# Time Share Hell

#### drnihili

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?

#### HallsofIvy

Homework Helper
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".

#### drnihili

So what do you suggest as an alternative in this case?

#### maximus

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.

#### drnihili

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?

#### quartodeciman

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.

#### drnihili

Originally posted by quartodeciman
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.
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. ;)

#### quartodeciman

I think I should study and become an eschatological lawyer. ;D

#### drnihili

Most lawyers I've know have just been scatological.

#### quartodeciman

:)

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?

#### drnihili

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.

#### JetBlckNewYr03

''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.

#### drnihili

Well, it is another infinity paradox. I'm not sure what you mean by "just another" though.

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.

#### drnihili

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.
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?

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.

#### drnihili

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?

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#### Guybrush Threepwood

Originally posted by drnihili

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?
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?[?] [?]

#### JetBlckNewYr03

Well, it is another infinity paradox. I'm not sure what you mean by "just another" though.
Just another created one where you can't really prove or dissprove it because there is no end result.

#### maximus

based on these posts, am i right in concluding that [oo] multipied by another number would equal another [oo]?

#### Guybrush Threepwood

Originally posted by maximus
based on these posts, am i right in concluding that [oo] multipied by another number would equal another [oo]?
yep. And [oo] divided by some arbitrary finite number still equals [oo]
(but you must look for the sigh though....)

#### drnihili

It depends on what number system you're working with.

Infinity isn't part of the Reals.

For the extended Reals you are correct.

For hyperreals the answer depends on which infinity you pick and what number you divide by, but the answer is well defined.

I'm not sure about surreals, though I suspect the act like the hyperreals.

If you're talking about Transfinite Cardinals, then division isn't well defined. Intuitively though, an infinite cardinal divided by a smaller cardinal yields the original. An infinite cardinal divided by itself could be anything from 1 to the original cardinal.

For ordinals you get different answers.

#### Antiproton

First, I would argue that the number of days left to infinity are a countably infinite set...you can map the number of days left to infinity one-to-one with the positive integers without incident. (These are days, mind you, not 'time', which is a whole different ball game) Just a point of fact.

Also, you have to argue order. Is there a natural well-ordering to 'time' in heaven and hell? Right now I'm stumped to come up with an argument for or against it (though I'll probably be thinking about it for the next month...thanks guys ). If there is no ordered arrangement of days in heaven/hell, as one might expect from such an ethereal existance (where are those philosphers when you need them), the deal could be perfectly fair. Your birthday could come every other 'Earth day'. Or You could have a string of 50 birthdays before another non-birthday. I'd say thats a pretty sweet deal.

Just throwing that out there, lets see what we can make of it.
Cheers.

#### drnihili

Yes, I purposely put the question in terms of days so we didn't have to get into uncountable infinities.

I'm assuming that days in heaven are ordered in the same way as on earth, otherwise talk of years makes no sense, in fact it's not clear how we would determine birthdays. And yes, different orderings give different intuitive answers since they demonstrate different bijections between the two sets. If the ordering is 50 birthdays and then 1 non-birthday, you seem to come out way ahead. But if the ordering is 364 non-birthdays and then 1 birthday, I do.

I've been puzzling over this stuff for years, glad to have the company. ;)

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