This is the
talk page for discussing improvements to the
Countable set article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Find sources: Google ( books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
Archives: 1 |
This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||
|
Archives:
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:
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)
Consider the set of natural numbers listed in the usual order. That is, starting with 1,2,3... and written from left to right. By Cantorian definition, this set is both "countable" and infinite. Therefore we should be able to place the elements of this set in 1 to 1 correspondence with the set of natural numbers listed in any other order. Let us choose to use 'reverse order' for this second set. Now suppose we place the first set above the second and try to match elements. Then we should find one and only one entry under the number "1" in the first set. What is this number? Clearly, there is no such number! The concept of "countably infinite sets" is inherently flawed. This concept is the basis of the pseudo-mathematics of Cantor and his followers.
What can be counted can not possibly be infinite, and what is infinite can not be counted. Carl Friedrich Gauss pointed this out when he wrote "I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics". The flaw in the 'diagonal argument' has precisely this defect. —Preceding unsigned comment added by 66.67.96.142 ( talk) 22:16, 23 May 2008 (UTC)
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)
It can make sense to include a short proof that any topological space satisfying certain simple conditions must be uncountable and perhaps that the reals do satisfy those conditions. That the reals are uncountable can be proved more simply from the fact that they are linearly ordered such that between any two points there is another and they satisfy the least-upper-bound property (you certainly DON'T need to talk about decimals, even though that's one way to do it). Michael Hardy ( talk) 17:29, 13 June 2008 (UTC)
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)
I am concerned about some recent edits that make use of the "proposition - proof" style of math textbooks. I feel these style elements reduce the accessibility of the article to the layman as they destroy the flow of the text and make heavy use of (standard) math notation. (They also are partially redundant to the section More formal introduction.) I would suggest to at least move the text after the section Gentle introduction and, preferably, to reword it into more accessible prose. — Tobias Bergemann ( talk) 11:29, 11 February 2009 (UTC)
I am responsible for the recent additions, but not for the definition (which is in my opinion the correct one). Also the terms injective, surjective, bijective have been around for some time, as has the notation for a function. I have only edited the section called Definition and Basic Properties. The label THEOREM is used many places in the article. I believe the intent (and certainly my intent) was to indicate a precise mathematically correct statement. Mathematics is proofs. It may be valuable in a wikipedia article to give heuristic arguments or give examples. But when a proof is given it I think it is useful for the reader to know "This is a proof, not an example and not a heuristic argument." That is the function of the labels Proof. The proofs given are all extremely short, though, in principle I see no reason not to include longer proofs. Increasingly Wikipedia is becoming a useful place for students to learn serious mathematics, as opposed to popularized versions (which are also very valuable). Indeed many mathematics articles discuss topics at a level not usually encountered until graduate school. This is a 'good thing.'
Achieving a balance between something for the layman (to use your phrase) and something useful for, say, a university level student, is very difficult and has not been achieved in this article. For example the section on the minimal model for set theory is inaccessible even to most professional mathematicians.
While Wikipedia is not a textbook I think it is very useful to have textbook like sections on particular topics like this one. I believe it enhances other entries. If you look, for example, at measure theory which is more highly rated than this article you might imagine that someone reading that would want to refer to an article on countable which had a quick summary of basic properties. I am open to moving the section later if others think that is appropriate. Johnfranks ( talk) 16:00, 11 February 2009 (UTC)
I moved the main content of the section "Definition and basic properties" later in the article. I combined the short section "Other results about uncountable sets" with it. I added references at the beginning. Johnfranks ( talk) 20:01, 11 February 2009 (UTC)
Is there a consensus that zero is a natural number? If so I am happy to go along. I have never seen this usage. Can someone give a reference? The recent edit declaring 0 a natural number broke one statement which assumed one could divide by a natural number. I think I have fixed it. I believe there is another place in this article (in a section I have not edited) that asserts that 0 is not in the natural numbers. Johnfranks ( talk) 22:35, 11 February 2009 (UTC)
I am removing this from the lede:
But countability has little to do with combinatorics (the mathematical study of counting). Rather, it is a property of certain sets in the mathematical area of set theory.
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)
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)
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)
In many textbooks, it is said that a set is countable if we can list the elements as . My question is: is it true that a set is countable if and only if there exists a Turing machine to enumerate (without termination) all the elements in the set? — Preceding unsigned comment added by Tmbu ( talk • contribs) 01:56, 9 March 2011 (UTC)
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)
Nevertheless, the given definition is wrong (If THIS f is also surjective...). You don't suppose the function f which showed you that S is countable to be surjective in case S is countably infinite. Example: f(n):=n+1 as function from N to N is injective. So this shows us N is countable. But f is not surjective. So N is not countably infinite? To correct the definition you have to find a new function which is bijective (in case f is only injective but not surjective). -- Jobu0101 ( talk) 20:37, 9 December 2014 (UTC)
1) What's with the Jainist thing in history? It would be cool if that were accurate, but I'm doubtful given some of the dates and the complete lack on non-Jainist material. 2) What's with all the "citation needed"'s on the Jainist material. It is plenty well cited, much more thoroughly cited than the rest of the article to say the least. If you wish to *dispute* the citations, which seems reasonable, do that clearly, don't just paste 'citation needed' on all of the clearly cited material. Also, text citations that aren't available online are valid citations, if that's what's drawing your ire. 73.170.107.159 ( talk) 06:45, 15 April 2015 (UTC)
Q can be defined as the set of all fractions a/b where a and b are integers and b > 0. This can be mapped onto the subset of ordered triples of natural numbers (a, b, c) such that a ≥ 0, b > 0, a and b are coprime, and c ∈ {0, 1} such that c = 0 if a/b ≥ 0 and c = 1 otherwise.
If a ≥ 0 and b > 0 then by definition a/b ≥ 0. — Preceding unsigned comment added by 24.84.237.14 ( talk) 03:48, 9 September 2015 (UTC)
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)
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)
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)
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)
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)
Since the phrasing was changed from the original quote by User:Trovatore I thought I'd pull some numbers in from Google n-grams. "Countably infinite" is at 5.6 units and rising in popularity, while "at most countable" is at 1.3 units and mostly flat. (81% vs 19%) So clearly the countably infinite terminology is the most common terminology and the right one to use for the article. But as far as the other convention, can I say that the convention appears in 20% of publications? Or is that original research? Regardless 20% doesn't seem "common" so maybe it should be described as "alternative". -- Mathnerd314159 ( talk) 17:21, 17 September 2021 (UTC)
Example of countable set 117.98.98.174 ( talk) 06:47, 4 May 2022 (UTC)
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)
Regarding the "citation needed" in the section "A note on terminology": A citation for what is asked for? The fact that the total number of letters in "at most countable" plus "countably infinite" exceeds bot the total number of letters in "at most countable" plus "countable" and the total number of letters in "countable" plus "countably infinite"? Hagman ( talk) 20:22, 14 November 2023 (UTC)
, although with respect to concision this is the worst of both worlds.[citation needed]. I'd also like to remark that checking the definition is generally a good idea in mathematics. - Jochen Burghardt ( talk) 21:19, 14 November 2023 (UTC)
This is the
talk page for discussing improvements to the
Countable set article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Find sources: Google ( books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
Archives: 1 |
This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||
|
Archives:
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:
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)
Consider the set of natural numbers listed in the usual order. That is, starting with 1,2,3... and written from left to right. By Cantorian definition, this set is both "countable" and infinite. Therefore we should be able to place the elements of this set in 1 to 1 correspondence with the set of natural numbers listed in any other order. Let us choose to use 'reverse order' for this second set. Now suppose we place the first set above the second and try to match elements. Then we should find one and only one entry under the number "1" in the first set. What is this number? Clearly, there is no such number! The concept of "countably infinite sets" is inherently flawed. This concept is the basis of the pseudo-mathematics of Cantor and his followers.
What can be counted can not possibly be infinite, and what is infinite can not be counted. Carl Friedrich Gauss pointed this out when he wrote "I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics". The flaw in the 'diagonal argument' has precisely this defect. —Preceding unsigned comment added by 66.67.96.142 ( talk) 22:16, 23 May 2008 (UTC)
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)
It can make sense to include a short proof that any topological space satisfying certain simple conditions must be uncountable and perhaps that the reals do satisfy those conditions. That the reals are uncountable can be proved more simply from the fact that they are linearly ordered such that between any two points there is another and they satisfy the least-upper-bound property (you certainly DON'T need to talk about decimals, even though that's one way to do it). Michael Hardy ( talk) 17:29, 13 June 2008 (UTC)
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)
I am concerned about some recent edits that make use of the "proposition - proof" style of math textbooks. I feel these style elements reduce the accessibility of the article to the layman as they destroy the flow of the text and make heavy use of (standard) math notation. (They also are partially redundant to the section More formal introduction.) I would suggest to at least move the text after the section Gentle introduction and, preferably, to reword it into more accessible prose. — Tobias Bergemann ( talk) 11:29, 11 February 2009 (UTC)
I am responsible for the recent additions, but not for the definition (which is in my opinion the correct one). Also the terms injective, surjective, bijective have been around for some time, as has the notation for a function. I have only edited the section called Definition and Basic Properties. The label THEOREM is used many places in the article. I believe the intent (and certainly my intent) was to indicate a precise mathematically correct statement. Mathematics is proofs. It may be valuable in a wikipedia article to give heuristic arguments or give examples. But when a proof is given it I think it is useful for the reader to know "This is a proof, not an example and not a heuristic argument." That is the function of the labels Proof. The proofs given are all extremely short, though, in principle I see no reason not to include longer proofs. Increasingly Wikipedia is becoming a useful place for students to learn serious mathematics, as opposed to popularized versions (which are also very valuable). Indeed many mathematics articles discuss topics at a level not usually encountered until graduate school. This is a 'good thing.'
Achieving a balance between something for the layman (to use your phrase) and something useful for, say, a university level student, is very difficult and has not been achieved in this article. For example the section on the minimal model for set theory is inaccessible even to most professional mathematicians.
While Wikipedia is not a textbook I think it is very useful to have textbook like sections on particular topics like this one. I believe it enhances other entries. If you look, for example, at measure theory which is more highly rated than this article you might imagine that someone reading that would want to refer to an article on countable which had a quick summary of basic properties. I am open to moving the section later if others think that is appropriate. Johnfranks ( talk) 16:00, 11 February 2009 (UTC)
I moved the main content of the section "Definition and basic properties" later in the article. I combined the short section "Other results about uncountable sets" with it. I added references at the beginning. Johnfranks ( talk) 20:01, 11 February 2009 (UTC)
Is there a consensus that zero is a natural number? If so I am happy to go along. I have never seen this usage. Can someone give a reference? The recent edit declaring 0 a natural number broke one statement which assumed one could divide by a natural number. I think I have fixed it. I believe there is another place in this article (in a section I have not edited) that asserts that 0 is not in the natural numbers. Johnfranks ( talk) 22:35, 11 February 2009 (UTC)
I am removing this from the lede:
But countability has little to do with combinatorics (the mathematical study of counting). Rather, it is a property of certain sets in the mathematical area of set theory.
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)
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)
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)
In many textbooks, it is said that a set is countable if we can list the elements as . My question is: is it true that a set is countable if and only if there exists a Turing machine to enumerate (without termination) all the elements in the set? — Preceding unsigned comment added by Tmbu ( talk • contribs) 01:56, 9 March 2011 (UTC)
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)
Nevertheless, the given definition is wrong (If THIS f is also surjective...). You don't suppose the function f which showed you that S is countable to be surjective in case S is countably infinite. Example: f(n):=n+1 as function from N to N is injective. So this shows us N is countable. But f is not surjective. So N is not countably infinite? To correct the definition you have to find a new function which is bijective (in case f is only injective but not surjective). -- Jobu0101 ( talk) 20:37, 9 December 2014 (UTC)
1) What's with the Jainist thing in history? It would be cool if that were accurate, but I'm doubtful given some of the dates and the complete lack on non-Jainist material. 2) What's with all the "citation needed"'s on the Jainist material. It is plenty well cited, much more thoroughly cited than the rest of the article to say the least. If you wish to *dispute* the citations, which seems reasonable, do that clearly, don't just paste 'citation needed' on all of the clearly cited material. Also, text citations that aren't available online are valid citations, if that's what's drawing your ire. 73.170.107.159 ( talk) 06:45, 15 April 2015 (UTC)
Q can be defined as the set of all fractions a/b where a and b are integers and b > 0. This can be mapped onto the subset of ordered triples of natural numbers (a, b, c) such that a ≥ 0, b > 0, a and b are coprime, and c ∈ {0, 1} such that c = 0 if a/b ≥ 0 and c = 1 otherwise.
If a ≥ 0 and b > 0 then by definition a/b ≥ 0. — Preceding unsigned comment added by 24.84.237.14 ( talk) 03:48, 9 September 2015 (UTC)
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)
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)
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)
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)
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)
Since the phrasing was changed from the original quote by User:Trovatore I thought I'd pull some numbers in from Google n-grams. "Countably infinite" is at 5.6 units and rising in popularity, while "at most countable" is at 1.3 units and mostly flat. (81% vs 19%) So clearly the countably infinite terminology is the most common terminology and the right one to use for the article. But as far as the other convention, can I say that the convention appears in 20% of publications? Or is that original research? Regardless 20% doesn't seem "common" so maybe it should be described as "alternative". -- Mathnerd314159 ( talk) 17:21, 17 September 2021 (UTC)
Example of countable set 117.98.98.174 ( talk) 06:47, 4 May 2022 (UTC)
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)
Regarding the "citation needed" in the section "A note on terminology": A citation for what is asked for? The fact that the total number of letters in "at most countable" plus "countably infinite" exceeds bot the total number of letters in "at most countable" plus "countable" and the total number of letters in "countable" plus "countably infinite"? Hagman ( talk) 20:22, 14 November 2023 (UTC)
, although with respect to concision this is the worst of both worlds.[citation needed]. I'd also like to remark that checking the definition is generally a good idea in mathematics. - Jochen Burghardt ( talk) 21:19, 14 November 2023 (UTC)