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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
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)
What's gimel used for? Is it used interchangeably with beth? Because they look similar? Awmorp 23:26, 7 June 2006 (UTC)
Any discussion is taking place at Talk:Beth_two#Merge_into_Beth_Numbers. Lambiam Talk 17:54, 2 April 2006 (UTC)
"sup" is not defined or linked in any way, as in:
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)
To Arthur Rubin: Given a cardinal and an ordinal we can define
where is any set of cardinality and
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)
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)
Arthur Rubin removed [1] the following recent addition to the article:
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)
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)
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)
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)
Got it, thanks! Robert Munafo ( talk) 23:09, 11 February 2008 (UTC)
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 and , the continuum hypothesis can be stated in this notation by saying ."
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)
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 and \aleph_1": By what definition? Isn't the definition of CH that there are no infinite cardinalities between and ? Isn't this irrelevant and shouldn't the inequality be ""? Then the logic makes sense that CH => . 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)
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)
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)
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)
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.
This is the
talk page for discussing improvements to the
Beth number article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Find sources: Google ( books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
![]() | This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
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
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)
What's gimel used for? Is it used interchangeably with beth? Because they look similar? Awmorp 23:26, 7 June 2006 (UTC)
Any discussion is taking place at Talk:Beth_two#Merge_into_Beth_Numbers. Lambiam Talk 17:54, 2 April 2006 (UTC)
"sup" is not defined or linked in any way, as in:
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)
To Arthur Rubin: Given a cardinal and an ordinal we can define
where is any set of cardinality and
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)
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)
Arthur Rubin removed [1] the following recent addition to the article:
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)
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)
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)
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)
Got it, thanks! Robert Munafo ( talk) 23:09, 11 February 2008 (UTC)
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 and , the continuum hypothesis can be stated in this notation by saying ."
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)
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 and \aleph_1": By what definition? Isn't the definition of CH that there are no infinite cardinalities between and ? Isn't this irrelevant and shouldn't the inequality be ""? Then the logic makes sense that CH => . 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)
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)
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)
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)
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.