text:
Talk:Cardinal number

Older discussion
Summarising:

The original assertion about the inaccessibles was inaccurate and has been amended by AxelBoldt. The page has moved from Cardinal to Cardinal number.

---

Someone should explain what the following types of cardinals are: inaccessible, Mahlo, indescribable, ineffable, partition, Ramsey, measurable, strongly compact, supercompact, extendible, huge.

 * Yeah, I second that. I want a description of indescribable numbers, and quick! The huge ones are easy though. They are just HUGE, man.

The definition of n-hyperinaccessible doesn't seem to be complete; so far, it only works for natural numbers n but I think we want it for all cardinal numbers n. --AxelBoldt 19:15, August 29, 2001 (UTC)

Does anyone have a more formal definition of Mahlo cardinals? 20:10, (unsigned by 62.8.212.xxx December 31, 2001 (UTC))

I think we should start with the everyday definition of cardinal (as opposed to ordinal) numbers. Only then should we progress to the more general (and abtruse) case described here. This is a deeply scary article :-)

A cardinal is 0-hyperinaccessible if it is a weakly inaccesible cardinal... It reminds me of undergraduate days. Pauk 20:37, September 27, 2002 (UTC)

Can I just say this article is beautifully written. Impeccably clear: it brings tears to my eyes to see this elegance and simplicity. From my student days I vaguely recall other kinds: hyper-hyper-Mahlo? Robinson? Gritchka 12:49, June 19, 2003 (UTC)

Algebra with cardinals?
Hi... I've been using the Wiki as a method of independent study on some areas of interest, such as this. I'm just a first-year calculus student with some curiosity and ambition, so some of the technical points are slightly beyond me. I'm posting this comment because I seem to remember seeing somewhere a list something like: Aleph-null + Alpeh-null = Alpeh-null 2 * Aleph-null = Aleph-null 2^Aleph-null = Aleph-one (my understanding is that this last one is somewhat murky -- cannot be [dis]proven) Anyway, I can do longer find this table, and I'm interesting in pursuing what kind of operations are legal on cardinals and what the results are. Can anyone help me out some? My inferences would expand this list to be something like: Alpeh-a + Aleph-b = Aleph-b, b >= a Also, I keep seeing things like: "Cantor's diagonal argument shows that 2^| X | > | X | for any set X." I could make a proof that shows that a set's cardinality is greater when I add one element to it; but adding one element wouldn't change the size of an infinite set. At what fundamental level is addition with transfinite cardinals and the power operation different? Also, if anyone is kind enough to field these questions, I'm curious would what happen if to transfinite cardinals were multiplied. Thanks, -David 02:29, September 1, 2004 (UTC)

 * This article does actually cover quite a bit of cardinal arithmetic already. You may also want to look at the PlanetMath article on cardinal arithmetic, which goes into a bit more detail. You are right that $$2^{\aleph_0} = \aleph_1$$ can neither be proved nor disproved in ZFC (see continuum hypothesis). The other equations you list are correct. I'm not sure I really understand your question about Cantor's Theorem. Addition or multiplication of two infinite cardinals is trivial in ZFC, as the result is merely the larger of the two cardinals. Cardinal exponentiation is more complicated. --Zundark 08:14, 1 Sep 2004 (UTC)

 * > I could make a proof that shows that a set's cardinality is greater when I add one element to it
 * What you can't do is make a valid proof of that.
 * Also, note that most of this stuff assumes at least some choice (Countable Choice, Axiom of Choice).
 * For example, "For all infinite X, |X cross X| = |X|." is equivalent to the Axiom of Choice.
 * See http://www.math.niu.edu/~rusin/known-math/99/infin_arith for a proof that it implies AC.
 * Even saying "If X is infinite then Aleph-null ≤ |X|." uses Countable Choice.

cut paragraph re axiom of choice
Removed this from the formal section as the previous definition does include the case "no axiom of choice". Some of this could still be useful, though:

"Note that without the axiom of choice there are sets which can not be well-ordered, and the informal definition of cardinal number given above does not work. It is still possible to define cardinal numbers (a mapping from sets to sets such that sets with the same cardinality have the same image), but it is slightly more complicated. One can also easily study cardinality without referring to cardinal numbers."

 * What's wrong with that paragraph? -- Schneelocke (cheeks clone)   17:42, 27 Nov 2004 (UTC)

Redundant article
With the information provided in this article, cardinality is redundant. It would make sense to merge one into the other, but I'm not sure which way... Fredrik | talk 23:30, 2 Dec 2004 (UTC)

"Makes no sense to me"

 * (every member of the set is a set of numbers of its own $$(a_0, a_1, ..., a_n),\;\; a_i \in \mathbb{N}$$, like an extended ordered pair)

This wording, from the history section, makes no sense to me. I would simply delete it, but destroying something just because I don't understand it seems dangerous :)

67.164.12.169 00:16, 14 October 2005 (UTC)

The wording here is saying that each member of the set (eg "apple banana cabbage") is a set of its own ("a" "p" "p" "l" "e"), ("b" "a" "n" "a" "n" "a"), ("c" "a" "b" "b" "a" "g" "e"), etc. The members of the "superset" each contain members of their own "subsets". In a sense, this is words versus letters. Words are units, letters are a smaller unit that are contained within words, and so on. --anon 70.92.174.251 04:21, 17 December 2005 (UTC)

One more question of cardinals in ZF

 * If the axiom of choice is not assumed and X does not have a well-ordering, the cardinality of X is defined to be the set of all sets which are equinumerous with X and have the least rank that a set equinumerous with X can have.

Is this indeed a set, not a class? (I'm not familiar with the definition of cardinal numbers without AC - I tried to find it here.) --Kompik 18:40, 29 November 2005 (UTC)

 * Yes it's a set, because for any ordinal r there is a set of all sets of rank r. This is the point of restricting to sets of least rank ("Scott's trick") - you can reduce any non-empty class to a representative set (but you need the Axiom of Regularity). --Zundark 20:15, 29 November 2005 (UTC)

Counting and cardinality
At a very basic level, there is an inconsistency in terminology which needs to be addressed for the benefit of the non-mathematical reader of this article.

In an eponymous wikipedia article, counting is defined as a matching process (tallying), while in this article counting numbers are equated to a set's cardinality. This latter includes the empty set (zero) that cannot be tallied. Material objects are required for the process of one-to-one matching.

It seems to me that it would be better to use the Peano definition of natural number, with counting numbers then being the set of all natural numbers that are successors. John Morgan 28 Jan 06

I'm not as troubled by this as John, but I have a suggestion if others are. During the discussion of the definition of cardinal numbers in terms of bijective functions, you can make the link clear with a small aside like: "If there is a bijective function between two sets, then the elements of those sets can be 'matched' 1-1." - Dennis K. 11 Aug 11 — Preceding unsigned comment added by 71.167.112.122 (talk) 12:14, 11 August 2011 (UTC)

Jaina mathematics
The following material was added April 8, 2006 by user:Jagged 85, and removed the same day by User:JRSpriggs with the brief edit summary "revert useless "history" about the Jains". Now, if the material is correct, it is hugely relevant to the article — but I find it hard to believe that it is correct, despite the numerous references added by Jagged 85. Someone should check these! Anyway, here is the material & references:
 * An early concept of infinite cardinal numbers is found in India in the works of Jaina mathematicians from the 4th century BC to the 2nd century CE. They classified all numbers into three groups: enumerable, innumerable and infinite. The highest enumerable number N of the Jains corresponds to the concept of aleph-null $$\aleph_0$$, the smallest infinite cardinal number. The Jains defined a system of infinite numbers, of which $$\aleph_0$$ is the smallest. In the Jaina work on the theory of sets, two basic types of infinite cardinal numbers are distinguished. On both physical and ontological grounds, a distinction was made between asmkhyata and ananata, between rigidly bounded and loosely bounded infinities.

 * ==References==
 * Jain, L. C. (1982). Exact Sciences from Jaina Sources.
 * Singh, N. (1987). Jain Theory of Actual Infinity and Transfinite Numbers.
 * Jain, L. C. (1973). "Set theory in the Jaina school of mathematics", Indian Journal of History of Science.
 * Agrawal, D. P. (2000). Ancient Jaina Mathematics: an Introduction, Infinity Foundation.
 * O'Connor, J. J. and E. F. Robertson (1998). "Georg Ferdinand Ludwig Philipp Cantor", MacTutor History of Mathematics archive.
 * O'Connor, J. J. and E. F. Robertson (1998). "Georg Ferdinand Ludwig Philipp Cantor", MacTutor History of Mathematics archive.

To Noe: This stuff is purely an attempt to gain credit for priority for the Jaina. Even if it is true historically (I do not know whether it is), it contains no useful mathematical information. It just clutters up the article with irrelavant stuff. If the authors of this material must put it in Wikipedia, let them put it into a purely historical article, NOT IN AN ARTICLE ON MATHEMATICS. JRSpriggs 11:55, 9 April 2006 (UTC)

 * To JJSpriggs: Like you, I think this stuff probably doesn't belong in the article, because it most likely is untrue. But you seem to think that even if it's true, it doesn't belong - and I must object to that. An article on a mathematical subject cannot be limited to "useful mathematical information" - e.g., the following isn't useful mathematical information either: "The cardinal numbers were invented by Georg Cantor [...] in 1874–1884".--Niels Ø 17:58, 9 April 2006 (UTC)
 * If it is true that the Jains had notions of transfinite numbers, then it deserves some mention in the history section, but it depends on what exactly their notion was. I recently reverted the Jainist addition at aleph number, and it's harder to justify its presense there: aleph number is about a specific concept that Cantor invented.  Even if the Jains had something, was it really equivalent to Cantor's definition?  Anyway, I followed some of those links, and read about infinity being the number of grains of sand in a cylinder the size of the Earth, and about distinguishing between infinity in 1, 2, 3, or more dimensions.  Neither of these concepts has anything to do with transfinite numbers, so I remain wholly unconvinced that this stuff belongs anywhere other than an article about ancient indian mathematics. -lethe talk [ +] 18:16, 9 April 2006 (UTC)
 * To Noe & Lethe: Well, I am not too keen on the historical stuff about Cantor either, but I realize that we must give credit to those who created the mathematics which we are using for three reasons: (1) because it is an ethical standard of our society (otherwise we are plagarists); (2) to provide a way for people to find additional reference materials on the subject which were not explicitly mentioned in the article; and (3) to disambiguate notions with similar names by qualifying them with the author's name.  I do not think that any of these reasons apply to the Jaina material because it was not a source used by Western mathematicians in developing the mathematics which we are writing about.  If the user who put in this stuff had stated a valid and original theorem or other mathematical fact which had been developed by the Jaina, then that would change this judgement. JRSpriggs 04:25, 10 April 2006 (UTC)
 * I agree with your suggestion that the Jaina material be put into "an article about ancient indian mathematics". Let them create ONE article called Jaina mathematics.  Then they can put a SINGLE pointer to that article in each mathematics article where they think that it would be appropriate.  It should be *Jaina mathematics in the "See also" section. JRSpriggs 06:00, 10 April 2006 (UTC)
 * I agree with your points. -lethe talk [ +] 11:31, 10 April 2006 (UTC)

I think the important point here is that it is inappropriate to insert mention of vaguely-related Jaina math concepts into very specific Cantor-formulated articles. For example, with transfinite number, clearly the article is about the concept originated by Cantor, and indeed that's what the term is used in modern mathematics to refer to. From the included material on Jaina mathematics, it's clear that their concepts are in fact distinct from Cantor's. Just because one author (in the given sources) has seen fit to call the Jaina number concept "transfinite" does not mean it is the same as Cantor's.  It certainly does not justify including material that is only vaguely related into the article.

I agree with the above proposal: in articles where Jaina mathematics are directly relevant, some mention (via a link) would be appropriate, but excessive mentions should be avoided. After all, other cultures have come up with their own mathematics also that has direct bearing on some of these concepts. Why ignore them and give Jaina mathematics all the attention? --C S (Talk) 06:37, 11 April 2006 (UTC)
 * I wouldn't object to these mentions in the article infinity (in addition to obviously any article on ancient indian mathematics). But any idea that they arrived at Cantor's definitions I will view with extreme skepticism.  I saw nothing of the sort in the few links I followed. -lethe talk [ +] 06:53, 11 April 2006 (UTC)

Why not to split article in two (linguistics and mathematics)?
Hi! Excuse my English! I'm a russian speaker. We have completely different terms for these notions in linguistics and mathematics (also as for term "ordinal number"). So, trying to decide what to do with interwiki, I realized, that it is also wrong in English to put both notions in one article. Wikipedia has disambig feature, so why not to use it? It seems to me, that it should be one common disambig article and two separate articles for two spheres, may be with linguistics one redirected to "Names of numbers in English". But I can't edit article myself, because I'm not sure I'm right. I don't know English well, so may be there are some weighty reasons for merging two notions? Dims 23:22, 23 September 2006 (UTC)

 * Zdorovo, tovarish! :)

 * I would very much disagree with moving the math part of this article to cardinal number (mathematics), as the main usage of cardinal numbers is in math. Perhaps the linguistics term needs to go to cardinal number (linguistics) while the math article staying where it is, at cardinal number. Anybody willing to create such a a linguistics article? Oleg Alexandrov (talk) 03:09, 24 September 2006 (UTC)

 * Oleg, this is similar to the dispute at Talk:Ordinal number over renaming that article. Notice that the ugly paragraph which you moved to the bottom of this article contains the disambiguation information which is usually at the top to allow people who got here by mistake to find their way without reading this whole article or becoming discouraged and giving up. JRSpriggs 03:33, 24 September 2006 (UTC)
 * Well, the linguistics concept is not much more different from the math one (at least that's what I can see with my untrained eyes). So the question remains the same, anybody willing to write a cardinal number (linguistics) article? (By the way, I doubt that many linguists actually look at the linguistics concept of a cardinal number). Oleg Alexandrov (talk) 03:39, 24 September 2006 (UTC)

 * I think there's a good deal less distinction between the linguistic and mathematical notions for the "cardinal" concepts than there is for the "ordinal" ones. The notion of an "ordinal number" considerably predates Cantor, and to the best of my knowledge always meant words like "first" and "seventeenth". I don't see that there was any explicable idea there that got generalized into Cantorian ordinals; AFAIK it was pretty much exclusively a linguistic concept, not one with an underlying noumenal referent. Note that the concept captured by Cantorian ordinals is not so much "order in which something occurs" as "length of a sequence"; that isn't something you'd express by st/rd/th-type words, and I don't know that it had a name before Cantor.
 * On the other hand the earlier notion of "cardinal number" -- that is, just, you know, ordinary numbers, counting how many of something there are, does generalize directly to Cantorian cardinals. --Trovatore 08:07, 24 September 2006 (UTC)

 * Perhaps adding a little note like "For the linguistic concept, see names of numbers in English." at the top (and deleting the linguistics section of the article altogether, or placing the material in that little section in the linked-to article) would work. -Grick(talk to me!) 09:34, 6 December 2006 (UTC)

An idea about cardinality of an infinite set
I've read over this site about cardinal number. And I don't know whether the sequence (aleph 0, aleph 1, aleph 2,...) exhausts all the cardinality of any infinite set X. How about your opinions? --Frejer

 * Yes, every infinite cardinal is of the form $$\aleph_\alpha$$ for some ordinal $$\alpha$$ (assuming the Axiom of Choice, of course). --Zundark 14:30, 3 November 2006 (UTC)

 * Of course, even assuming the Axiom of Choice (which most everybody is inclined to do), it can be impossible to prove which ordinal number a is. Without the continuum hypothesis, examples like $$2^{\aleph_0}$$ could be equal to $$\aleph_1$$, $$\aleph_{42}$$, or even $$\aleph_\omega$$.  However, the axiom of choice requires it be SOME $$\aleph_\alpha$$, where $$\alpha$$ is an ordinal number.  Eebster the Great (talk) 04:19, 8 May 2008 (UTC)

 * There are still limits on what $$2^\omega$$ can be; per König's theorem (set theory), $$2^\omega$$ has confinality greater than ω and so cannot equal $$\aleph_\omega$$. — Carl (CBM · talk) 11:46, 8 May 2008 (UTC)

 * But $$2^{\omega_0}$$ ≠ $$2^{\aleph_0}$$; one is ordinal and the other cardinal. But I might be wrong; is there an upper limit to $$\alpha$$ in $$\aleph_\alpha = 2^{\aleph_0}$$? Eebster the Great (talk) 21:11, 16 May 2008 (UTC)

 * I tend to follow the usual convention of using ordinals and cardinal interchangeably in exponents. What I was saying is that the cofinality of $$2^{\aleph_0}$$ can't be ω. There is no upper limit on what $$2^{\aleph_0}$$ can be; it can be any cardinal with an uncountable cofinality. — Carl (CBM · talk) 21:19, 16 May 2008 (UTC)
 * So for example it could be $${2^{\aleph_0}}^+$$ ? :-) --Trovatore (talk) 09:00, 17 May 2008 (UTC)

I'm under the impression due to a statement by Cohen that without the AC the continuum can be larger than any aleph. Does anyone think there are that many real numbers? Breath of the Dying (talk) 13:26, 24 September 2009 (UTC)

 * Even without AC, there is still an aleph that is not less than the continuum (see Hartogs number). What exactly did Cohen say? --Zundark (talk) 14:05, 24 September 2009 (UTC)

 * Well look Cohen just keeps on saying he disbelieved the continuum hypothesis because the powerset operation is very "rich", so I had that thought. I personally think it's pointless attempting to imagine how "big" these cardinals are, since nowhere in the real world do we see them. They're just a product of axiomatic set theory, i.e. logic.Breath of the Dying (talk) 16:42, 24 September 2009 (UTC)

 * Easton's theorem indicates that the continuum can be made larger than any given aleph by appropriate use of forcing. However, as Zundark said, the continuum cannot be made larger than all alephs simultaneously because it cannot be greater than its own Hartogs number. JRSpriggs (talk) 10:04, 25 September 2009 (UTC)

Linguistic usage link
I have replaced the linguistics section with a link to Names of numbers in English at the top of the article. The content of this article is about infinite cardinals in mathematics. However, to those unfamiliar with mathematics, "cardinal number" refers to the linguistic meaning, so the link must be at the top of the article, not hidden at the bottom. Many other Wikipedia articles use the same "this article ..." link.

If you disagree with what I did, or think it violated Wikipedia policy, please tell me on my talk page. Pcu123456789 04:14, 27 January 2007 (UTC)

 * I modified the "Common use meaning" slightly not because it was wrong, but only because I didn't immediately see the connection between readin', 'riting & 'rithmatic. Numbers are math not language! (yes, there are a million things wrong with that thought. lol) Reference materials are not only about containing the "data" but also about being able to find the data we're intending to look for.  For example, how could I ever discover the meaning of Phthongometer if I didn't know (or forgot) that it started with a PH and the search options were too narrowly confined.
 * Anyway I stole this modified definition from Japanese counter word: Ordinal numbers which for the first time in 44 years made sense to me. (I'm adding this copy from ordinal discussion page) J-puppy (talk) 22:10, 26 March 2009 (UTC)
 * I don't like the current note. Cardinal numbers always indicate quantity, whether in specialized mathematical usage or ordinary usage. I can't presently think of a better way of putting it, though. Algebraist 22:14, 26 March 2009 (UTC)
 * Cardinal numbers always indicate quantity? wow and all this time I thought that such a fancy name must have a special meaning. I can't presently think of a better way of putting it either. Hopefully someone cleverer than me will. Thank you for teaching me. =) J-puppy (talk) 22:27, 26 March 2009 (UTC)

Doubts over the recently added 'Simplified Explanation' section.
This section was added about an hour ago. It does not cover infinite cardinals and is therefore incorrect or incomplete. The introductory paragraph already mentions the natural numbers, hence this new section is redundant. I would suggest removing this new section. Aaron McDaid (talk - contribs) 10:44, 27 May 2007 (UTC)

 * Maybe you are right, perhaps the simplified section does contain inaccuracies, I don't know. I came to this Wikipedia page to find out what a cardinal number was, andn after reading the entire page became totally confused.  I may be thick, but the article is far to complex for me to understand.  Whiulst it now makes more sense (after I found out what a cardinal number is elsewhere on the net), it is not useful to a beginner.  We need a simple explanation.  By all means correct my text if it is wrong, but do not over-complicate it to the point of being un-readable to the mathematically ignorant.  Thanks for commenting rather than the more rash instant removal.
 * Although 11:17, 27 May 2007 (UTC)

 * Simplified Mathematics
 * To: JRSpriggs

 * You have reversed the "simplified Explanation in the article Cardinal Numbers.

 * I came to this Wikipedia page to find out what a cardinal number was, andn after reading the entire page became totally confused. I may be thick, but the article is far to complex for me to understand. Whilst it now makes more sense (after I found out what a cardinal number is elsewhere on the net), it is not useful to a beginner. We need a simple explanation. By all means correct my text if it is wrong, but do not over-complicate it to the point of being un-readable to the mathematically ignorant. Commenting rather than the more rash instant removal would have been more polite.

 * I am not a mathematician, but I do like to understand maths. Unfortunately, all too often the mathematical topics in Wikipedia go into so much detail that the novice is left completely confused by massive in-depth examination of every minute detail. For all maths topics we need a simplified explanation, as if you were teaching a child of 7. Then we can have the more complex explanations that only a trained mathematician will understand.

 * By all means remiove a simple explanation if you are willing to clean up the original over-complicated article to make it understandable. Otherwise just pressing the delete button is lazy and arrogant. How would you like it if somone did that to you? - I guess you know already because it seems to happen all the time to everyone.

 * I hope you will take the time to make the article more understandable.

 * Thanks in advance for any help you can give in this area. Although 13:58, 28 May 2007 (UTC)

 * Retrieved from "http://en.wikipedia.org/wiki/User_talk:JRSpriggs"

 * I think this is the crucial sentence which could be improved:
 * ''While for finite sets the size is given by a natural number - the number of elements - cardinal numbers (cardinality) can also classify degrees of infinity.
 * We need to mention, as this does, that cardinal numbers describe the size of sets - natural numbers (1,2,3) for finite sets and others for infinite sets. And also that not all infinite sets are the same size. Perhaps give an example about the relative sizes of naturals, rationals and reals? I think the fact that infinite sets can be compared for size is the interesting concept and which could be explained better. Any ideas? Aaron McDaid (talk - contribs) 22:37, 28 May 2007 (UTC)

 * The information you are asking for is already in the lead, if you read it with an open mind. However, the examples and explanations must and do come later.
 * The added section which I removed was not helpful because it ignores what cardinal numbers are, and instead talks about what they are not. This will only confuse most people. I have been working (as needed) on this article for over a year, so do not treat my attempts to protect it as interference with your supposedly needed, but actually haphazard changes. JRSpriggs 07:46, 29 May 2007 (UTC)

 * Hi, it wasn't me that made the original big change. I agree that the new paragraph made the article worse not better, but that doesn't mean the current article is perfect. The current article is correct, and perfect for people who already know the subject like myself but just maybe it could be a little clearer in the introduction. For example the phrase 'degrees of infinity' isn't as clear as it could be - it's just talking about the sizes of infinite sets. It could simply be used to say that cardinal numbers measure the size of sets, both finite sets and infinite sets? The word 'cardinality' is introduced without explaining that it's just the technical term for 'size'. Aaron McDaid (talk - contribs) 18:43, 29 May 2007 (UTC)

I added the following paragraph:
 * The cardinal numbers are: $$0, 1, 2, 3, \cdots n, \cdots ; \aleph_0, \aleph_1, \aleph_2, \cdots \aleph_{\alpha}, \cdots $$. That is, they are the natural numbers (finite cardinals) followed by the aleph numbers (infinite cardinals). The aleph numbers are indexed by ordinal numbers. The natural numbers and aleph numbers are subclasses of the ordinal numbers. If the axiom of choice fails, the situation becomes more complicated — there are additional infinite cardinals which are not alephs.

I hope that this satisfies you. JRSpriggs 05:00, 30 May 2007 (UTC)

Thanks for the improvement. Although 21:33, 31 May 2007 (UTC)

Linguistics, again
I tend to think that some of the linguistic material should be moved back to this page; at minimum, there should be two articles, one for cardinal number (mathematics) and cardinal number (linguistics). The article on Names of numbers in English doesn't cover the field adequately; cardinal numbers appear in languages other than English. Linguists broadly distinguish four different kinds of numbers:

 * cardinal numbers
 * ordinal numbers
 * fractional numbers, and
 * serial numbers, used in counting.

- Smerdis of Tlön 17:29, 5 November 2007 (UTC)

Move proposal
I just noticed a move proposal at Talk:Cardinal (Catholicism) that may concern editors of this page. People who watch this page might want to comment there. Sam Staton 14:16, 4 December 2007 (UTC)
 * To clarify, that proposal is about the disambiguation page cardinal, not this page. — Carl (CBM · talk) 14:20, 4 December 2007 (UTC)

Original definition
I remember that I read somewhere that Cantor's original definition of a "cardinal number" was the set of all sets equipotent with a given set. Something like:
 * $$Card(X) = \{ x | \exists f: x \to X, f \mbox{ is bijective} \} \,$$

of course now we know that this monster (except when X is the empty set) is not a set, but a proper class, but with this definition it makes sense to call Cantor's paradox a paradox... Does anyone know where is the reference for this? It would be appropriate to include (if sourced) in the article. Albmont (talk) 14:22, 14 May 2009 (UTC)

 * In article New Foundations, this definition is attributed (without sources!) to Gottlob Frege. Albmont (talk) 17:41, 14 May 2009 (UTC)
 * This seems to confirm my belief that Cantor did not define ordinals or cardinals to be sets at all: for him they were entirely new objects. The definition you give seems to be (a rather anachronistic version of) something Frege says in the Grundlagen. Algebraist 18:26, 14 May 2009 (UTC)
 * See Scott's trick. Each proper equivalence class is replaced by its subset consisting of the elements having minimal rank. Thus cardinal numbers can be defined as sets without assuming the axiom of choice. JRSpriggs (talk) 00:01, 15 May 2009 (UTC)
 * Seen Scott's trick. It sounds like a circular definition to me. Yickes, these math articles in Wikipedia are circular! Maybe they should all be moved to Wikibooks and organized in a hierarchical (sp?) way, with article index α rigorously based on all articles index x for every ordinal number x < α... (Wikipedia is not well-founded). Albmont (talk) 14:20, 15 May 2009 (UTC)
 * The circularity you see is from confusing an internal definition of ordinals or cardinals with an external one. On the one hand, if one already knows what the ordinals are, one can define the cumulative hierarchy by transfinite iteration. On the other hand, if one starts with a model of ZFC, one can define within it (representatives for) ordinals. But the former construction is an external, meta-level construction, while the latter is an internal, object-level definition. — Carl (CBM · talk) 16:19, 15 May 2009 (UTC)

Different definitions
Hmm, should this page discuss different definitions of cardinals (Von Neumann, Frege-Russell, Scott-Potter)? -- Schneelocke (talk) 23:35, 27 November 2009 (UTC)
 * They're all mentioned briefly. A proper discussion seems a good idea, though. Algebraist 23:38, 27 November 2009 (UTC)

 * There are other articles such as https://en.wikipedia.org/wiki/Measurable_cardinal that speak of the "subsets" of a cardinal number in such a way that "the subsets of the cardinal K" are treated as if they are the same things as the subsets of a set S that "has cardinality K". It would be useful to reconcile this terminology with the definition of cardinality given in the current article.  The definition in the current article treats a cardinal as an equivalence relation defined on a collection of sets.  From this point of view, a cardinal K is a collection of ordered pairs of sets.   So if the set S "has cardinality K", this indicates that S is an member of at least one ordered pair in the collection K. So "the subsets of K" are not the same set as "the subsets of S". Tashiro~enwiki (talk) 17:32, 28 March 2017 (UTC)

 * A cardinal is not an equivalence relationship; it is an equivalence class — a class of sets which are equinumerous to each other.
 * Those other articles are using "cardinal number Κ" as a shorthand (like using "crown" for King Henry VIII) for the "initial ordinal of Κ", one member of the equivalence class. In this case, talking about its subsets in this way is appropriate. JRSpriggs (talk) 20:18, 28 March 2017 (UTC)

Why the class of sets equinumerous to a given nonempty set is not a set
In an edit summary, said "Formal definition: "too large to be a set" is not defined anywhere on Wikipedia, so this should be clarified. Also, there's no argument for why that class is too large.". See Limitation of size. Suppose X is any nonempty set (the empty set does not have this problem since { {} } is a set). If
 * $$Z = \{ Y \, \vert \, Y \approx X \} \,$$

were a set, then for some fixed a∈X let
 * $$W = \{ b \, \vert \, b = Z \lor ( b \in X \land b \neq a ) \} \,.$$

We get
 * $$W \approx X \,$$
 * $$W \in Z \,$$
 * $$Z \in W \,.$$

Consequently
 * $$A = \{ W, Z \} \,$$

is a nonempty set containing no element disjoint from A itself, which is a violation of the axiom of foundation. JRSpriggs (talk) 08:04, 23 October 2010 (UTC)
 * By the way, if Z∈X already (so that W defined above does not work), then just set W=X. JRSpriggs (talk) 08:10, 23 October 2010 (UTC)

Looking back over this I see that I have not explained why the class Z is too large to be a set, merely that it cannot be well-founded. To see that it is too large, let us define a class function
 * $$F (c) = X \times \{ c \} = \{ \langle b, c \rangle \, \vert \, b \in X \} \approx X \,$$

whose domain is the universe V and each value of which is a distinct member of the class Z. Thus Z is at least as large as the universe and so cannot be a set. The reason this argument does not apply to the empty set is that {}×{c}={} so the values in that case are all equal instead of being distinct. JRSpriggs (talk) 11:15, 23 October 2010 (UTC)

Incorrect statement in article
The article says:
 * If the axiom of choice holds and $$\aleph_0 \leq \kappa$$ and $$\mu < \kappa \,$$ and $${\mu}^{\lambda} = \kappa \,,$$ then $$\aleph_0 \leq \lambda < \kappa \,$$ and for any cardinal $$\nu \,,$$ $$1 < \nu \leq \kappa \,$$ implies $${\nu}^{\lambda} = \kappa \,.$$

But this is wrong. For example, if $$\mu=\beth_\omega$$ and $$\lambda=\aleph_0$$ and $$\kappa=(\beth_\omega)^{\aleph_0}$$ and $$\nu=2$$, then $$\nu^\lambda<\kappa$$. --Zundark (talk) 16:12, 25 January 2011 (UTC)
 * To Zundark: Thank you for pointing that out. It was my mistake. I have now reverted myself. JRSpriggs (talk) 20:03, 25 January 2011 (UTC)

least rank-definition
However, if we restrict from this class to those equinumerous with X that have the least rank, then it will work (this is a trick due to Dana Scott: it works because the collection of objects with any given rank is a set).

I think that this trick is much older. It is mentioned in Suppes, "Axiomatic Set Theory", (1960), but I'm triyng to find a better reference. kismalac 22:02, 14 April 2011 (UTC)
 * I don't know what that proves. Scott was active in 1960. --Trovatore (talk) 22:20, 14 April 2011 (UTC)
 * It doesn't prove anything. By "better" I meant "earlier". kismalac 06:27, 15 April 2011 (UTC)

Remaking of the History section
The section history is very awful and has incorrect affirmations.

I will read the only historical source. The most important historical sources, the work of Cantor, Dauben, etc. is not cited and I have serious about the interpretation of the very only source.

I will work, when I have some time, on the Portuguese wikipedia and re-work the historical section.

The article is very weak and would be remaked in most part.

Apologize the bad English.

--Gonzalcg (talk) 23:40, 22 June 2012 (UTC)

For example, the statement:

 * "Cantor first established cardinality as an instrument to compare finite sets; e.g. the sets {1,2,3} and {2,3,4} are not equal, but have the same cardinality: three."

is false.

--Gonzalcg (talk) 00:20, 23 June 2012 (UTC)

 * Would prefer a citation for this. — Arthur Rubin  (talk) 08:27, 23 June 2012 (UTC)

Citation
Of course: G. Cantor, 1932. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. E. Zermelo, Ed. Berlin: Springer; reprinted Hildesheim: Olms, 1962; Berlin/Heidelberg/New York: Springer, 1980. http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN237853094&IDDOC=49439

But who actually must charge with the onus of the proof is the people which have written the above affirmation.

Does not matter. In page 125 of the cited paper by Deiser, the analysis of the research of Cantor 1873-74 this work is considered the first. In the first paper by Cantor, he has considered only infinite sets. See p. 115 of the work above, in which Cantor has begun with algebraic reals, i.e. an infinite set.

I don't understand, please explain to me: why you and others do not want that the mistakes were corrected? Why systematically my editions are reverted?

I insist: the history section is awful and it must be totally rewritten. The reader must be warned that the contents is not reliable.

Apologize the bad English

--Gonzalcg (talk) 21:28, 23 June 2012 (UTC)

Some problems of the section
The section contributes to the following misunderstandings:

 * 1) Cantor was the first to consider the one-to-one correspondence as a way of quantity (because previous authors are not cited).

 * 2) Cantor first has formulated the one-to-one correspondence for finite sets and then has extended it for infinite ones.

 * 3) Cantor was the first to think about how to extend the notion of quantity based on one-to-one correspondences to infinite sets.

 * 4) Cantor has given a suitable notion of cardinal number

 * 5) The current notion of cardinal number is essentially the Cantor concept.

If you think that some of the above (1-5) statements would be true, please talk me.

--Gonzalcg (talk) 17:46, 24 June 2012 (UTC)

 * I think it's correct to say that (one of the) current concepts of cardinal number is a formalization of the Cantor concept, although that may also be true of some of the other contemporaneous concepts of cardinal number. The history section needs sources, but I do not believe Deiser represents current thinking among mathematical historians.  — Arthur Rubin  (talk) 18:18, 24 June 2012 (UTC)

 * Perhaps Deiser does not represents. The problem of the section is bigger that this: what is the Cantorian concept of cardinal number? In both senses: the Cantorian double line definition,, and the mathematical use of the notion of cardinal. In other words: what was the main contribution of Cantor for the concept (and not the theory) of cardinal numbers? Dauben, Hallett, Hao Wang, Fraenkel, Bar Hillel, etc. can help us in this task, but the work of Cantor is the primary source. Moreover, at least Galileo's paradox and the work of Bolzano should be cited. — Preceding unsigned comment added by Gonzalcg (talk • contribs) 19:11, 24 June 2012 (UTC)
 * I certainly think you're correct that the history section is garbled, and in particular that 1, 2, and 3 above are wrong (I'd have to check to see if the text actually says those things, exactly, but it definitely says some wrong things). Numbers 4 and 5 are a little more iffy.  On 4, suitable for what, exactly?  And on 5, it depends on what you mean by "essentially".  I think it is essentially the same concept; the use of initial ordinals, which Cantor did not use, is a convenient way of coding that concept. --Trovatore (talk) 16:19, 27 June 2012 (UTC)

 * I corrected History section to avoid making the inaccurate statement 2) above. Weux (talk) 19:47, 29 April 2013 (UTC)

The cause
The first edition of the section has more mistakes and there is not citation nor sources:

http://en.wikipedia.org/w/index.php?title=Cardinal_number&diff=4362151&oldid=4352334

Note that the "1)" above is explicitly asserted:
 * "Cantor invented the one-to-one mapping, which easily showed that two sets had the same cardinality if there was a one-to-one correspondence between the members of the set."

Further corrections were not enough to avoid misunderstandings. In similar cases it is advised to rewrite all the section.

Notice also that the above (false) statement: "Cantor first established cardinality as an instrument to compare finite sets; e.g. the sets {1,2,3} and {2,3,4} are not equal, but have the same cardinality: three." already was in http://en.wikipedia.org/w/index.php?title=Cardinal_number&oldid=4362151

It was not deleted. But my contributions were deleted almost immediately, although they are serious and well founded.

Apologize the bad English

--Gonzalcg (talk) 18:43, 24 June 2012 (UTC)

SIMPLIFY INTRO!
I had to look stuff up to follow your intro. Go to any length you want in the body of the article, but keep it simple in the intro. This was written by a mathematician, to a mathematician...hardly appropriate for an encyclopedia. Pb8bije6a7b6a3w (talk) 15:22, 10 June 2014 (UTC)

 * What stuff did you look up? -Jochen Burghardt (talk) 17:32, 10 June 2014 (UTC)

 * I guessed that bijection was a problematic term and put the idea in simpler terms. --Mark viking (talk) 20:48, 5 September 2014 (UTC)

last chances with my life........i will never came back here in my one whole human one ............but i always be with my world in tack with me to breath with my favorite beautiful views......this are only way to feel my refreshing into my love life in lives....... — Preceding unsigned comment added by 203.198.248.59 (talk) 04:34, 25 October 2014 (UTC)

 * I wholeheartedly concur. Please DO keep in mind Wikipedia is used by elementary school children as well as those in higher learning.  As such, you must NOT begin with collegiate-level mathematical language, as 90% of readers are NOT familiar with it.  Instead, begin with a simple definition and a simple example suitable for elementary school children. Then, move the higher-level description to the Formal Definition section where it belongs.  DO NOT repeate the Formal Definition section in the intro. Clepsydrae (talk) 16:21, 22 January 2022 (UTC)

Truncated subtraction
It is possible to define a total subtraction operation on cardinal numbers that extends the truncated subtraction operation on the finite cardinal numbers. Using the same variable names as in the section, it is defined by
 * σ ∸ μ = inf {κ | μ + κ ≥ σ}.

If μ < σ, this is the unique cardinal number κ such that μ + κ = σ. If μ ≥ σ, σ ∸ μ = 0. This can then be used to define (truncated) multiset subtraction, extending set subtraction. Since OR is a no-no, this cannot be added without a reference to published work. The idea is so obvious that it is unlikely no one has written it down, but I could not find it described. --Lambiam 13:03, 14 December 2020 (UTC)

Sourcing for our definition of cardinal arithmetic
In the discussion on Talk:Infinity plus one, I'm worried about the lack of sourcing for our definitions in the Cardinal arithmetic section. I'd be grateful if someone could confirm they are established in the literature. — Charles Stewart (talk) 14:41, 21 January 2022 (UTC)
 * Added a citation there from the book that happens to be on my shelf here, at least for addition, multiplication and exponentiation.
 * Since addition and multiplication are trivial, "Cardinal arithmetic" in works by Shelah and other set theorists usually refers to cardinal exponentiation; but introductory sections like the one in Schindler (2014) :I cited there use it to refer to all the operations. When I am home tonight I can check my copy of Kunen's Set Theory, which is more widely used than Schindler's I think. Felix QW (talk) 14:57, 21 January 2022 (UTC)
 * Thanks very much! I note also that Thomas Forster's list of research avenues has one on cardinal arithmetic and permutation models: "Cardinal arithmetic is that part of set theory for which equipollence is a congruence relation" . That's a nice way of putting it: it's an unconventional source, but maybe it's usable attributed. — Charles Stewart (talk) 15:24, 21 January 2022 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Cardinal number


text:
Chinese Islamic cuisine

Chinese Islamic cuisine consists of variations of regionally popular foods that are typical of Han Chinese cuisine, in particular to make them halal. Dishes borrow ingredients from Middle Eastern, Turkic, Iranian and South Asian cuisines, notably mutton and spices. Much like other northern Chinese cuisines, Chinese Islamic cuisine uses wheat noodles as the staple, rather than rice. Chinese Islamic dishes include clear-broth beef noodle soup and chuanr.

The Hui (ethnic Chinese Muslims), Bonan, Dongxiang, Salar and Uyghurs of China, as well as the Dungans of Central Asia and the Panthays of Burma, collectively contribute to Chinese Islamic cuisine.

History
Due to the large Muslim population in Western China, many Chinese restaurants cater to or are run by Muslims. Northern Chinese Islamic cuisine originated in China proper. It is heavily influenced by Beijing cuisine, with nearly all cooking methods identical and differs only in material due to religious restrictions. As a result, northern Islamic cuisine is often included in home Beijing cuisine though seldom in east coast restaurants.

During the Yuan dynasty, halal and kosher methods of slaughtering animals and preparing food was banned and forbidden by the Mongol emperors, starting with Genghis Khan who banned Muslims and Jews from slaughtering their animals their own way and made them follow the Mongol method.

Among all the [subject] alien peoples only the Hui-hui say "we do not eat Mongol food." [Cinggis Qa'an replied:] "By the aid of heaven we have pacified you; you are our slaves. Yet you do not eat our food or drink. How can this be right?" He thereupon made them eat. "If you slaughter sheep, you will be considered guilty of a crime." He issued a regulation to that effect ... [In 1279/1280 under Qubilai] all the Muslims say: “if someone else slaughters [the animal] we do not eat." Because the poor people are upset by this, from now on, Musuluman [Muslim] Huihui and Zhuhu [Jewish] Huihui, no matter who kills [the animal] will eat [it] and must cease slaughtering sheep themselves, and cease the rite of circumcision.

Traditionally, there is a distinction between Northern and Southern Chinese Islamic cuisine despite both using lamb and mutton. Northern Chinese Islamic cuisine relies heavily on beef, but rarely ducks, geese, shrimp or seafood, while southern Islamic cuisine is the reverse. The reason for this difference is due to availability of the ingredients. Oxen have been long used for farming and Chinese governments have frequently strictly prohibited the slaughter of oxen for food. However, due to the geographic proximity of the northern part of China to minority-dominated regions that were not subjected to such restrictions, beef could be easily purchased and transported to Northern China. At the same time, ducks, geese and shrimp are rare in comparison to Southern China due to the arid climate of Northern China.

A Chinese Islamic restaurant can be similar to a Mandarin restaurant with the exception that there is no pork on the menu and the dishes are primarily noodle/soup based.

In most major eastern cities in China, there are very limited Islamic/Halal restaurants, which are typically run by migrants from Western China (e.g., Uyghurs). They primarily offer inexpensive noodle soups only. These restaurants are typically decorated with Islamic motifs such as Islamic writing.

Another difference is that lamb and mutton dishes are more commonly available than in other Chinese restaurants, due to the greater prevalence of these meats in the cuisine of Western Chinese regions. (Refer to image 1.)

Other Muslim ethnic minorities like the Bonan, Dongxiang, Salar and Tibetan Muslims have their own cuisines as well. Dongxiang people operate their own restaurants serving their cuisine.

Many cafeterias (canteens) at Chinese universities have separate sections or dining areas for Muslim students (Hui or Western Chinese minorities), typically labeled "qingzhen". Student ID cards sometimes indicate whether a student is Muslim and will allow access to these dining areas or will allow access on special occasions such as the Eid feast following Ramadan.

Several Hui restaurants serving Chinese Islamic cuisine exist in Los Angeles. San Francisco, despite its huge number of Chinese restaurants, appears to have only one whose cuisine would qualify as halal.

Many Chinese Hui Muslims who moved from Yunnan to Burma (Myanmar) are known as Panthays operate restaurants and stalls serving Chinese Islamic cuisine such as noodles. Chinese Hui Muslims from Yunnan who moved to Thailand are known as Chin Haw and they also own restaurants and stalls serving Chinese Islamic food.

In Central Asia, Dungan people, descendants of Hui, operate restaurants serving Chinese Islamic cuisine, which is respectively referred to as Dungan cuisine there. They cater to Chinese businessmen. Chopsticks are used by Dungans. The cuisine of the Dungan resembles northwestern Chinese cuisine.

Most Chinese regard Hui halal food as cleaner than food made by non-Muslims so their restaurants are popular in China. Hui who migrated to Northeast China (Manchuria) after the Chuang Guandong opened many new inns and restaurants to cater to travelers, which were regarded as clean.

The Hui who migrated to Taiwan operate Qingzhen restaurants and stalls serving Chinese Islamic cuisine in Taipei and other big cities.

The Thai Department of Export Promotion claims that "China's halal food producers are small-scale entrepreneurs whose products have little value added and lack branding and technology to push their goods to international standards" to encourage Thai private sector halal producers to market their products in China.

A 1903-started franchise serving Muslim food is Dong Lai Shun in Hankou.

400 meters have to be kept as a distance from each restaurant serving beef noodles to another of its type if they belong to Hui Muslims, since Hui have a pact between each other in Ningxia, Gansu and Shaanxi.

Halal restaurants are checked up upon by clerics from mosques.

Halal food manufacture has been sanctioned by the government of the Ningxia Autonomous Region.

Lamian
Lamian (, Dungan: Ламян) is a Chinese dish of hand-made noodles, usually served in a beef or mutton-flavored soup (湯麪, даңмян, tāngmiàn), but sometimes stir-fried (炒麪, Чаомян, chǎomiàn) and served with a tomato-based sauce. Literally, 拉, ла (lā) means to pull or stretch, while 麪, мян (miàn) means noodle. The hand-making process involves taking a lump of dough and repeatedly stretching it to produce a single very long noodle.

Words that begin with L are not native to Turkic — läghmän is a loanword as stated by Uyghur linguist Abdlikim: It is of Chinese derivation and not originally Uyghur.

Beef noodle soup
Beef noodle soup is a noodle soup dish composed of stewed beef, beef broth, vegetables and wheat noodles. It exists in various forms throughout East and Southeast Asia. It was created by the Hui people during the Qing dynasty of China.

In the west, this food may be served in a small portion as a soup. In China, a large bowl of it is often taken as a whole meal with or without any side dish.

Chuanr
Chuanr (Chinese: 串儿, Dungan: Чўанр, Pinyin: chuànr (shortened from "chuan er"), "kebab"), originating in the Xinjiang (新疆) province of China and in recent years has been disseminated throughout the rest of that country, most notably in Beijing. It is a product of the Chinese Islamic cuisine of the Uyghur (维吾尔) people and other Chinese Muslims. Yang rou chuan or lamb kebabs, is particularly popular.

Suan cai
Suan cai is a traditional fermented vegetable dish, similar to Korean kimchi and German sauerkraut, used in a variety of ways. It consists of pickled Chinese cabbage. Suan cai is a unique form of pao cai due to the material used and the method of production. Although suan cai is not exclusive to Chinese Islamic cuisine, it is used in Chinese Islamic cuisine to top off noodle soups, especially beef noodle soup.

Nang
Nang (Chinese: 馕, Dungan: Нәң) is a type of round unleavened bread, topped with sesame. It is similar to South and Central Asia naan.

title: Chinese Islamic cuisine


text:
Cantonese cuisine

Cantonese or Guangdong cuisine, also known as Yue cuisine ( or ), is the cuisine of Guangdong province of China, particularly the provincial capital Guangzhou, and the surrounding regions in the Pearl River Delta including Hong Kong and Macau. Strictly speaking, Cantonese cuisine is the cuisine of Guangzhou or of Cantonese speakers, but it often includes the cooking styles of all the speakers of Yue Chinese languages in Guangdong.

The Teochew cuisine and Hakka cuisine of Guangdong are considered their own styles. However, scholars may categorize Guangdong cuisine into three major groups based on the region's dialect: Cantonese, Hakka and Chaozhou cuisines. Neighboring Guangxi's cuisine is also considered separate despite eastern Guangxi being considered culturally Cantonese due to the presence of ethnic Zhuang influences in the rest of the province.

Cantonese cuisine is one of the Eight Great Traditions of Chinese cuisine. Its prominence outside China is due to the large number of Cantonese emigrants. Chefs trained in Cantonese cuisine are highly sought after throughout China. Until the late 20th century, most Chinese restaurants in the West served largely Cantonese dishes.

Background
Guangzhou (Canton) City, the provincial capital of Guangdong and the centre of Cantonese culture, has long been a trading hub and many imported foods and ingredients are used in Cantonese cuisine. Besides pork, beef and chicken, Cantonese cuisine incorporates almost all edible meats, including offal, chicken feet, duck's tongue, frog legs, snakes and snails. However, lamb and goat are less commonly used than in the cuisines of northern or western China. Many cooking methods are used, with steaming and stir-frying being the most favoured due to their convenience and rapidity. Other techniques include shallow frying, double steaming, braising and deep frying.

Compared to other Chinese regional cuisines, the flavours of most traditional Cantonese dishes should be well-balanced and not greasy. Apart from that, spices should be used in modest amounts to avoid overwhelming the flavours of the primary ingredients, and these ingredients in turn should be at the peak of their freshness and quality. There is no widespread use of fresh herbs in Cantonese cooking, in contrast with their liberal use in other cuisines such as Sichuanese, Vietnamese, Lao, Thai and European. Garlic chives and coriander leaves are notable exceptions, although the former are often used as a vegetable and the latter are usually used as mere garnish in most dishes.

Sauces and condiments
In Cantonese cuisine, ingredients such as sugar, salt, soy sauce, rice wine, corn starch, vinegar, scallion and sesame oil suffice to enhance flavour, although garlic is heavily used in some dishes, especially those in which internal organs, such as entrails, may emit unpleasant odours. Ginger, chili peppers, five-spice powder, powdered black pepper, star anise and a few other spices are also used, but often sparingly.

Dried and preserved ingredients
Although Cantonese cooks pay much attention to the freshness of their primary ingredients, Cantonese cuisine also uses a long list of preserved food items to add flavour to a dish. This may be influenced by Hakka cuisine, since the Hakkas were once a dominant group occupying imperial Hong Kong and other southern territories.

Some items gain very intense flavours during the drying/preservation/oxidation process and some foods are preserved to increase their shelf life. Some chefs combine both dried and fresh varieties of the same items in a dish. Dried items are usually soaked in water to rehydrate before cooking. These ingredients are generally not served a la carte, but rather with vegetables or other Cantonese dishes.

Traditional dishes
A number of dishes have been part of Cantonese cuisine since the earliest territorial establishments of Guangdong. While many of these are on the menus of typical Cantonese restaurants, some simpler ones are more commonly found in Cantonese homes. Home-made Cantonese dishes are usually served with plain white rice.

Deep fried dishes
There are a small number of deep-fried dishes in Cantonese cuisine, which can often be found as street food. They have been extensively documented in colonial Hong Kong records of the 19th and 20th centuries. A few are synonymous with Cantonese breakfast and lunch, even though these are also part of other cuisines.

Soups
Old fire soup, or lou fo tong, is a clear broth prepared by simmering meat and other ingredients over a low heat for several hours. Chinese herbs are often used as ingredients. There are basically two ways to make old fire soup – put ingredients and water in the pot and heat it directly on fire, which is called bou tong ; or put the ingredients in a small stew pot, and put it in a bigger pot filled with water, then heat the bigger pot on fire directly, which is called dun tong. The latter way can keep the most original taste of the soup.

Soup chain stores or delivery outlets in cities with significant Cantonese populations, such as Hong Kong, serve this dish due to the long preparation time required of slow-simmered soup.

Seafood

Due to Guangdong's location along the South China Sea coast, fresh seafood is prominent in Cantonese cuisine, and many Cantonese restaurants keep aquariums or seafood tanks on the premises. In Cantonese cuisine, as in cuisines from other parts of Asia, if seafood has a repugnant odour, strong spices and marinating juices are added; the freshest seafood is odourless and, in Cantonese culinary arts, is best cooked by steaming. For instance, in some recipes, only a small amount of soy sauce, ginger and spring onion is added to steamed fish. In Cantonese cuisine, the light seasoning is used only to bring out the natural sweetness of the seafood. As a rule of thumb, the spiciness of a dish is usually negatively correlated to the freshness of the ingredients.

Noodle dishes
Noodles are served either in soup broth or fried. These are available as home-cooked meals, on dim sum side menus, or as street food at dai pai dongs, where they can be served with a variety of toppings such as fish balls, beef balls, or fish slices.

Siu mei

Siu mei is essentially the Chinese rotisserie style of cooking. Unlike most other Cantonese dishes, siu mei solely consists of meat, with no vegetables.

Lou mei
Lou mei is the name given to dishes made from internal organs, entrails and other left-over parts of animals. It is widely available in southern Chinese regions.

Siu laap

All Cantonese-style cooked meats, including siu mei, lou mei and preserved meat can be classified as siu laap. Siu laap also includes dishes such as:

A typical dish may consist of offal and half an order of multiple varieties of roasted meat. The majority of siu laap is white meat.

Little pot rice

Little pot rice are dishes cooked and served in a flat-bottomed pot (as opposed to a round-bottomed wok). Usually this is a saucepan or braising pan (see clay pot cooking). Such dishes are cooked by covering and steaming, making the rice and ingredients very hot and soft. Usually the ingredients are layered on top of the rice with little or no mixing in between. Many standard combinations exist.

Banquet/dinner dishes
A number of dishes are traditionally served in Cantonese restaurants only at dinner time. Dim sum restaurants stop serving bamboo-basket dishes after the yum cha period (equivalent to afternoon tea) and begin offering an entirely different menu in the evening. Some dishes are standard while others are regional. Some are customised for special purposes such as Chinese marriages or banquets. Salt and pepper dishes are one of the few spicy dishes.

Dessert
After the evening meal, most Cantonese restaurants offer tong sui, a sweet soup. Many varieties of tong sui are also found in other Chinese cuisines. Some desserts are traditional, while others are recent innovations. The more expensive restaurants usually offer their specialty desserts. Sugar water is the general name of dessert in Guangdong province. It is cooked by adding water and sugar to some other cooking ingredients.

Delicacies
Certain Cantonese delicacies consist of parts taken from rare or endangered animals, which raises controversy over animal rights and environmental issues. This is often due to alleged health benefits of certain animal products. For example, the continued spreading of the idea that shark cartilage can cure cancer has led to decreased shark populations even though scientific research has found no evidence to support the credibility of shark cartilage as a cancer cure.

title: Cantonese cuisine


text:
Teochew cuisine

Teochew cuisine, also known as Chiuchow cuisine, Chaozhou cuisine or Teo-swa cuisine, originated from the Chaoshan region in the eastern part of China's Guangdong Province, which includes the cities of Chaozhou, Shantou and Jieyang. Teochew cuisine bears more similarities to that of Fujian cuisine, particularly Southern Min cuisine, due to the similarity of Teochew's and Fujian's culture, language, and their geographic proximity to each other. However, Teochew cuisine is also influenced by Cantonese cuisine in its style and technique.

Background
Teochew cuisine is well known for its seafood and vegetarian dishes. Its use of flavouring is much less heavy-handed than most other Chinese cuisines and depends much on the freshness and quality of the ingredients for taste and flavour. As a delicate cuisine, oil is not often used in large quantities and there is a relatively heavy emphasis on poaching, steaming and braising, as well as the common Chinese method of stir-frying. Teochew cuisine is also known for serving congee (or mue), in addition to steamed rice or noodles with meals. The Teochew mue is rather different from the Cantonese counterpart, being very watery with the rice sitting loosely at the bottom of the bowl, while the Cantonese dish is more a thin gruel.

Authentic Teochew restaurants serve very strong oolong tea called Tieguanyin in very tiny cups before and after the meal. Presented as gongfu tea, the tea has a thickly bittersweet taste, colloquially known as gam gam.

A condiment that is popular in Fujian and Taiwanese cuisine and commonly associated with cuisine of certain Teochew groups is shacha sauce. It is made from soybean oil, garlic, shallots, chilies, brill fish and dried shrimp. The paste has a savoury and slightly spicy taste. As an ingredient, it has multiple uses: as a base for soups, as a rub for barbecued meats, as a seasoning for stir-fried dishes, or as a component for dipping sauces.

In addition to soy sauce (widely used in all Chinese cuisines), Teochew people also use fish sauce in their cooking.

Teochew chefs often use a special stock called superior broth. This stock remains on the stove and is continuously replenished. Portrayed in popular media, some Hong Kong chefs allegedly use the same superior broth that is preserved for decades. This stock can as well be seen on Chaozhou TV's cooking programmes.

There is a notable feast in Teochew cuisine called jiat dot. A myriad of dishes are often served, which include shark fin soup, bird's nest soup, lobster, steamed fish, roasted suckling pig and braised goose.

Teochew chefs take pride in their skills of vegetable carving, and carved vegetables are used as garnishes on cold dishes and on the banquet table.

Teochew cuisine is also known for a late night meal known as meh siao or daa laang  among the Cantonese. Teochew people enjoy eating out close to midnight in restaurants or at roadside food stalls. Some dai pai dong-like eateries stay open till dawn.

Unlike the typical menu selections of many other Chinese cuisines, Teochew restaurant menus often have a dessert section.

Many people of Teochew origin, also known as Teochiu or Teochew people, have settled in Hong Kong and places in Southeast Asia like Malaysia, Singapore, Cambodia and Thailand. Influences they bring can be noted in Singaporean cuisine and that of other settlements. A large number of Teochew people have also settled in Taiwan, evident in Taiwanese cuisine. Other notable Teochew diaspora communities are in Vietnam, Cambodia and France. A popular noodle soup in both Vietnam and Cambodia, known as hu tieu, originated from the Teochew. There is also a large diaspora of Teochew people (most were from Southeast Asia) in the United States - particularly in California. There is a Teochew Chinese Association in Paris called L'Amicale des Teochews en France.

title: Teochew cuisine


text:
Co-NP

In computational complexity theory, co-NP is a complexity class. A decision problem X is a member of co-NP if and only if its complement $\textsf{NP}\ \overset{?}{=}\ \textsf{co-NP}$ is in the complexity class NP. The class can be defined as follows: a decision problem is in co-NP if and only if for every no-instance we have a polynomial-length "certificate" and there is a polynomial-time algorithm that can be used to verify any purported certificate.

That is, co-NP is the set of decision problems where there exists a polynomial $\overline{X}$ and a polynomial-time bounded Turing machine M such that for every instance x, x is a no-instance if and only if: for some possible certificate c of length bounded by $p(n)$, the Turing machine M accepts the pair $(x, c)$.

Complementary Problems
While an NP problem asks whether a given instance is a yes-instance, its complement asks whether an instance is a no-instance, which means the complement is in co-NP. Any yes-instance for the original NP problem becomes a no-instance for its complement, and vice versa.

Unsatisfiability
An example of an NP-complete problem is the Boolean satisfiability problem: given a Boolean formula, is it satisfiable (is there a possible input for which the formula outputs true)? The complementary problem asks: "given a Boolean formula, is it unsatisfiable (do all possible inputs to the formula output false)?". Since this is the complement of the satisfiability problem, a certificate for a no-instance is the same as for a yes-instance from the original NP problem: a set of Boolean variable assignments which make the formula true. On the other hand, a certificate of a yes-instance for the complementary problem (whatever form it might take) would be equally as complex as for the no-instance of the original NP satisfiability problem.

co-NP-completeness
A problem L is co-NP-complete if and only if L is in co-NP and for any problem in co-NP, there exists a polynomial-time reduction from that problem to L.

Tautology Reduction
Determining if a formula in propositional logic is a tautology is co-NP-complete: that is, if the formula evaluates to true under every possible assignment to its variables.

Relationship to other classes

P, the class of polynomial time solvable problems, is a subset of both NP and co-NP. P is thought to be a strict subset in both cases. Because P is closed under complementation, and NP and co-NP are complementary, it cannot be strict in one case and not strict in the other: if P equals NP, it must also equal co-NP, and vice versa.

NP and co-NP are also thought to be unequal. If so, then no NP-complete problem can be in co-NP and no co-NP-complete problem can be in NP. This can be shown as follows. Suppose for the sake of contradiction there exists an NP-complete problem $p(n)$ that is in co-NP. Since all problems in NP can be reduced to $\mathcal{X}$, it follows that for every problem in NP, we can construct a non-deterministic Turing machine that decides its complement in polynomial time; i.e., $\mathcal{X}$. From this, it follows that the set of complements of the problems in NP is a subset of the set of complements of the problems in co-NP; i.e., $\textsf{NP} \subseteq \textsf{co-NP}$. Thus $\textsf{co-NP} \subseteq \textsf{NP}$. The proof that no co-NP-complete problem can be in NP if $\textsf{co-NP} = \textsf{NP}$ is symmetrical.

co-NP is a subset of PH, which itself is a subset of PSPACE.

Integer Factorization
An example of a problem that is known to belong to both NP and co-NP (but not known to be in P) is Integer factorization: given positive integers m and n, determine if m has a factor less than n and greater than one. Membership in NP is clear; if m does have such a factor, then the factor itself is a certificate. Membership in co-NP is also straightforward: one can just list the prime factors of m, all greater or equal to n, which the verifier can confirm to be valid by multiplication and the AKS primality test. It is presently not known whether there is a polynomial-time algorithm for factorization, equivalently that integer factorization is in P, and hence this example is interesting as one of the most natural problems known to be in NP and co-NP but not known to be in P.

title: Co-NP


text:
Chuck Yeager

Brigadier General Charles Elwood Yeager (, February 13, 1923 –December 7, 2020) was a United States Air Force officer, flying ace, and record-setting test pilot who in October 1947 became the first pilot in history confirmed to have exceeded the speed of sound in level flight.

Yeager was raised in Hamlin, West Virginia. His career began in World War II as a private in the United States Army, assigned to the Army Air Forces in 1941. After serving as an aircraft mechanic, in September 1942, he entered enlisted pilot training and upon graduation was promoted to the rank of flight officer (the World War II Army Air Force version of the Army's warrant officer), later achieving most of his aerial victories as a P-51 Mustang fighter pilot on the Western Front, where he was credited with shooting down 11.5 enemy aircraft (the half credit is from a second pilot assisting him in a single shootdown). On October 12, 1944, he attained "ace in a day" status, shooting down five enemy aircraft in one mission.

After the war, Yeager became a test pilot and flew many types of aircraft, including experimental rocket-powered aircraft for the National Advisory Committee for Aeronautics (NACA). Through the NACA program, he became the first human to officially break the sound barrier on October 14, 1947, when he flew the experimental Bell X-1 at Mach 1 at an altitude of 45000 ft, for which he won both the Collier and Mackay trophies in 1948. He then went on to break several other speed and altitude records in the following years. In 1962, he became the first commandant of the USAF Aerospace Research Pilot School, which trained and produced astronauts for NASA and the Air Force.

Yeager later commanded fighter squadrons and wings in Germany, as well as in Southeast Asia during the Vietnam War. In recognition of his achievements and the outstanding performance ratings of those units, he was promoted to brigadier general in 1969 and inducted into the National Aviation Hall of Fame in 1973, retiring on March 1, 1975. His three-war active-duty flying career spanned more than 30 years and took him to many parts of the world, including the Korean War zone and the Soviet Union during the height of the Cold War.

Yeager is referred to by many as one of the greatest pilots of all time, and was ranked fifth on Flying list of the 51 Heroes of Aviation in 2013. Throughout his life, he flew more than 360 different types of aircraft over a 70-year period, and continued to fly for two decades after retirement as a consultant pilot for the United States Air Force.

Early life and education
Yeager was born February 13, 1923, in Myra, West Virginia, to farming parents Albert Hal Yeager (1896–1963) and Susie Mae Yeager (Sizemore; 1898–1987). When he was five years old, his family moved to Hamlin, West Virginia. Yeager had two brothers, Roy and Hal Jr., and two sisters, Doris Ann (accidentally killed at age one by four-year-old Roy playing with a firearm) and Pansy Lee.

He attended Hamlin High School, where he played basketball and football, receiving his best grades in geometry and typing. He graduated from high school in June 1941.

His first experience with the military was as a teen at the Citizens Military Training Camp at Fort Benjamin Harrison, Indianapolis, Indiana, during the summers of 1939 and 1940. On February 26, 1945, Yeager married Glennis Dickhouse, and the couple had four children. Glennis Yeager died in 1990, predeceasing her husband by 30 years.

His cousin, Steve Yeager, was a professional baseball catcher.

World War II
Yeager enlisted as a private in the U.S. Army Air Forces (USAAF) on September 12, 1941, and became an aircraft mechanic at George Air Force Base, Victorville, California. At enlistment, Yeager was not eligible for flight training because of his age and educational background, but the entry of the U.S. into World War II less than three months later prompted the USAAF to alter its recruiting standards. Yeager had unusually sharp vision (a visual acuity rated 20/10), which once enabled him to shoot a deer at 600 yd.

At the time of his flight training acceptance, he was a crew chief on an AT-11. He received his pilot wings and a promotion to flight officer at Luke Field, Arizona, where he graduated from Class 43C on March 10, 1943. Assigned to the 357th Fighter Group at Tonopah, Nevada, he initially trained as a fighter pilot, flying Bell P-39 Airacobras (being grounded for seven days for clipping a farmer's tree during a training flight), and shipped overseas with the group on November 23, 1943.

Stationed in the United Kingdom at RAF Leiston, Yeager flew P-51 Mustangs in combat with the 363d Fighter Squadron. He named his aircraft Glamorous Glen after his girlfriend, Glennis Faye Dickhouse, who became his wife in February 1945. Yeager had gained one victory before he was shot down over France in his first aircraft (P-51B-5-NA s/n 43-6763) on March 5, 1944, on his eighth mission. He escaped to Spain on March 30, 1944, with the help of the Maquis (French Resistance) and returned to England on May 15, 1944. During his stay with the Maquis, Yeager assisted the guerrillas in duties that did not involve direct combat; he helped construct bombs for the group, a skill that he had learned from his father. He was awarded the Bronze Star for helping a navigator, Omar M. "Pat" Patterson, Jr., to cross the Pyrenees.

Despite a regulation prohibiting "evaders" (escaped pilots) from flying over enemy territory again, the purpose of which was to prevent resistance groups from being compromised by giving the enemy a second chance to possibly capture him, Yeager was reinstated to flying combat. He had joined another evader, fellow P-51 pilot 1st Lt Fred Glover, in speaking directly to the Supreme Allied Commander, General Dwight D. Eisenhower, on June 12, 1944. "I raised so much hell that General Eisenhower finally let me go back to my squadron" Yeager said. "He cleared me for combat after D Day, because all the free Frenchmen – Maquis and people like that – had surfaced". Eisenhower, after gaining permission from the War Department to decide the requests, concurred with Yeager and Glover. In the meantime, Yeager shot down his second enemy aircraft, a German Junkers Ju 88 bomber, over the English Channel.

Yeager demonstrated outstanding flying skills and combat leadership. On October 12, 1944, he became the first pilot in his group to make "ace in a day," downing five enemy aircraft in a single mission. Two of these victories were scored without firing a single shot: when he flew into firing position against a Messerschmitt Bf 109, the pilot of the aircraft panicked, breaking to port and colliding with his wingman. Yeager said both pilots bailed out. He finished the war with 11.5 official victories, including one of the first air-to-air victories over a jet fighter, a German Messerschmitt Me 262 that he shot down as it was on final approach for landing.

In his 1986 memoirs, Yeager recalled with disgust that "atrocities were committed by both sides", and said he went on a mission with orders from the Eighth Air Force to "strafe anything that moved". During the mission briefing, he whispered to Major Donald H. Bochkay, "If we are going to do things like this, we sure as hell better make sure we are on the winning side". Yeager said, "I'm certainly not proud of that particular strafing mission against civilians. But it is there, on the record and in my memory". He also expressed bitterness at his treatment in England during World War II, prompting descriptions of the British as "arrogant" and "nasty" on Twitter.

Yeager was commissioned a second lieutenant while at Leiston, and was promoted to captain before the end of his tour. He flew his 61st and final mission on January 15, 1945, and returned to the United States in early February 1945. As an evader, he received his choice of assignments and, because his new wife was pregnant, chose Wright Field to be near his home in West Virginia. His high number of flight hours and maintenance experience qualified him to become a functional test pilot of repaired aircraft, which brought him under the command of Colonel Albert Boyd, head of the Aeronautical Systems Flight Test Division.

Test pilot – breaking the sound barrier
Yeager remained in the U.S. Army Air Forces after the war, becoming a test pilot at Muroc Army Air Field (now Edwards Air Force Base), following graduation from Air Materiel Command Flight Performance School (Class 46C). After Bell Aircraft test pilot Chalmers "Slick" Goodlin demanded US$150000 1947 to break the sound "barrier", the USAAF selected the 24-year-old Yeager to fly the rocket-powered Bell XS-1 in a NACA program to research high-speed flight. Under the National Security Act of 1947, the USAAF became the United States Air Force (USAF) on September 18.

Such was the difficulty, that the answers to many of the inherent challenges were like "Yeager better have paid-up insurance". Two nights before the scheduled flight date, Yeager broke two ribs when he fell from a horse. He was worried that the injury would remove him from the mission and reported that he went to a civilian doctor in nearby Rosamond, who taped his ribs. Besides his wife who was riding with him, Yeager told only his friend and fellow project pilot Jack Ridley about the accident. On the day of the flight, Yeager was in such pain that he could not seal the X-1's hatch by himself. Ridley rigged up a device, using the end of a broom handle as an extra lever, to allow Yeager to seal the hatch.

Yeager broke the sound barrier on October 14, 1947, in level flight while piloting the X-1 Glamorous Glennis at Mach 1.05 at an altitude of 45000 ft over the Rogers Dry Lake of the Mojave Desert in California. The success of the mission was not announced to the public for nearly eight months, until June 10, 1948. Yeager was awarded the Mackay Trophy and the Collier Trophy in 1948 for his mach-transcending flight, and the Harmon International Trophy in 1954. The X-1 he flew that day was later put on permanent display at the Smithsonian Institution's National Air and Space Museum. During 1952, he attended the Air Command and Staff College.

Yeager continued to break many speed and altitude records. He was one of the first American pilots to fly a Mikoyan-Gurevich MiG-15, after its pilot, No Kum-sok, defected to South Korea. Returning to Muroc, during the latter half of 1953, Yeager was involved with the USAF team that was working on the X-1A, an aircraft designed to surpass Mach 2 in level flight. That year, he flew a chase aircraft for the civilian pilot Jackie Cochran as she became the first woman to fly faster than sound.

On November 20, 1953, the U.S. Navy program involving the D-558-II Skyrocket and its pilot, Scott Crossfield, became the first team to reach twice the speed of sound. After they were bested, Ridley and Yeager decided to beat rival Crossfield's speed record in a series of test flights that they dubbed "Operation NACA Weep". Not only did they beat Crossfield by setting a new record at Mach 2.44 on December 12, 1953, but they did it in time to spoil a celebration planned for the 50th anniversary of flight in which Crossfield was to be called "the fastest man alive".

The new record flight, however, did not entirely go to plan, since shortly after reaching Mach 2.44, Yeager lost control of the X-1A at about 80000 ft due to inertia coupling, a phenomenon largely unknown at the time. With the aircraft simultaneously rolling, pitching, and yawing out of control, Yeager dropped 51000 ft in less than a minute before regaining control at around 29000 ft. He then managed to land without further incident. For this feat, Yeager was awarded the Distinguished Service Medal (DSM) in 1954.

Military command
Yeager was foremost a fighter pilot and held several squadron and wing commands. From 1954 to 1957, he commanded the F-86H Sabre-equipped 417th Fighter-Bomber Squadron (50th Fighter-Bomber Wing) at Hahn AB, West Germany, and Toul-Rosieres Air Base, France; and from 1957 to 1960 the F-100D Super Sabre-equipped 1st Fighter Day Squadron at George Air Force Base, California, and Morón Air Base, Spain.

He was a full colonel in 1962, after completion of a year's studies and final thesis on STOL aircraft at the Air War College. He became the first commandant of the USAF Aerospace Research Pilot School, which produced astronauts for NASA and the USAF, after its redesignation from the USAF Flight Test Pilot School. He had only a high school education, so he was not eligible to become an astronaut like those he trained. In April 1962, Yeager made his only flight with Neil Armstrong. Their job, flying a T-33, was to evaluate Smith Ranch Dry Lake in Nevada for use as an emergency landing site for the North American X-15. In his autobiography, he wrote that he knew the lake bed was unsuitable for landings after recent rains, but Armstrong insisted on flying out anyway. As Armstrong suggested that they do a touch-and-go, Yeager advised against it, telling him "You may touch, but you ain't gonna go!" When Armstrong did touch down, the wheels became stuck in the mud, bringing the plane to a sudden stop and provoking Yeager to fits of laughter. They had to wait for rescue.

Yeager's participation in the test pilot training program for NASA included controversial behavior. Yeager reportedly did not believe that Ed Dwight, the first African American pilot admitted into the program, should be a part of it. In the 2019 documentary series Chasing the Moon, the filmmakers made the claim that Yeager instructed staff and participants at the school that "Washington is trying to cram the nigger down our throats. [President] Kennedy is using this to make 'racial equality,' so do not speak to him, do not socialize with him, do not drink with him, do not invite him over to your house, and in six months he'll be gone." In his autobiography, Dwight details how Yeager's leadership led to discriminatory treatment throughout his training at Edwards Air Force Base.

Between December 1963 and January 1964, Yeager completed five flights in the NASA M2-F1 lifting body. An accident during a December 1963 test flight in one of the school's NF-104s resulted in serious injuries. After climbing to a near-record altitude, the plane's controls became ineffective, and it entered a flat spin. After several turns, and an altitude loss of approximately 95,000 feet, Yeager ejected from the plane. During the ejection, the seat straps released normally, but the seat base slammed into Yeager, with the still-hot rocket motor breaking his helmet's plastic faceplate and causing his emergency oxygen supply to catch fire. The resulting burns to his face required extensive and agonizing medical care. This was Yeager's last attempt at setting test-flying records.

In 1966, Yeager took command of the 405th Tactical Fighter Wing at Clark Air Base, the Philippines, whose squadrons were deployed on rotational temporary duty (TDY) in South Vietnam and elsewhere in Southeast Asia. There he flew 127 missions. In February 1968, Yeager was assigned command of the 4th Tactical Fighter Wing at Seymour Johnson Air Force Base, North Carolina, and led the McDonnell Douglas F-4 Phantom II wing in South Korea during the Pueblo crisis.

Yeager was promoted to brigadier general and was assigned in July 1969 as the vice-commander of the Seventeenth Air Force.

From 1971 to 1973, at the behest of Ambassador Joseph Farland, Yeager was assigned as the Air Attache in Pakistan to advise the Pakistan Air Force which was led by Abdur Rahim Khan (the first Pakistani to break the sound barrier). He arrived in Pakistan at a time when tensions with India were at a high level. One of Yeager's jobs during this time was to assist Pakistani technicians in installing AIM-9 Sidewinders on PAF's Shenyang F-6 fighters. He also had a keen interest in interacting with PAF personnel from various Pakistani Squadrons and helping them develop combat tactics. In one instance in 1972, while visiting the No. 15 Squadron "Cobras" at Peshawar Airbase, the Squadron's OC Wing Commander Najeeb Khan escorted him to K2 in a pair of F-86Fs after Yeager requested a visit to the second highest mountain on Earth. After hostilities broke out in 1971, he decided to stay in West Pakistan and continued overseeing the PAF's operations. Yeager recalled "the Pakistanis whipped the Indians' asses in the sky... the Pakistanis scored a three-to-one kill ratio, knocking out 102 Russian-made Indian jets and losing 34 airplanes of their own". During the war, he flew around the western front in a helicopter documenting wreckages of Indian aircraft of Soviet origin which included Sukhoi Su-7s and MiG-21s. These aircraft were transported to the United States after the war for analysis. Yeager also flew around in his Beechcraft Queen Air, a small passenger aircraft that was assigned to him by the Pentagon, picking up shot-down Indian fighter pilots. The Beechcraft was later destroyed during an air raid by the IAF at a Pakistani airbase when Yeager was not present. Edward C. Ingraham, a U.S. diplomat who had served as political counselor to Ambassador Farland in Islamabad, recalled this incident in the Washington Monthly of October 1985: "After Yeager's Beechcraft was destroyed during an Indian air raid, he raged to his cowering colleagues that the Indian pilot had been specifically instructed by Indira Gandhi to blast his plane. 'It was', he later wrote, 'the Indian way of giving Uncle Sam the finger'". Yeager was incensed over the incident and demanded U.S. retaliation.

Post-retirement and in popular culture
On March 1, 1975, Yeager retired from the Air Force at Norton Air Force Base, California.

Yeager made a cameo appearance in the movie The Right Stuff (1983). He played "Fred", a bartender at "Pancho's Place", which was most appropriate, because he said, "if all the hours were ever totaled, I reckon I spent more time at her place than in a cockpit over those years". Sam Shepard portrayed Yeager in the film, which chronicles in part his famous 1947 record-breaking flight.

Yeager has been referenced several times in the shared Star Trek universe, including having a namesake fictional type of starship, a dangerous starship formation-maneuver named after him called the "Yeager Loop" (most notably mentioned in the Star Trek: The Next Generation episode "The First Duty"), and appearing in archival footage within the opening title sequence for the series Star Trek: Enterprise (2001–2005). For Enterprise, executive producer Rick Berman said that he envisaged the lead character, Captain Jonathan Archer, as being "halfway between Chuck Yeager and Han Solo".

For several years in the 1980s, Yeager was connected to General Motors, publicizing ACDelco, the company's automotive parts division. In 1986, he was invited to drive the Chevrolet Corvette pace car for the 70th running of the Indianapolis 500. In 1988, Yeager was again invited to drive the pace car, this time at the wheel of an Oldsmobile Cutlass Supreme. In 1986, President Reagan appointed Yeager to the Rogers Commission that investigated the explosion of the Space Shuttle Challenger.

During this time, Yeager also served as a technical adviser for three Electronic Arts flight simulator video games. The games include Chuck Yeager's Advanced Flight Trainer, Chuck Yeager's Advanced Flight Trainer 2.0, and Chuck Yeager's Air Combat. The game manuals feature quotes and anecdotes from Yeager and were well received by players. Missions feature several of Yeager's accomplishments and let players challenge his records. Chuck Yeager's Advanced Flight Trainer was Electronic Art's top-selling game for 1987.

In 2009, Yeager participated in the documentary The Legend of Pancho Barnes and the Happy Bottom Riding Club, a profile of his friend Pancho Barnes. The documentary was screened at film festivals, aired on public television in the United States, and won an Emmy Award.

On October 14, 1997, on the 50th anniversary of his historic flight past Mach 1, he flew a new Glamorous Glennis III, an F-15D Eagle, past Mach 1. The chase plane for the flight was an F-16 Fighting Falcon piloted by Bob Hoover, a longtime test, fighter, and aerobatic pilot who had been Yeager's wingman for the first supersonic flight. At the end of his speech to the crowd in 1997, Yeager concluded, "All that I am ... I owe to the Air Force". Later that month, he was the recipient of the Tony Jannus Award for his achievements.

On October 14, 2012, on the 65th anniversary of breaking the sound barrier, Yeager did it again at the age of 89, flying as co-pilot in a McDonnell Douglas F-15 Eagle piloted by Captain David Vincent out of Nellis Air Force Base.

Awards and decorations
In 1973, Yeager was inducted into the National Aviation Hall of Fame, arguably aviation's highest honor. In 1974, Yeager received the Golden Plate Award of the American Academy of Achievement. In December 1975, the U.S. Congress awarded Yeager a silver medal "equivalent to a noncombat Medal of Honor ... for contributing immeasurably to aerospace science by risking his life in piloting the X-1 research airplane faster than the speed of sound on October 14, 1947". President Gerald Ford presented the medal to Yeager in a ceremony at the White House on December 8, 1976.

Yeager never attended college and was often modest about his background, but is considered by many, including Flying Magazine, the California Hall of Fame, the State of West Virginia, National Aviation Hall of Fame, a few U.S. presidents, and the United States Army Air Force, to be one of the greatest pilots of all time. Air & Space/Smithsonian magazine ranked him the fifth greatest pilot of all time in 2003. Regardless of his lack of higher education, West Virginia's Marshall University named its highest academic scholarship the Society of Yeager Scholars in his honor. He was the chairman of Experimental Aircraft Association (EAA)'s Young Eagle Program from 1994 to 2004, and was named the program's chairman emeritus.

In 1966, Yeager was inducted into the International Air & Space Hall of Fame. He was inducted into the International Space Hall of Fame in 1981. He was inducted into the Aerospace Walk of Honor 1990 inaugural class.

Yeager Airport in Charleston, West Virginia, is named in his honor. The Interstate 64/Interstate 77 bridge over the Kanawha River in Charleston is named in his honor. He also flew directly under the Kanawha Bridge and West Virginia named it the Chuck E. Yeager Bridge. On October 19, 2006, the state of West Virginia also honored Yeager with a marker along Corridor G (part of U.S. Highway 119) in his home Lincoln County, and also renamed part of it the Yeager Highway.

Yeager was an honorary board member of the humanitarian organization Wings of Hope. On August 25, 2009, Governor Arnold Schwarzenegger and Maria Shriver announced that Yeager would be one of 13 California Hall of Fame inductees in The California Museum's yearlong exhibit. The induction ceremony was on December 1, 2009, in Sacramento, California. Flying Magazine ranked Yeager number 5 on its 2013 list of The 51 Heroes of Aviation; for many years, he was the highest-ranked living person on the list.

The Civil Air Patrol, the volunteer auxiliary of the USAF, awards the Charles E. "Chuck" Yeager Award to its senior members as part of its Aerospace Education program.

Other achievements

 * 1940–1949 – Harmon Trophy: Citation of Honorable Mention
 * 1947 – Collier Trophy and Mackay Trophy, for breaking the sound barrier for the first time.
 * 1953 – Harmon Trophy
 * 1976 – Congressional Silver Medal

Personal life
Yeager named his plane after his wife, Glennis, as a good-luck charm: "You're my good-luck charm, hon. Any airplane I name after you always brings me home." Yeager and Glennis moved to Grass Valley, California, after his retirement from the Air Force in 1975. The couple prospered as a result of Yeager's best-selling autobiography, speaking engagements, and commercial ventures. Glennis Yeager died of ovarian cancer in 1990. They had four children (Susan, Don, Mickey, and Sharon). Yeager's son Mickey (Michael) died unexpectedly in Oregon, on March 26, 2011.

Yeager appeared in a Texas advertisement for George H. W. Bush's 1988 presidential campaign. In 2000, Yeager met actress Victoria Scott D'Angelo on a hiking trail in Nevada County. The pair started dating shortly thereafter, and married in August 2003. A bitter dispute arose between Yeager, his children, and D'Angelo. The children contended that she, at least 35 years Yeager's junior, had married him for his fortune. Yeager and D'Angelo both denied the charge. Litigation ensued, in which his children accused D'Angelo of "undue influence" on Yeager, and Yeager accused his children of diverting millions of dollars from his assets. In August 2008, the California Court of Appeal ruled for Yeager, finding that his daughter Susan had breached her duty as trustee.

Yeager lived in Grass Valley, Northern California and died in the afternoon of December 7, 2020 (National Pearl Harbor Remembrance Day), at age 97, in a Los Angeles hospital. Following his death, President Donald Trump issued a statement of condolences stating Yeager "was one of the greatest pilots in history, a proud West Virginian, and an American original who relentlessly pushed the boundaries of human achievement".

title: Chuck Yeager