Talk:Beth number

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[edit] Difference between aleph numbers and beth numbers

I don't see any difference between anAleph number and a Bet number. Can they be used interchangeably? Then why have both terms? 65.74.59.37 08:41, 21 January 2006 (UTC)Chani

They are different if (and only if) GCH fails. It's not usual to assume GCH, so they can't usually be used interchangeably. --Zundark 11:53, 21 January 2006 (UTC)

[edit] Gimel numbers

What's gimel used for? Is it used interchangeably with beth? Because they look similar? Awmorp 23:26, 7 June 2006 (UTC)

I have no idea. I've never seen it outside of Wikipedia. — Arthur Rubin | (talk) 23:59, 7 June 2006 (UTC)
I think that gimel numbers are defined analogously to aleph numbers and beth numbers, but with \gimel_{\alpha+1}=\gimel_\alpha^{\mathrm{cf}(\gimel_\alpha)}. However, I'm not aware of any published reference for this, so I've removed the mention of them from the article (which in any case was confusing due to the lack of explanation). --Zundark 08:21, 8 June 2006 (UTC)

[edit] Merging Beth2 into this article

Any discussion is taking place at Talk:Beth_two#Merge_into_Beth_Numbers. LambiamTalk 17:54, 2 April 2006 (UTC)

  • Merge completed (yesterday), as best I could. If I missed anything relevant that should have been merged, the last version before merge is still there. — Arthur Rubin | (talk) 14:43, 25 July 2006 (UTC)

[edit] Clarification needed

"sup" is not defined or linked in any way, as in:

\beth_{\lambda}=\sup\{ \beth_{\alpha}:\alpha<\lambda \}.

It would be helpful if it was; I can't be the only reader who finds this complete double-Dutch. --Michael C. Price talk 03:07, 14 February 2007 (UTC)

If the axiom of choice holds, then these cardinals can be identified with their initial ordinals. In that case, the supremum is the least ordinal greater than all those in the set. If the axiom of choice does not hold, then the "obvious" generalization is the direct limit of the sequence of representative sets. So if A is the initial set, then
\beth_{\lambda} (|A|) = | \bigcup_{\alpha < \lambda} B_{\alpha} |
where
\beth_{\alpha} (|A|) = | B_{\alpha} |
and the Bs are nested. JRSpriggs 08:44, 14 February 2007 (UTC)
Even if the axiom of choice holds, the direct limit of cardinals is not always defined, I'm afraid. The direct limit for a strictly increasing set of cardinals works, which applies here, but the limit of ω1 (the ordinal corresponding to \aleph_1) \aleph_0's can be \aleph_0 or \aleph_1. The cardinality of the union of the specific nested set of sets makes sense, but "direct limit" of cardinals, in general, does not. — Arthur Rubin | (talk) 15:45, 14 February 2007 (UTC)

[edit] spelling for beth

Who decided to spell the word as "beth", when it's actually pronounced "bet"? Is there a lot of history for this spelling? Do mathematicians pronounce it like the female name beth? Ariel. (talk) 11:01, 22 January 2008 (UTC)

Beth is the traditional English spelling, and is usually pronounced beth in Britain (but, apparently, bayth in America). I would guess that the spelling is based on Tiberian vocalization, but I don't know for sure. It dates from 1650 according to Merriam-Webster - the OED probably has details, if someone feels like checking them. But all this is something that should be covered in the article on the letter, not in this article - mathematicians merely adopted the existing conventions, which long predate the theory of cardinal numbers. --Zundark (talk) 12:49, 22 January 2008 (UTC)
So you are saying mathematicians pronounce it the same way as the hebrew pronunciation? Because in that case it should be noted in the article to pronounce it bet or bais (bays), and not like it looks. Ariel. (talk) 13:12, 22 January 2008 (UTC)
I concur that the pronunciation is as in the Hebrew, but the spelling I've usually seen is Beth, in spite of the fact that I usually refer to the Hebrew letter as Bet. — Arthur Rubin | (talk) 13:56, 22 January 2008 (UTC)
At least in America. — Arthur Rubin | (talk) 14:03, 22 January 2008 (UTC)
No, I wasn't saying that mathematicians pronounce it as in Hebrew, and I don't know how you read my comment that way. Also, the only type of Hebrew I mentioned was Tiberian Hebrew, and in Tiberian Hebrew the pronunciation is beth, so I especially don't see how you read my comment as suggesting bet or bays. I think that the mathematical pronunciation probably varies as much as the non-mathematical pronunciation, and that we therefore can't specify it. But if someone has a good reference for mathematical pronunciation, go ahead and add it to the article. --Zundark (talk) 15:00, 22 January 2008 (UTC)
I've always heard it "beth" as a cardinal, though I say "bet" for the letter in other contexts. CRGreathouse (t | c) 02:53, 23 January 2008 (UTC)
I'm afraid I've heard it pronouced "bayt", but my father speaks Hebrew, as well as working in logic and set theory, so that might be a special case. My parents still spelled it "Beth", though, when we weren't using the symbol. — Arthur Rubin | (talk) 03:01, 23 January 2008 (UTC)

[edit] Counterfactuals

Arthur Rubin removed [1] the following recent addition to the article:

  • The minimum number of possible worlds that might exist (David Lewis, "Counterfactuals")

The addition was so unusual I decided to look it up to see if there was any merit. First, I saw that David Lewis has an article on Wikipedia: David Kellogg Lewis. Second, in looking for any mention of David Lewis, Counterfactuals, and something cardinality-related, I came across a similarly-titled paper which references Dr. Lewis and proves logical (as in mathematical logic) results relating to the cardinality of algebras: see T15 and T16. Donald Nute's Counterfactuals

I have no opinion on the paper or the suitability of either Nute's paper or Lewis' book for this article, but it does seem that they're related in some way.

CRGreathouse (t | c) 13:49, 8 February 2008 (UTC)

Hmmm. Perhaps I was mistaken, but I'm not sure Lewis's definition of "possible worlds" is notable, even if Lewis is notable. Even so, we would need a detailed reference for the statement. Also, under the circumstances, the question of whether Beth 2 means the same thing to Lewis as it does to us seems open. — Arthur Rubin | (talk) 19:06, 8 February 2008 (UTC)
Nute makes explicit reference to 2^(2^stuff), where stuff is a cardinal, so Nute's theorems have beth_2 when stuff is countably infinite. I haven't seen the Lewis paper, but Nute references it so it's not too much of a stretch. Of course this doesn't mean it's notable -- though our list of places that beth_2 appears is limited, so I'd love to keep it as an example if reasonable.
Partially related: have you seen any reasonable examples of beth_3? The article could use one/some if there are any outside of the obvious contrived examples.
CRGreathouse (t | c) 21:10, 8 February 2008 (UTC)

[edit] References

I am Robert Munafo. I just noticed this article acknowledges me as a source. However, it bears so little relation to the material on my web pages (e.g. Aleph-1, Power set, etc.) that I wonder if the citation is still relevant. I wouldn't mind if it gets removed. I don't know much about the legal requirements of GNU Free Documentation License to be able to decide on my own. —Preceding unsigned comment added by Mrob27 (talk • contribs) 02:12, 11 February 2008 (UTC)

It turns out that the reason for this acknowledgement is that this article is partly derived from the old "Beth two" article, and the original version of that article was based very closely on your page 2 to the Power of C. But none of that old text exists in the current article, so the acknowledgement is no longer relevant. (By the way, your page incorrectly says that aleph_2 is the same as 2c only if GCH holds. But, in fact, it can hold without GCH, as it's much weaker.) --Zundark (talk) 09:25, 11 February 2008 (UTC)

Good, then I'l take out the reference to my own web page (smile) (And by the way, thanks for the correction about 2c and GCH) —Preceding unsigned comment added by Mrob27 (talk • contribs) 14:04, 11 February 2008 (UTC)

Now that I've done that, the article has no references. Can someone who knows how, add a tag that says we need to add references? (I'm still new to all the magic Wiki-text stuff.) Robert Munafo (talk) 14:08, 11 February 2008 (UTC)

You can add {{unreferenced}} to the top of the page if you like. It will add a banner to the page asking for citations. (By the way, welcome; I like your page on big numbers a lot.) CRGreathouse (t | c) 15:42, 11 February 2008 (UTC)

Got it, thanks! Robert Munafo (talk) 23:09, 11 February 2008 (UTC)

[edit] Comparison to Aleph Numbers

This section says that "Assuming the axiom of choice, infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable, and so, since by definition no infinite cardinalities are between \aleph_0 and \aleph_1, the continuum hypothesis can be stated in this notation by saying \beth_1=\aleph_1."

If I'm reading this correctly, it states that assuming the axiom choice implies the continuum hypothesis and equivalence of aleph and beth numbers. Since this is clearly incorrect (the continuum hypothesis was proved independant from ZFC), I think it should be reworded in a way that makes it clear that aleph and beth numbers are only equal assuming the continuum hypothesis.Eebster the Great (talk) 18:55, 20 April 2008 (UTC)

It's saying that, assuming AC, the equality \beth_1=\aleph_1 is equivalent to CH. It's not saying that this equality (or CH) follows from AC. I've reworded it to make this a little clearer. --Zundark (talk) 22:20, 20 April 2008 (UTC)