# The different sizes of infinity

**URL:** <https://boards.straightdope.com/t/the-different-sizes-of-infinity/553474>\
**Category:** Factual Questions\
**Created:** [September 12, 2010, 4:03am UTC](https://boards.straightdope.com/t/the-different-sizes-of-infinity/553474 "2010-09-12T04:03:14Z")\
**Posts on this page:** 10\
**Page:** 3

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**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\
**Post date:** [September 24, 2017, 9:20pm UTC](https://boards.straightdope.com/t/the-different-sizes-of-infinity/553474/41 "2017-09-24T21:20:24Z")

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> [@Channing\_Idaho\_Banks](#):
>
> I have read this a few times but I’m srill trying to get it. Still not sure if all infinities are equal, or only some of them.

All infinities are equal, but some infinities are more equal than others.

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**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\
**Post date:** [September 24, 2017, 9:54pm UTC](https://boards.straightdope.com/t/the-different-sizes-of-infinity/553474/42 "2017-09-24T21:54:57Z")

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> [@David\_Marcus](#):
>
> Actually, I read the Wikipedia article before posting my question. It wasn’t obvious to me that being not solvable is the same as the roots not being in any extension of the rationals. It seems that being not solvable means there is no formula for the roots that only involves the coefficients. This sounds like it is weaker than the roots not being in any extension. I confess to being ignorant of almost all of Galois theory. I’m a probabilist by training. Of course, I knew that quintics and higher can’t be solved in general.

Although it is not immediately obvious, there is a radical closure of a field and it is a field. What is not obvious is that if you take two complicated radicals and add them, multiply, divide them, the result can still be expressed as aa complicated radical, but it is true. And the solution to x^5 - x - 1 =0 is not in the radical closure of the rationals.

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**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\
**Post date:** [September 24, 2017, 10:27pm UTC](https://boards.straightdope.com/t/the-different-sizes-of-infinity/553474/43 "2017-09-24T22:27:56Z")

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I have no comment on infinities.

I am impressed that after a 7 year hiatus the OP returns within 10 minutes of this thread being bumped by a regular. Talk about bird-dogging your handiwork. Wow.

All the more impressive when you notice the OP’s sole contribution to SDMB other than his posts in this thread was, a few years ago, to resurrect a different 6 year-old zombie: [http://boards.straightdope.com/sdmb/showthread.php?p=13795647#post13795647](http://boards.straightdope.com/sdmb/showthread.php?p=13795647#post13795647)

Understand I’m not complaining; I’m just remarking that it’s unusual and noteworthy for that.

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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:** [September 24, 2017, 11:39pm UTC](https://boards.straightdope.com/t/the-different-sizes-of-infinity/553474/44 "2017-09-24T23:39:20Z")

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We can always use more mathematicians around here. Why not stick around, and take part in other discussions, too?

Aside: I just noticed that two of the current participants in this discussion are named after fictional mathematicians. Who’s smarter, Heinlein’s supergenius or Asimov’s supergenius?

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**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\
**Post date:** [September 25, 2017, 12:08am UTC](https://boards.straightdope.com/t/the-different-sizes-of-infinity/553474/45 "2017-09-25T00:08:08Z")

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I was also impressed with Big Ed’s math chops from his comments back in 2010. That’s more substantive commentary than I’ve heard from him on any topic during my 14 year\* membership. Obviously a bright dude. I wonder how Cecil found him?

========

- I just noticed this month is my Doperversary. Yaay me. 🙂

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**Author:** ![Ludovic](https://avatars.discourse-cdn.com/v4/letter/l/7ab992/32.png) [@Ludovic](https://boards.straightdope.com/u/Ludovic)\
**Post date:** [September 25, 2017, 12:38am UTC](https://boards.straightdope.com/t/the-different-sizes-of-infinity/553474/46 "2017-09-25T00:38:05Z")

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> [@David\_Marcus](#):
>
> There are numbers in the set that are arbitrarily close to zero, but the set itself isn’t doing anything.

That’s the beauty of it!

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**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\
**Post date:** [September 25, 2017, 4:18am UTC](https://boards.straightdope.com/t/the-different-sizes-of-infinity/553474/47 "2017-09-25T04:18:31Z")

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> [@Chronos](#):
>
> Some of the details, including at least some very important ones, were lost in that article.
> 
> That would only mean that p is intermediate between aleph-0 and t, not that it’s intermediate between aleph-0 and c. You’d only get the former if you had also already proven that t ≤ c.
> 
> And if they had proven that t ≤ c, then they obviously wouldn’t be able to prove that t ≠ p, since that (as mentioned) would disprove the continuum hypothesis, and it’s already been shown that the continuum hypothesis is undecidable.
> 
> I’m guessing that the truth behind the article is that mathematicians expected the question of whether t = p to be undecidable, and that the surprise was just that it is decidable, not that it’s decidable in that particular direction (since being decidable in the other direction would be impossible).

t is obviously ≤ c, once you see the definitions. You are correct that mathematicians essentially were surprised by the fact that ZFC decided the question of whether t = p, and that it was long clear that ZFC could not (if consistent) prove p \< t.

For what it’s worth, it’s not that hard to define p and t. Given two subsets f and g of the naturals, let’s say f \<= g if there are at most finitely many things in f which aren’t in g. We’ll say a collection S of subsets of the naturals is “inconsistent” if there is no infinite set which is \<= each thing in S. We’ll say S is “finitely consistent” if no finite subset of S is inconsistent. And we’ll say S is a “tower” if for each f and g in S, either f \<= g or g \<= f.

p is the minimum size of a finitely consistent but overall inconsistent collection. t is the minimum size of a finitely consistent but overall inconsistent tower.

Since the latter condition is more stringent than the former, we have that p \<= t automatically. Furthermore, since these are collections of subsets of naturals, we also have that p and t are both \<= c automatically.

Now, I’ve been a bit glib, in that to know that these values are well-defined at all, we need to show that finitely consistent but overall inconsistent collections/towers actually exist at all, but this can be done without too much difficulty. We can also show that no countable collection will work, and thus find that p and t are both \> aleph\_0.

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**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\
**Post date:** [September 25, 2017, 4:50am UTC](https://boards.straightdope.com/t/the-different-sizes-of-infinity/553474/48 "2017-09-25T04:50:34Z")

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In fact, let’s go ahead and show why no countable collection will work: Suppose you had A\_1, A\_2, A\_3, … . Let’s turn this into B\_1, B\_2, B\_3, where each B\_n is the intersection of A\_1, A\_2, through A\_n. Clearly, each B\_n is a superset of the following B\_{n + 1}. What’s more, the As are finitely consistent iff the Bs are all infinite sets, and the As are overall inconsistent if and only if the Bs are overall inconsistent.

So let’s presume the Bs are all infinite sets, and show that they cannot be overall inconsistent (i.e., that there is some infinite set which is \<= each B\_n).

Note that our sequence B\_1, B\_2, B\_3, …, may be eventually constant. If it’s eventually constant, then its eventual constant value is some infinite set \<= each of the Bs, and thus the Bs are not overall inconsistent.

Otherwise, there are infinitely many occasions on which B\_{n +1} has shed some of the elements in the prior B\_n. On each such occasion, pick one element which was shed; then group all of these choices of shed elements into some infinite set Shed. Note that Shed \<= each B\_n (since by the time of B\_n, at most n of the elements in Shed have already been shed, which is to say, there are most finitely many elements of Shed not in B\_n). Thus, the Bs are not overall inconsistent.

This concludes the proof that there can be no countable collection which is finitely consistent but overall inconsistent.

Now let’s go ahead and show why there CAN be a finitely consistent but overall inconsistent collection at all, and indeed even such a tower.

Given an infinite set of naturals X, define f(X) as any infinite set you like which is obtained from X by shedding infinitely many elements.

Now let A\_1 be the set of all naturals, let A\_2 be f(A\_1), let A\_3 be f(A\_2), and so on. Then define A\_{omega} as some infinite set which is \<= each A\_n for finite n. Then define A\_{omega + 1} as f(A\_{omega}), define A\_{omega + 2} as f(A\_{omega + 1}), and so on. Then define A\_{omega \* 2} as some infinite set which is \<= each An for n \< omega \* 2. Keep going in this way until, at some point, at some limit ordinal, you’re unable to find an infinite set \<= each previous A\_n. [This must happen eventually, because the A values keep getting distinctly smaller in the \<= ordering, and eventually run out of distinct possibilities to take on]. At that point, you’ve constructed a finitely consistent but overall inconsistent tower. Hooray!

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**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\
**Post date:** [September 25, 2017, 3:18pm UTC](https://boards.straightdope.com/t/the-different-sizes-of-infinity/553474/49 "2017-09-25T15:18:26Z")

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> [@LSLGuy](#):
>
> I am impressed that after a 7 year hiatus the OP returns within 10 minutes of this thread being bumped by a regular. Talk about bird-dogging your handiwork. Wow.

I assume that David Marcus had “subscribed” to the thread so that he got an email message when CIB posted his reply. I didn’t know that was a feature the SDMB had, but there it is down under “Additional Options.”

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**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\
**Post date:** [September 25, 2017, 5:37pm UTC](https://boards.straightdope.com/t/the-different-sizes-of-infinity/553474/50 "2017-09-25T17:37:07Z")

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Agree that’s almost certainly how it happened. I use the other thread notification features myself.

It was just a bit surprising to me for somebody then a newbie to post a thread, care enough to set up notifications, and then 7 years later, having all but ignored SDMB the whole time, have the notification trigger, and then have him be at the ready to read the bump, compose an update, and post it. All in 10 minutes flat.

It just struck me as an odd / incongruous combination of utter disinterest and total interest. It’s not like SDMB hasn’t had a few other choice math threads in the last almost-decade he might have chosen to contribute to. Yet he did not.

Odd is not bad; it’s merely noteworthy.

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