Talk:Beth number

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)

Moreover, the definition of the Bet numbers requires the Axiom of Choice, while the definition of the Aleph numbers only uses the Replacement Schema and not the Axiom of Choice. Hence you cannot even use the Bet numbers without assuming AC. Yann Pequignot (talk) 11:01, 13 June 2010 (UTC)

Without AC, it is no longer true that every cardinality includes a well-ordered set, but it is still true that every set has a cardinality. So there isn't any difficulty in defining Beth numbers without AC. — Carl (CBM · talk) 12:25, 13 June 2010 (UTC)

Actually, it's not necessarily true that every set has a "cardinality" (to be precise, in vNBG with Urelements, that there is a (class) function (F) with domain the universe such that F(x) = F(y) if and only if x is equipollent to y). In set theories without urelements, one can take F(x) to be the set of all sets equipollent to x with minimal rank. Even then, we cannot necessarily have that F(x) is equipollent to x.

We note in the article that the von Neumann universes $V_{\omega +\alpha }\!$ have cardinality $\beth _{\alpha }\!$. This appears to be correct (for most definitions of cardinality), even without choice. — Arthur Rubin (talk) 15:46, 13 June 2010 (UTC)

There is a difference between (1) each set having a cardinality, (2) the existence of a uniform selection of representatives for the cardinalities, and (3) each cardinality containing a well-ordered set. I was referring to (1) and you are referring to (2). I think the person I responded to was referring to (3). — Carl (CBM · talk) 15:55, 13 June 2010 (UTC)

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)

Merging Beth2 into this article

Any discussion is taking place at Talk:Beth_two#Merge_into_Beth_Numbers. Lambiam^Talk 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)

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)

Also, a description says an equation (I noted with '^[clarify]') will use alpha, but alpha is not used.--Dchmelik (talk) 07:35, 1 November 2008 (UTC)

It doesn't say that the equation will use $\alpha$, it says that there is an $\alpha$ such that the equation holds for any $\beta \geq \alpha$. In other words, the equation holds for every sufficiently large ordinal $\beta$. I've reworded it a little, so that it is hopefully clearer. --Zundark (talk) 10:03, 1 November 2008 (UTC)

[Maybe it is because I have not studied much number theory, but] it seems no clearer: α,β,k,μ were re-quantified, so they are not defined as earlier: it seems like $\exists {\alpha }{,}\forall {\beta ,k,\mu }{\in }\infty P(\beta ,k,\mu .))$ Variables were re-quantified--so maybe not the same, and there is no P(α.)--Dchmelik (talk) 14:59, 3 November 2008 (UTC)

I don't know how you read it that way. The basic form (ignoring the fact that κ and μ are required to be cardinals, and α and β are required to be ordinals) is $\forall \kappa \forall \mu \exists \alpha \forall \beta (\beta \geq \alpha \rightarrow P(\beta ,\kappa ,\mu ))$. --Zundark (talk) 14:09, 6 November 2008 (UTC)

To Arthur Rubin: Given a cardinal $\kappa \,$ and an ordinal $\alpha \,,$ we can define

$\beth _{\alpha }(\kappa )\,=\,\vert W_{\alpha }(A)\vert \,$

where $A\,$ is any set of cardinality $\kappa \,$ and

$W_{\alpha }(A)\,=\,(\{0\}\times A)\cup (\{1\}\times \bigcup _{\beta <\alpha }{\mathcal {P}}(W_{\beta }(A))\,)\,.$

This is modeled on the alternative definition of the Von Neumann universe. I hope this clears it up. Sorry about the long delay in responding. JRSpriggs (talk) 13:01, 1 November 2008 (UTC)

Better would be

$W_{\alpha }(A)\,=\,(\{0\}\times A)\cup \bigcup _{\beta <\alpha }\{\langle \beta +1,B\rangle \,:\,B\subseteq W_{\beta }(A)\land \vert B\vert \neq 1\}\,$

which takes out singletons to compensate for the union with previous stages. JRSpriggs (talk) 02:23, 17 February 2010 (UTC)

Well, this is all an interesting conversation, but the core issue still remains: the linked article supremum doesn't go beyond calculus; perhaps the above conversation can be captured in this article or a new article supremum (cardinal numbers) or something like that? linas (talk) 18:11, 21 April 2012 (UTC)

As I said above, there is no problem when the axiom of choice holds. Since almost all of us believe in the axiom of choice, it is not worth the trouble to do more than we have already. As for Arthur Rubin's objection, it does not apply since the cardinal at each stage is larger than at any of the earlier stages. JRSpriggs (talk) 11:47, 22 April 2012 (UTC)

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 refer to it as bet too. Faster than Thunder (talk) 23:05, 17 December 2021 (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)

Not to worry, mathematicians butcher eastern european surnames with aplomb. Hardly limited to hebrew. linas (talk) 18:15, 21 April 2012 (UTC)

Zundark created this page and there are no move logs for this page. Faster than Thunder (talk) 03:00, 14 January 2022 (UTC)

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)

The footnote on page 90 of David Lewis's book "Counterfactuals" states in part that the cardinality of possible worlds in a framework proposed by W.V.O. Quine equals beth 2 which he identifies with the cardinality of the power set of reals. He contrasts this with the cardinality of sentences in a finite language equal to beth 1 which he identifies with the cardinality of reals. Lewis considers propositions as sets of possible worlds. So, the above comports with his comment on page 107 of his book "On the plurality of worlds" that the lowest reasonable estimate of the cardinality of propositions is beth 3. —Preceding unsigned comment added by Kmarkus (talk • contribs) 16:54, 26 August 2010 (UTC)

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 2^c 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 2^c 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)

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)

I'm very confused. "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": By what definition? Isn't the definition of CH that there are no infinite cardinalities between $\aleph _{0}$ and $\aleph _{1}$? Isn't this irrelevant and shouldn't the inequality be "$\beth _{1}>\aleph _{0}$"? Then the logic makes sense that CH => $\beth _{1}=\aleph _{1}$. Otherwise CH = "The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers." doesn't seem to support the conclusion.

-- — Preceding unsigned comment added by Schmmd (talk • contribs) 23:16, 26 January 2011 (UTC)

It all makes sense now. --Schmmd (talk) 06:03, 27 January 2011 (UTC)

I'm glad it makes sense now, but I want to make it clear that "[T]here are no infinite cardinalities between $\aleph _{0}$ and $\aleph _{1}$?" is definitely not the statement of the Continuum Hypothesis. The cardinality $\aleph _{1}$ is just defined to be the smallest cardinal greater than $\aleph _{0}$. The cardinal $\aleph _{1}$ must exist since the Axiom of Choice implies that the cardinal numbers are well-ordered.

Rather, CH is the claim that $\aleph _{1}$ is the cardinality of the power set (set of all subsets) of $\aleph _{0}$. I.e., that $\aleph _{1}$ is the cardinality of the set of all real numbers. — Preceding unsigned comment added by 2601:200:C082:2EA0:6115:5F27:A5AC:8057 (talk) 18:06, 9 April 2024 (UTC)

Definable real numbers

Laurusnobilis (talk · contribs) added the definable real numbers to the list of sets with cardinality Beth number#Beth null (i.e. Aleph null). I reverted on the grounds that the definable reals are not a well-defined set. Laurusnobilis reverted me, saying that they were defined in the article on them.

According to the lead of that article, "A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds in the standard model of set theory (see Kunen 1980:153). ... Note that this definition cannot be expressed in the language of set theory itself.".

In order to be able to say that a set is denumerable (i.e. of cardinality Aleph null), it must first be a set. Since there is no way of defining the collection of definable real numbers in the language of set theory, there is no way to use the axiom of separation (or of replacement) to extract it from the real numbers. Furthermore, the collection would vary depending on the model of set theory within which one tried to define it. Also the article itself provides alternative definitions which would lead to different sets, and provides an "unambiguously described number" which cannot be in the set although most people would consider that number to be a definable real number (if the set of such existed).

Thus, as I said, this notion is not well-defined. So I will revert again. JRSpriggs (talk) 10:13, 4 January 2016 (UTC)

'Beth' or 'bet'?

ב's name in Hebrew is bet, not Beth, so can you replace the 'Beths' with 'bets'? Faster than Thunder (talk) 06:54, 5 December 2021 (UTC)

The relevant question isn't what the Hebrew name is but what the English name is, or more specifically, what name is used in English-language mathematics. As discussed in the section above (you've posted there), this is "beth", not "bet". - CRGreathouse (t | c) 04:23, 19 January 2022 (UTC)

GCH on limit ordinals

Let’s say that GCH holds on an ordinal x if the only cardinality <=|V[x]| that is greater than all |V[y]| with y < x is |V[x]|. Then is it true the last comment in your Definition section implies that GCH holds on each limit ordinal x, whether or not it holds on successor ordinals. 2601:589:4B80:14B0:3151:D61C:9B27:F1AA (talk) 17:31, 15 March 2022 (UTC)

Clarification of "sup"

It would be extremely useful if someone knowledgeable about this subject made it clear in this article that the supremum of a family of beth cardinals is the union of those cardinals.

The article Limit cardinal makes this clear; it should be clear here, too.