# Two infinities............and beyond! (HA HA, geddit!)

**URL:** <https://boards.straightdope.com/t/two-infinities-and-beyond-ha-ha-geddit/141441>\
**Category:** Factual Questions\
**Created:** [December 9, 2002, 4:19am UTC](https://boards.straightdope.com/t/two-infinities-and-beyond-ha-ha-geddit/141441 "2002-12-09T04:19:32Z")\
**Posts on this page:** 11\
**Page:** 1

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**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\
**Post date:** [December 9, 2002, 4:19am UTC](https://boards.straightdope.com/t/two-infinities-and-beyond-ha-ha-geddit/141441/1 "2002-12-09T04:19:32Z")

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I’ve seen the proof that there are two different infinities (one is infinitetly larger than the other) and my question is does the larger infinity have any mathematical application outside of the proof?

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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:** [December 9, 2002, 4:32am UTC](https://boards.straightdope.com/t/two-infinities-and-beyond-ha-ha-geddit/141441/2 "2002-12-09T04:32:29Z")

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Yes, definitely. But not in any real, practical sense.

The smallest infinite cardinal is the size of the natural numbers. It’s also the size of any set whose elements you can list. An example of such a set would be the set of all computer programs.

The next largest infinite cardinal\* is the size of the real numbers. Because there are strictly more real numbers than computer programs, it follows that most real numbers are not the output of any computer program.

Actually, there are many more than two infinite cardinals. When you’ve got one, you can always construct a larger one.

\*This is contingent on your acceptance of the continuum hypothesis, but I’ll only get into that if you’re interested.

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**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\
**Post date:** [December 9, 2002, 4:32am UTC](https://boards.straightdope.com/t/two-infinities-and-beyond-ha-ha-geddit/141441/3 "2002-12-09T04:32:57Z")

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I’m not sure I understand exactly what your question is, but will [this](http://www.gap-system.org/~john/analysis/Lectures/L5.html) do?

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**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\
**Post date:** [December 9, 2002, 4:46am UTC](https://boards.straightdope.com/t/two-infinities-and-beyond-ha-ha-geddit/141441/4 "2002-12-09T04:46:56Z")

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Actually I forgot to consider the set of natural numbers, set I am considering as the larger infinity is the set of (real) numbers between 0 and 1. And I suppose the question should of been does this number have any other use other than to describe the number of numbers in the set of real numbers btween 0 and 1?

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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:** [December 9, 2002, 4:49am UTC](https://boards.straightdope.com/t/two-infinities-and-beyond-ha-ha-geddit/141441/5 "2002-12-09T04:49:27Z")

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[0, 1] has the same number of elements as **R** , so what I said applies to that cardinality as well.

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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:** [December 9, 2002, 4:55am UTC](https://boards.straightdope.com/t/two-infinities-and-beyond-ha-ha-geddit/141441/6 "2002-12-09T04:55:55Z")

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> [@](#):
>
> And I suppose the question should of been does this number have any other use other than to describe the number of numbers in the set of real numbers btween 0 and 1?

Well, off the top of my head, I can think of two different ways to describe the number of real numbers between 0 and 1:

1. It’s also the cardinality of the set of functions mapping the natural numbers to the set {0,1} (or any finite set with more than one element, for that matter).

2. It’s also the cardinality of the power set of the natural numbers (the set of all subsets of the natural numbers).

Does that answer your question?

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**Author:** ![MC\_Master\_of\_Ceremonies](https://avatars.discourse-cdn.com/v4/letter/m/bb73d2/32.png) [@MC\_Master\_of\_Ceremonies](https://boards.straightdope.com/u/MC_Master_of_Ceremonies)\
**Post date:** [December 9, 2002, 4:56am UTC](https://boards.straightdope.com/t/two-infinities-and-beyond-ha-ha-geddit/141441/7 "2002-12-09T04:56:53Z")

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I suppose, it does answer my question but are there any mathematical operations that utlize the different infinties (number theory and sets is something I don’t know a huge deal about).

Also I would like to hear the continium hypothesis, if possible, as I’m not sure if I am familair with it.

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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:** [December 9, 2002, 5:12am UTC](https://boards.straightdope.com/t/two-infinities-and-beyond-ha-ha-geddit/141441/8 "2002-12-09T05:12:20Z")

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The continuum hypothesis basically says that any infinite subset of the reals is either the same size as the reals, or the same size as the naturals.

You could do worse than to read about it [here](http://mathworld.wolfram.com/ContinuumHypothesis.html).

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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:** [December 9, 2002, 5:29am UTC](https://boards.straightdope.com/t/two-infinities-and-beyond-ha-ha-geddit/141441/9 "2002-12-09T05:29:24Z")

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> [@](#):
>
> …are there any mathematical operations that utlize the different infinties

Oh yes, definitely; the bulk of modern set theory deals with sets of various infinite cardinalities.

For example, you can do arithmetic (addition, multiplication, and exponentiation) with infinte cardinals. Addition and multiplication is damn simple. If x and y are infinite cardinals:

x+y = x\*y = the maximum of {x,y}.

Exponentiation is another story, which leads to the continuum hypothesis. Cantor showed that the cardinality of the natural numbers is strictly less than the cardinality of the reals. If we let omega=cardinality of the natural numbers, then we can write this as:

omega \< 2[sup]omega[/omega]

(2[sup]omega[/sup] can be shown to equal the cardinality of the reals).

And that’s an example of “infinite cardinal” exponentiation.

And here’s where it gets messy. To shorten typing, I’ll use “c” to denote the cardinality of the reals. We know c is bigger than omega–the question is, how much bigger?

We also know that the cardinals are well ordered (every collection of cardinals has a smallest cardinal). This means the cardinals go in order:

omega, omega\_1, omega\_2, omega\_3,…

(or aleph\_0, aleph\_1, aleph\_2,… as you’ll commonly see).

(I’ll stop there for simplicity) with no cardinals in between (it gets a little different further on up when you get to limit cardinals, but let’s ignore them for now).

Cantor thought that c should be equal to omega\_1–the very next cardinal after omega.

It turns out that’s not necessarily true, however. It’s consistent with the standard axioms of set theory that c could instead be equal to omega\_2, or omega\_3, or omega\_5475847594375, or omega\_(omega+1), or, really, damn near any other cardinal (technically, not ANY cardinal, but “almost” any of them).

The Continuum Hypothesis (CH) is the assumption that c = omega\_1 (Cantor’s original belief). This makes things pretty simple; in fact, it makes things so simple that set theorists now commonly believe that CH is actually false. In fact, common belief is that c could be really huge–it could be so big that it has uncountably many cardinals that come before it, for example.

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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:** [December 9, 2002, 5:37am UTC](https://boards.straightdope.com/t/two-infinities-and-beyond-ha-ha-geddit/141441/10 "2002-12-09T05:37:53Z")

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> [@](#):
>
> If we let omega=cardinality of the natural numbers, then we can write this as:
> 
> omega \< 2omega[/omega]
> 
> (2[sup]omega can be shown to equal the cardinality of the reals).

[/sup]  
Damn, I really screwed up that coding. That should read:

If we let omega=cardinality of the natural numbers, then we can write this as:

omega \< 2[sup]omega[/sup]

(2[sup]omega[/sup] can be shown to equal the cardinality of the reals).

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**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\
**Post date:** [December 9, 2002, 2:55pm UTC](https://boards.straightdope.com/t/two-infinities-and-beyond-ha-ha-geddit/141441/11 "2002-12-09T14:55:51Z")

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Regarding applications of different infinities.

The _technique_ used by Cantor to show that the cardinality of the reals is greater than that of the integers is called _diagonalization_. Diagonalization was used by Goedel to prove a lot of important properties about the limits of what can be proven Mathematically. It was then used by Turing to prove that many important questions about computer programs are unsolvable, the most famous being the Halting Problem. (You can’t write a computer program that determines if other programs halt, or any other interesting runtime correctness property.) Then, starting in the '60s it was further used to proved relationships about classes of complexity of problems. E.g., we know that the set of problems that can be solved in linear time is a proper subset of those solvable in quadratic time. Now you are getting into reasonably practical matters. But again, it is mostly the technique used not the actual answer.