# Another infinity question

**URL:** <https://boards.straightdope.com/t/another-infinity-question/180190>\
**Category:** Factual Questions\
**Created:** [June 6, 2003, 10:46pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190 "2003-06-06T22:46:20Z")\
**Posts on this page:** 19\
**Page:** 4

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [June 10, 2003, 2:18pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/61 "2003-06-10T14:18:04Z")

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Actually, the demon only switches the light on and off a countable number of times in this example.

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**Author:** ![zwaldd](https://avatars.discourse-cdn.com/v4/letter/z/a9adbd/32.png) [@zwaldd](https://boards.straightdope.com/u/zwaldd)\
**Post date:** [June 10, 2003, 2:55pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/62 "2003-06-10T14:55:27Z")

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> [@](#):
>
> \*Originally posted by ultrafilter \*  
> \*\*Actually, the demon only switches the light on and off a countable number of times in this example. \*\*

Which premise specifies that? I thought he always hits the switch halfway to the two minute mark.

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [June 10, 2003, 4:06pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/63 "2003-06-10T16:06:55Z")

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I’m not sure exactly what you’re asking, **zwaldd** , but the demon hits the switch for the nth time at time t = sum(1/2[sup]k[/sup], 0 \< k \< n - 1). So as close as you get to two minutes, the demon has only hit the switch a finite number of times. Does that make sense?

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**Author:** ![green\_dragon\_1](https://avatars.discourse-cdn.com/v4/letter/g/13edae/32.png) [@green\_dragon\_1](https://boards.straightdope.com/u/green_dragon_1)\
**Post date:** [June 10, 2003, 4:08pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/64 "2003-06-10T16:08:36Z")

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It’s a countable process embedded in an uncountable field; if the field was countable, there _would_ be a ‘most recent event’ and we wouldn’t still be here rattling on about it 🙂

**Shade** hit the nail bang on the head - we cannot determine what happens at 2 minutes because there is no most recent event from which to draw our conclusion. Say the lamp is on at 1min 59.99999999999 seconds - the next time he hits the switch it will be off, and then on, and then on…there is no point where “if he hits the switch _one more time_ then he will get to two minutes”, and so, for the last time, **we cannot determine what happens at 2 minutes!**

And **zwaldd** , I should have been more specific. In the context of the problem, his speed tends to infinity as time tends to 2 minutes (as we don’t know what happens AT two minutes we can’t specify his speed there; for all we know, he takes a coffee break! Mmmmm, coffee…)

His speed is finite and calculable at each stage of the problem, but it tends asymptotically to infinity. The ‘to do infinite things in a finite time you need to move at an infinite speed’ comment was just supposed to illustrate my point, but I admit that wasn’t very clear - thanks for the oppourtunity to clear that up.

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [June 10, 2003, 4:25pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/65 "2003-06-10T16:25:59Z")

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I’m not sure exactly what you’re asking, **zwaldd** , but the demon hits the switch for the nth time at time t = sum(1/2[sup]k[/sup], 0 \< k \< n - 1). So as close as you get to two minutes, the demon has only hit the switch a finite number of times. Does that make sense?

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**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\
**Post date:** [June 10, 2003, 4:27pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/66 "2003-06-10T16:27:27Z")

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> [@](#):
>
> The next biggest infinity is the ‘continuum of the reals’ and is just the size of the set of all real numbers.

Actually, that’s a larger cardinality than that of the natural numbers, but not necessarily the next largest. The next largest is omega-1 (or aleph-1); in ZFC, it is consistent that the reals have cardinality omega-1 (the continuum hypothesis), but it’s also consistent for the reals to have just about any larger cardinality, as well. The issue is independent of ZFC.

> [@](#):
>
> If someone is finished performing actions in two minutes, those actions must be countable. True or false?

That’s somewhat debatable, depending on what you require. For something like the OP, we define the “times of the events” by taking a continuous, order preserving function f:omega -\> [0,2] (omega = the natural numbers). So f(1) is the time of the first event, f(2) the time of the second, and so on.

If we require a similar setup for an uncountable number of events, such that the times of the events are well ordered, and are given by some continuous, order preserving function g:A -\> [0,2], then it can’t be done. If we take the smallest uncountable ordinal, omega-1, then there is no continuous, order preserving function g:omega-1 -\> [0,2]. If this function is continuous, then it must be eventually constant, so it can’t be order preserving (or even 1-1).

So, subject to those restrictions, if we have an infinite number of actions performed in _any_ length of time (not necessarily finite, either), the number of actions must be countable.

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**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\
**Post date:** [June 10, 2003, 4:28pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/67 "2003-06-10T16:28:36Z")

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[Here is a link](http://www.seop.leeds.ac.uk/archives/fall1999/entries/spacetime-supertasks/#2) to a page on supertasks, which has some discussion on Thomson’s lamp in particular (see sections 2.3 and 3.1). In the later section, it discusses “physical” models of the lamp, and shows two, one where the “model” predicts on at the end, and a second where the “model” predicts off at the end.

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**Author:** ![zwaldd](https://avatars.discourse-cdn.com/v4/letter/z/a9adbd/32.png) [@zwaldd](https://boards.straightdope.com/u/zwaldd)\
**Post date:** [June 10, 2003, 4:31pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/68 "2003-06-10T16:31:38Z")

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> [@](#):
>
> \*Originally posted by ultrafilter \*  
> \*\*I’m not sure exactly what you’re asking, **zwaldd** , but the demon hits the switch for the nth time at time t = sum(1/2[sup]k[/sup], 0 \< k \< n - 1). So as close as you get to two minutes, the demon has only hit the switch a finite number of times. Does that make sense? \*\*

Makes sense, but his task is not complete after a finite number of switches. The only way he can stop hitting the switch while staying true to his task is if he finds an interval where the halfway point to two minutes is exactly the same as the entire way to two minutes. There is only one theoretical moment where this occurs, and that moment is infinity, because half of infinity is infinity.

The OP asks whether the light is on or off at the end of two minutes, not at a certain point close to two minutes. I agree that at any given point prior to two minutes, the demon has only hit the switch a finite number of times. At two minutes however, he has hit the switch an infinite number of times. Infinite is not countable.

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [June 10, 2003, 4:34pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/69 "2003-06-10T16:34:12Z")

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> [@](#):
>
> \*Originally posted by green\_dragon \*  
> \*\*It’s a countable process embedded in an uncountable field; if the field was countable, there _would_ be a ‘most recent event’ and we wouldn’t still be here rattling on about it 🙂  
> \*\*

Not necessarily. Since the demon only acts at rational times, we can ignore the irrationals, but the rationals are still self-dense.

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**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\
**Post date:** [June 10, 2003, 4:36pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/70 "2003-06-10T16:36:53Z")

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Quoting myself:

> [@](#):
>
> If we require a similar setup for an uncountable number of events, such that the times of the events are well ordered, and are given by some continuous, order preserving function g:A -\> [0,2], then it can’t be done.

I forgot to include “for some well ordered set A”, but that’s probably clear from context.

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**Author:** ![zwaldd](https://avatars.discourse-cdn.com/v4/letter/z/a9adbd/32.png) [@zwaldd](https://boards.straightdope.com/u/zwaldd)\
**Post date:** [June 10, 2003, 4:45pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/71 "2003-06-10T16:45:40Z")

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> [@](#):
>
> \*Originally posted by zwaldd \*  
> **There is only one theoretical moment where this occurs, and that moment is infinity, because half of infinity is infinity.**

Getting back to my original point…the only way there could be an interval of infinite duration, where halfway to the next interval = the entire way to the next interval, is if the passage of time has completely stopped, i.e., the next interval will never arrive. This is the only condition under which the demon can stop hitting the switch and still be on task.

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**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\
**Post date:** [June 10, 2003, 5:06pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/72 "2003-06-10T17:06:06Z")

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**zwaldd** :

> [@](#):
>
> Infinite is not countable.

Oh. As it turns out, you’re using a definition of countable that’s entirely different from what I and the others are using. You’re taking “countable” to mean “finite”, while in mathematics, “countable” means “finite, or having cardinality equal to that of the counting numbers”.

So, for example, the infinite set of integers is countable.

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**Author:** ![zwaldd](https://avatars.discourse-cdn.com/v4/letter/z/a9adbd/32.png) [@zwaldd](https://boards.straightdope.com/u/zwaldd)\
**Post date:** [June 10, 2003, 5:23pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/73 "2003-06-10T17:23:02Z")

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> [@](#):
>
> \*Originally posted by Cabbage \*  
> \*\*in mathematics, “countable” means “finite, or having cardinality equal to that of the counting numbers”.
> 
> So, for example, the infinite set of integers is countable. \*\*

I stand corrected.

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**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\
**Post date:** [June 10, 2003, 5:28pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/74 "2003-06-10T17:28:57Z")

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> [@](#):
>
> Actually, that’s a larger cardinality than that of the natural numbers, but not necessarily the next largest. The next largest is omega-1 (or aleph-1); in ZFC, it is consistent that the reals have cardinality omega-1 (the continuum hypothesis), but it’s also consistent for the reals to have just about any larger cardinality, as well. The issue is independent of ZFC.

Without using the Continuum Hypothesis, is it even possible to prove that Aleph[sub]1[/sub] exists? That is to say: Aleph[sub]1[/sub] is defined as the lowest cardinal which is strictly greater than Aleph[sub]0[/sub], correct? How do we know that there exists any single cardinal which is the minimum of that set?

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**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\
**Post date:** [June 10, 2003, 7:38pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/75 "2003-06-10T19:38:35Z")

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You do not need the Continuum Hypothesis to prove Aleph[sub]1[/sub] exists. I think you need the Axiom of Choice – in the form of the Well Ordering Principle.

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**Author:** ![green\_dragon\_1](https://avatars.discourse-cdn.com/v4/letter/g/13edae/32.png) [@green\_dragon\_1](https://boards.straightdope.com/u/green_dragon_1)\
**Post date:** [June 10, 2003, 11:11pm UTC](https://boards.straightdope.com/t/another-infinity-question/180190/76 "2003-06-10T23:11:03Z")

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Drifting away from the point a little here guys…not that a little revision of last year’s calculus class isn’t helpful 🙂 (and needed :()

**ultrafilter** , point taken about the irrationals. Hey, I had to slip up somewhere, thought I did pretty well back there on page 1 😃

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**Author:** ![Jpeg\_Jones](https://avatars.discourse-cdn.com/v4/letter/j/4bbf92/32.png) [@Jpeg\_Jones](https://boards.straightdope.com/u/Jpeg_Jones)\
**Post date:** [June 11, 2003, 12:08am UTC](https://boards.straightdope.com/t/another-infinity-question/180190/77 "2003-06-11T00:08:09Z")

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On.

Definitely on.

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**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\
**Post date:** [June 11, 2003, 2:01am UTC](https://boards.straightdope.com/t/another-infinity-question/180190/78 "2003-06-11T02:01:43Z")

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> [@](#):
>
> You do not need the Continuum Hypothesis to prove Aleph1 exists. I think you need the Axiom of Choice – in the form of the Well Ordering Principle.

Actually, you don’t need the axiom of choice, either. You can construct the ordinals (and cardinals) without the axiom of choice. You _do_ need the axiom of choice to well order an _arbitrary_ set, and to define the cardinality of an _arbitrary_ set.

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**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\
**Post date:** [June 11, 2003, 5:05am UTC](https://boards.straightdope.com/t/another-infinity-question/180190/79 "2003-06-11T05:05:29Z")

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Thanks **Cabbage** , I wasn’t sure.

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