Talk:Countable set

message: There are lesser-known sequences for counting the rationals other than Cantor's mapping. For example, the following sequence assigns each natural to a unique positive rational:
:S(0) = 0 S(1) = 1 S(2n) = S(n) + 1 for all n > 0 S(2n+1) = 1/S(2n) for all n > 0
This sequence is based on Farey sequences and Stern-Brocot trees. It can be extended to cover the negative rationals as well. — Loadmaster (talk) 17:40, 3 February 2008 (UTC)
thread_title:  Countable rationals 

message: The new section Countable set#Topological proof of the uncountability of the real numbers seem to be out of place in this article. It is about topology, not really about countable sets. Also, from my limited knowledge of topology, the proof appears to be incorrect. Specifically, in the indiscrete topology one-point sets are not open. JRSpriggs (talk) 17:16, 13 June 2008 (UTC)
replies: text: Not disagreeing, but the indiscrete topology doesn't satisfy the hypotheses because it is non-Hausdorff.  (It is in fact mentioned as a counterexample in the indicated section.) siℓℓy rabbit (talk) 17:21, 13 June 2008 (UTC)
thread_title:  Why drag topology into this article? 

message: How does the proof of the assertion 'Q (the set of all rational numbers) is countable' contradict the intuition that 'the cardinality of Q is larger than that of N'?  Either some details are left out or it doesn't.
(0,1,0), (1,1,0), (1,1,1) and so on are not members of N.  They are rather triples composed of members of N.  Is there assumed another bijection from these tuples to N?  If this bijection exists and is assumed, it may be useful to explicitly state it.  Johanatan (talk) 23:55, 20 October 2008 (UTC)
replies: text: Since the article begins by defining a countable set to be one with an injection from it to N, why are people bending over backwards to created bijections when it is much simpler to create injections?  For example the nonnegative rationals m/n are obviously a subset of N2, namely those pairs (m,n) for which n is nonzero and m and n are relatively prime (gcd is 1).  And there are lots of very simple injections of N2 into N, for example f(m,n) = 2m3n, that don't require a diagram to explain.  --Vaughan Pratt (talk) 15:55, 17 August 2009 (UTC)
thread_title: Some details missing about bijection from Q to N

message: I am removing this from the lede:
Of course infinite countable sets have little to do with finite combinatorics, but that is because they are infinite, not because of countability. On the other hand countability is an important concern in infinitary combinatorics, and so it seems wrong to me to say that countability has little to do with combinatorics overall. For example, Martin's axiom uses countability in an central way. — Carl (CBM · talk) 11:04, 28 October 2009 (UTC)
thread_title:  Combinatorics 

message: This sections discusses how the set N and 2N have the same cardinality.  Not bad so far.  However, in the paragraph just below the line "1 ↔ 2, 2 ↔ 4, 3 ↔ 6, 4 ↔ 8, ...." it suggests that "(t)his gives an example of a set which is of the same size as one of its proper subsets".  Not quite.  If 2N were strictly a subset of N, it would be skipping every other number/element in the set, so would DEFINITELY NOT be the exact same size -- it would be exactly half the size!  Infinity math or not, if it doesn't include all of the same elements it cannot be both a subset and equal in size.  It's only when considering 2N as a strictly independent set that it would have the same cardinality.
KhyranLeander (talk) 21:24, 19 August 2010 (UTC)
replies: text: The cardinality of a set doesn't change, whether you are considering it as the subset of some other set, or not. So the set N = {1, 2, 3, ...} and the set 2N = {2, 4, 6, ...} have the same cardinality, by definition, since the function f(x) = 2x, is a one-to-one correspondence between the two sets. And this is so despite the fact that one set is a prober subset of the other. That's one of the properties of the cardinality of infinite sets that set them apart from finite sets, where the cardinality of a proper subset is always strictly smaller. Since the "size" of N is infinite, in order for your statement that 2N is  "exactly half the size" of N, to have any meaning, the notion of "half of infinity" needs to be defined. It turns out this can be done, in a reasonable way, and under the usual definitions, half of infinity equals infinity. In which case both the statements "2N and N have the same size", and  "2N is half the size of N" are true. Paul August ☎ 22:32, 19 August 2010 (UTC)
thread_title:  Under 'More Formal Introduction' 

message: How about adding a short analogy to the introduction. Such as a countable set being a set of temperature readings, that may go on for ever but every individual reading is clearly spaced. Where as an uncountable set you can mine more and more numbers simply by increasing the resolution, like the level of detail if you're zooming in on fractal, or dividing 1 by x with x>1. You will allways end up with a number, regardless of what you put in unless the number is infenite in size, so there is really no end to how small 1/x can get. (Axioms that suppose a largest uncountable non infenite number aside)  —Preceding unsigned comment added by 194.132.104.253 (talk) 09:33, 29 October 2010 (UTC)
replies: text: We usually try to avoid analogies in mathematics since they can easily be misunderstood. If you have a precise example, that could be added at the appropriate place in the article, i.e. further down rather than in the lead. JRSpriggs (talk) 10:59, 29 October 2010 (UTC)
thread_title:  An analogy in the introduction? 

message: Under definition it states : "If f is also surjective and therefore bijective, then S is called countably infinite."  A function being surjective does NOT imply it is bijective, however the reverse is true. For instance if a function is bijective it is also therefore surjective.  But a surjective function is not always bijective, in the case when more than one element of the first set maps to the same element of the second set.  — Preceding unsigned comment added by 24.63.136.210 (talk) 14:22, 2 February 2012 (UTC)
replies: text: The function f is defined to be injective. So it is surjective if and only if it is bijective. --Zundark (talk) 14:52, 2 February 2012 (UTC)
replies: text: Thanks for the clarification. I am going to go ahead and specify the reason why it is bijective in paranthesis next to it to avoid confusion. 24.63.136.210 (talk) 02:13, 3 February 2012 (UTC)

text: I guess most the the 'citation needed' originate from me. I gave a detailled reason for each of it. If it doesn't appear when you move the mouse pointer over it, you can see it in the source code. - Jochen Burghardt (talk) 07:50, 15 April 2015 (UTC)
thread_title:  Definition wrong? 

message: The lede makes the misleading statement:"Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a natural number." This statement has gone too far attempting to use simple concepts to explain what a countable set is. First, both element and set should be linked to the appropriate article, but that is a minor point. There are two major problems, imho, with the statement: #1. Time should NOT be brought in to the discussion ("one at a time", "may never finish"). This is so blatantly blindingly obvious that it should not need to be said. #2. "associated with a natural number" is wrong. So, if I have a set of, say 10^10^100 elements and associate EACH and EVERY one of them with the natural number 1, then the set is countable???? WRONG! Each element of the set must be able to be set into a one-to-one correspondence with a UNIQUE natural number. There are usually many ways to set up such a correspondence. That is, which natural number is assigned to label a particular element will depend on how the "numbering" or "counting" is chosen. (Take the set consisting of the three elements {A,B,C}. They may be put into correspondence with 1,2,3 or 2,1,3 or 6,4,2, or even 2,3,5 or 100,10,1000.) A countable set is one in which each element can be assigned a unique natural number. It turns out that some infinite sets have this property, countable sets may be finite or infinite. (some, actually most, infinite sets are not countable.)72.172.10.197 (talk) 13:48, 10 October 2015 (UTC)
thread_title:  Problem with lede 

message: Wikipedia at the moment doesn't include Booleans as a countable set or indeed as a number system. Should Booleans be added, as a subset of the natural numbers with their own arithmetic just like any of the other sets listed in Template:Number_systems? 80.42.172.137 (talk) 11:43, 29 November 2016 (UTC)
replies: text: What do you mean by "Booleans"? JRSpriggs (talk) 18:09, 30 November 2016 (UTC)
thread_title:  Booleans 

message: The section Countable_set#Some_technical_details outlines a proof of the proposition 'The algebraic numbers A are countable'. The outline of the proof relies on the set of n-degree polynomials to be countable, since their coefficient space can be one-to-one mapped to the set of rational numbers. As far as I can see, this only holds if each polynomial coefficient is integer (or rational, but that makes no difference). So it should either be stated that the polynomial coefficients are integer (or explained how the proof holds for non-integer (actually irrational) coefficients, in which case I missed something). Thanks. Lklundin (talk) 08:29, 12 September 2018 (UTC)
replies: text: Thanks for the remark. The coefficients must be rational; by multiplying with the least common denominator, they can always be made integer. Admitting arbitrary real numbers as coefficients would make e.g. \pi algebraic; admitting algebraic numbers as coefficients would use them in their own definition; both wouldn't make sense. - Jochen Burghardt (talk) 10:49, 12 September 2018 (UTC)
replies: text: Thanks (also because I no longer doubt that I understand the underlying math). Although the text now mentions 'integer coefficients' twice which is possibly one more than necessary, it is very clear now. Thanks again. Lklundin (talk) 15:00, 12 September 2018 (UTC)
thread_title:  Set of polynomials countable ? 

message: I've removed the "encyclopedic tone" cleanup tag. The specific objection (from the edit summary when the tag was added) was to the use of "you" and "we". There were only five occurrences of "you" in the article, and I've removed all five by rephrasing. The use of "we" is much more pervasive, but it is not clear that it is really a problem. The Manual of Style exhorts us to avoid "we" but admits that this can create problems with the readability of mathematics articles. I doubt seriously that removing "we" would improve this article. If anyone thinks it would, please demonstrate your point by actually making the desired changes; retagging the article is unlikely to accomplish anything useful.
—Syrenka V (talk) 11:17, 16 April 2019 (UTC)
thread_title:  Encyclopedic tone 

message: Readers arriving at this page to learn what denumerable means would benefit from a special note to confirm whether or not it is synonymous with countable and/or enumerable. Because of its de prefix, the word gives the strong sense that it might mean the opposite of numerable, or somehow refers to being capable of having its numerability removed. It is especially unevocative when used in the term non-denumerable, adding to the value that could be provided to readers by having its meaning deliberately spelled out unambiguously.  Gwideman (talk) 13:31, 22 February 2021 (UTC)
replies: text: Originally, I would have guessed that "denumerable" requires the mapping f:S→N to be computable, while "countable" does not. However, that is apparently wrong — a quick web search showed that 5 of 5 URLs use "denumerable" to mean the same as "countable":     . I hesitate to  add a corresponding note before a native English speaker has commented on this issue. - Jochen Burghardt (talk) 15:22, 22 February 2021 (UTC)
replies: text:  actually all five of your links seem to use "denumerable" to mean "countably infinite" rather than "countable".  I also had a vague recollection that "denumerable" typically meant "countably infinite".  However I'm not completely sure that it's not also sometimes used to mean just "countable" (that is, including finite sets).  In any case I think it's a fairly uncommon word to use in research mathematics today, irrespective of meaning. --Trovatore (talk) 21:13, 20 August 2021 (UTC)

text: It's discussed in the note on terminology. AFAICT it's ambiguous just like countable. Mathnerd314159 (talk) 18:16, 11 September 2022 (UTC)
thread_title:  "Denumerable" deserves a special note 

message: Example of countable set 117.98.98.174 (talk) 06:47, 4 May 2022 (UTC)
thread_title:  Examples of countable sets 

message: sadly, the intro, as is the case for so many wiki articles on math, is not written at a level appropriate for a general encylopedia
it is written at much high level, and uses way to much jargon (injective function !!)
please delte and re write
thanks  — Preceding unsigned comment added by 72.43.78.113 (talk) 15:10, 11 September 2022 (UTC)
replies: text: I have edited the lead. D.Lazard (talk) 17:06, 11 September 2022 (UTC)
thread_title: intro is bad