mass of an object becomes infinite as it approaches the speed of light. Without that, no theory of relativity. And without that, no GPS or Sky Sports (satellites would never be where they were intended to be). More generally, all sorts of things are valid that have no basis "in the real world". Music, for instance. It's unique among the arts, in that it doesn't depend on "the real world" one bit for inspiration. |
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thought electronics needed only one and zero
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If we were to count the numbers of stars in the sky it may seem infinite, but that's IMPOSSIBLE
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Hmm I will look into some of those from my comfortable armchair, thanks SFB.
First thought is that music is completely countable. As are satellite positions. Theory yes, but theory subsequently proved to be actual and measurable. But undeniably, theoretical maths helping the real world. Not infinite numbers though. |
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Bob Dylan when starting out as a young man was asked what kind of musician he was ,
he thought about it , and then replied a 'mathematical musician' |
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what kind of music he made ? he answered...
mathematical music |
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also of course one of his most intriguing lyrics in one of his greatest songs
' ah but I was so much older then, im younger than that now' |
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That's the spirit. Much better than Roger Waters out of Pink Floyd:
So you run and you run to catch up with the sun but it's sinking Racing around to come up behind you again. The sun is the same in a relative way but you're older, Shorter of breath and one day closer to death. Miserable git. |
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beautiful song
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its beginning to look as though the concept of infinity is on very shaky ground
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Why? What changed?
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nothing
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What's the largest natural number?
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