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Topic: Re: Transfinite Subway (Read 3091 times) |
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Sir Col
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Re: Transfinite Subway
« Reply #25 on: Oct 15th, 2003, 12:47am » |
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The reason for my question was because I see a similarity with the unanswered problem of the cardinality of the real numbers. Cantor showed that they were not denumerable, and therefore greater than [smiley=aleph.gif]0, but couldn't show which degree of infinity their cardinality, c, is in. If, in the same way, the cardinality of the countable ordinals exceeds [smiley=aleph.gif]0, why does it necessarily have to be in the next degree of infinity?
Like many things in transfinite mathematics, I imagine your rigorous proof would be too difficult for me to follow, but if you can give me an indication of the outline, I'd be delighted.
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Icarus
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Re: Transfinite Subway
« Reply #26 on: Oct 15th, 2003, 3:33pm » |
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I doubt it's that hard to follow: Let x = card([omega]1). if y < x, then there must be a subset A of [omega]1 of cardinality y. A is well-ordered, so it corresponds to some ordinal [alpha], and so card([alpha]) = y. Since card([alpha]) < card([omega]1), [alpha] < [omega]1. Since [omega]1 is the least uncountable ordinal, [alpha] is countable, so y must be finite or [smiley=varaleph.gif]0. Since this holds for all cardinals y < x, x cannot be any greater than [smiley=varaleph.gif]1. But since [omega]1 is uncountable, x cannot be any less either. [therefore] card([omega]1) = [smiley=varaleph.gif]1.
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Sir Col
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Re: Transfinite Subway
« Reply #27 on: Oct 15th, 2003, 4:38pm » |
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on Oct 15th, 2003, 3:33pm, Icarus wrote:| I doubt it's that hard to follow. |
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Excellent, that made perfect sense. Thanks, Icarus!
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william wu
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Re: Transfinite Subway
« Reply #28 on: Oct 16th, 2003, 1:14am » |
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Executive Summary of the past two months:
"You don't f*ck around with the infinite."
- Charlie (Harvey Keitel), Mean Streets (Scorcese 1977)
lol
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| « Last Edit: Oct 16th, 2003, 3:30pm by william wu » |
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Barukh
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Re: Transfinite Subway
« Reply #29 on: Oct 16th, 2003, 6:36am » |
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Despite the concise William's summary, I still have a question: Does THUD&BLUNDER's argument hold for every arrangment of leaving passangers, or just in case they leave at random?
In the countable case (a variation of "Impish Pixie") there were arrangements of withdrawn balls with probabilities < 1 for the urn to be empty.
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Sir Col
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Re: Transfinite Subway
« Reply #30 on: Oct 16th, 2003, 10:57am » |
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I have a question too...
on Oct 11th, 2003, 9:56pm, THUDandBLUNDER wrote:| It now follows by transfinite induction that the subway is empty when it reaches [omega]1. |
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Regardless of the state of the train at the penultimate stop, there can never be a negative number of passengers. So the number of passengers on-baord will be [ge]0. At this stop, [smiley=aleph.gif]0 passengers embark. Hence when the train pulls into Hilbert Station, won't there be at least [smiley=aleph.gif]0 passengers on board?
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wowbagger
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Re: Transfinite Subway
« Reply #31 on: Oct 16th, 2003, 11:46am » |
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on Oct 16th, 2003, 10:57am, Sir Col wrote:| Regardless of the state of the train at the penultimate stop, there can never be a negative number of passengers. |
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See James's question and T&B's answer.
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Sir Col
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Re: Transfinite Subway
« Reply #32 on: Oct 16th, 2003, 12:01pm » |
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But this is where the ordinal position and cardinality part ways. The next ordinal after [omega] is [omega]+1, even though they have the same cardinality. So [omega]–1 must be before [omega].
Besides this (as I'm probably showing off my ignorance again), if there is no station prior to Hilbert Station the train will never arrive; it's simple induction of counting. The whole experiment is similar to Hilbert's hotel paradox. Although not put quite as 'eloquently' as Harvey Keitel, I think that Hilbert said something similar.
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Icarus
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Re: Transfinite Subway
« Reply #33 on: Oct 16th, 2003, 5:42pm » |
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on Oct 16th, 2003, 12:01pm, Sir Col wrote:| But this is where the ordinal position and cardinality part ways. The next ordinal after [omega] is [omega]+1, even though they have the same cardinality. So [omega]–1 must be before [omega]. |
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This is where the infinite in whatever form and the finite part ways: [omega] - 1 is undefined, as there is no ordinal x such that x + 1 = [omega].
Now, there is a unique ordinal x such that 1 + x = [omega]. Namely, [omega] itself (addition is not commutative for infinite ordinals). But this is uninteresting, and our notation for subtraction doesn't lend itself to bidirectional interpretation in this way. So subtraction is usually defined for ordinals on the right. Here, though, not all subtractions are possible.
As for arriving at the Hilbert Hotel, no it will not in any finite amount of time, unless it actually travels infinitely fast - instead of finite speed increasing without bound. (This is different from filling the Hotel itself or any of the other problems around here that I recall, as in those cases the infinity to be reached is [smiley=varaleph.gif]0. Thr uncountable infinity [smiley=varaleph.gif]1 is impossible to reach in even an unbounded finite fashion.)
Barukh - The argument T&B reproduced here does not make any assumptions at all about how the passengers choose to disembark. It holds for all possible cases.
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| « Last Edit: Oct 16th, 2003, 5:55pm by Icarus » |
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