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Time Share Hell

  1. Jun 23, 2003 #1
    Imagine that you and I find ourselves standing before the Pearly
    Gates awaiting our assignment to Heaven or Hell. St. Peter looks
    at us and says ''Well neither of you have been good enough to get
    into Heaven, but you haven't been bad enough to spend the rest of
    eternity in Hell either. So I'll let you share one spot in Heaven
    and one in Hell. When one of you is in Heaven, the other will be
    in Hell and you can switch back and forth. You two work out a
    schedule for switching, just make sure it's fair in the end.''
    While we are trying to agree on a schedule, I propose the
    following to you. ''You can spend your birthday in Heaven, I'll
    spend it in Hell. The other 364 days per year, I'll be in Heaven
    and you'll be in Hell. And just to show you how fair I'm being,
    you can be in Heaven on Leap Day too.''

    Note that by the end of eternity, we both spend /aleph_0 days in Heaven and the same in Hell. So from the standpoint of cardinality, my proposal is fair. Likewise, my proposal is fair with respect to ordinality. Nonetheless, I think most people would not agree to my proposal, feeeling that they were not getting as much time in Heaven as I was. Would you agree to my proposal? And what mathematical basis would you use to justify your agreement or non-agreement?
  2. jcsd
  3. Jun 23, 2003 #2


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    It's also true that the cardinality of the interal [0,1] is the same as the cardinality of the set of all real numbers. Cardinality is simply not a good way to measure "size".
  4. Jun 23, 2003 #3
    So what do you suggest as an alternative in this case?
  5. Jun 23, 2003 #4
    well, the scenerio is mathamatically true. but from the human perspective of time, it is not. humans will not experience eternity the same way. we will live in the present moment throughout eternity. therefore at no present moment (anywhere in the eternity) will the other man say to you "we have spent an equal number of days in heaven". you will always have spent more.
  6. Jun 23, 2003 #5
    But notice I've posted this in the Mathematics forum, not Philosophy. What I'm interested in are the mathematical bases for rejecting my proposal. Are there any extant mathematically sound bases for rejecting the proposal? If so, what are they?
  7. Jun 23, 2003 #6
    I don't think I would agree with this deal. If I and you agreed to timeshare a job 24x365.25 hours per year, I wouldn't like the idea of 10 days and nights on and 10 days and nights off. That would kill me. Likewise, 364 or so continous days and nights in Hell would be too awful to contemplate. On the other hand, a day and night in jail followed by a day and night free and alternating might be somewhat tolerable, even though the sentence takes twice as long to complete. Part of the pain of punishment is its continuous unrelenting nature.

    But an eschatologist might complain about St. Peter's sentence (it's supposed to be Jesus Christ, by the way). Essential to the Heaven experience is its endless exstasy and joy; essential to the Hell experience is its endless misery and hopelessness (no days or nights off). That is why christians postpone the final judgement to the end segment of eternity. Then everything is decided and done once and for all. A medieval eschatologist might insist on two future reservations for heaven, with certain finite (though possibly long) amounts of suffering in Purgatory for both persons ahead of that. But I risk getting excommunicated for heresy, so I will shut up now.
  8. Jun 23, 2003 #7
    For you I have special deal. You can have one minute of each day in Heaven instead of one day each year. And I'll still throw in all of leap day. ;)
  9. Jun 23, 2003 #8
    I think I should study and become an eschatological lawyer. ;D
  10. Jun 23, 2003 #9
    Most lawyers I've know have just been scatological.
  11. Jun 25, 2003 #10

    Your problem would seem to require some new kind of measure, or the punishment concept fails. It's rather like compounding interest on a money deposit for an eternity. What does it mean to have so much wealth that paying off all the debts in the galaxy doesn't deplete it at all?
  12. Jun 25, 2003 #11
    Well a new measure is what I'm working on. A measure for "generated sets" i.e. sets conidered as being generated by some function on the naturals or perhaps the positive reals.

    I've got the basics worked out, but am now looking at connections to things like hyperreals and surreals. What I'm trying to see is whether some existing notion already gets the intuitive answer, and if so, how it relates to the stuff I'm doing. For example, how about this "solution".

    You should reject my offer because the hyperreal representing the days I spend in heaven is greater than the hyperreal representing the days you spend in heaven. This is because any member of the sequence of days I have spent in heaven is greater than the corresponding member of your sequence.

    Mind you, I know very little about hyperreals as yet, so this may be misguided. But that's the sort of thing I'm looking for.
  13. Jun 25, 2003 #12
    ''You can spend your birthday in Heaven, I'll
    spend it in Hell. The other 364 days per year, I'll be in Heaven
    and you'll be in Hell. And just to show you how fair I'm being,
    you can be in Heaven on Leap Day too.''

    let x=eternity
    your time in heaven every year = 364x
    the other guys time in heaven = 1x and on leap years 2x
    judging by the coeffiecent on the variable, i wouldn't say that's a fair deal, unless x = 0, which it doesn't by equalling infiniti. this is just another infiniti paradox.
  14. Jun 25, 2003 #13
    Well, it is another infinity paradox. I'm not sure what you mean by "just another" though.
  15. Jun 25, 2003 #14
    I suspect that this deals with two different infinities both contained in an infinite set. Example: We have the infinite set of all real numbers. I'll take the infinite set of all non-prime numbers, and you can take the set of all prime numbers. While both infinite, mine is larger than yours. So too, I believe this is the case in your example. Though we need someone more specialized in set theory to give the concise explination.
  16. Jun 25, 2003 #15
    On what grounds do you say that the set of non-primes is greater than the set of primes? The standard answer from a set theoretic perspective would say they are the same size. Are you unfamiliar with cardinality or do you have a different theoretical basis for your answer?
  17. Jun 25, 2003 #16
    Mainly from the fact that as we go on in numbers, primes thin out. However as I said, since my knowledge is limited, it would be better for a specialist in this area to come in.
  18. Jun 25, 2003 #17
    Well I agree with your intuition. Unfortunately extant theory contradicts it as far as I'm aware. Another way of posing by question would be to ask if anyone knows of any mathematical basis for the claim that there are fewer primes than there are naturals?
    Last edited: Jun 25, 2003
  19. Jun 26, 2003 #18
    how about this....
    I can say that it's not fair because for every finite time interval of the eternity (bigger than 4 days) I spend more time in hell than you. Is that making some sense?[?] [?]
  20. Jun 26, 2003 #19
    Just another created one where you can't really prove or dissprove it because there is no end result.
  21. Jun 27, 2003 #20
    based on these posts, am i right in concluding that [oo] multipied by another number would equal another [oo]?
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