Talk:Cardinality

Size not cardinality not ordinality

Frank Griffin is putting in edits such that he assumes size is identical to cardinality. That is a common misunderstanding. The highest level set theory mathematicians as well as the original set theory mathematicians such as Cantor claim otherwise. This is a common problem with Wikipedia presenting Set Theory more flowery than it is. One excuse that is used is that cardinality is only technically not size. No it is not at all size. It is size plus 2 lies, than 'infinity' is defined, as is stated, as that which applies to the finite by proof, Does Apply to the infinite (never-ending) without proof No but rather That which applies to the finite by proof Does/Does not (such as an infinity + 1 does not = the another infinity but = the same infinity) Apply to the matching infinite Obviously this is not valid if not universal, but subjective, not general math but set theory. And the lie if something can't be proven true, by the normally used proof, and so not because of only probabilistically true (such as prime number conjectures) then it is false. This is used once per cardinality and once with ordinality. The diagonal argument does not directly prove ≠ size. again the set of natural numbers can be proven the same size as the set of binary strings, binary numbers from 0 to before 1, by a binary tree argument. The opponents of these arguments confuse the list arguments with the tree arguments, thus apparently winning arguments confusing their opponents. Obviously one cannot have lack of positive proof to be negative proof.

Now in formal set theory proofs these matters are handled in the Definitions. Thus Cardinal definition instead of size axioms. Replace with axioms and you get various statements that are in the edits; Your <=> techniques apply to = and ≤ and ≥ but not other relations, (that would produce unacceptable results). ≠ comes from the diagonal argument disproving your = proof, plus ≥ proof.

Since the highest level and original set theorists and Cantor are in agreement with me (size is not cardinality not ordinality ) you are in violation of Wikipedia policy Unless you state these matters are your (Wikipedia's) math system rather than ZFC set theory.

I will let you get away with it If you retitle the article 'controversy over Cantor's theory'to be a general anti-set-theory article and put the corrections there. Victor Kosko (talk) 03:47, 15 January 2026 (UTC)

Victor Kosko, It's not at all clear what you're advocating for here. Can you be more specific about which parts of the article you have issues with and what exactly you want to change them to? – Farkle Griffen (talk) 04:51, 15 January 2026 (UTC)

Axiom of choice in the lead

@David Eppstein, what does it add to mention AC so early in the lead? The expected reader of that section doesn't know what it is, and only adds unexplained technical terms. Also, is it fair to call it a "lie" to not mention alternative foundations of mathematics to start? It doesn't seem like a lie-to-children to give a proof by contradiction without tagging it for assuming the law of excluded middle. – Farkle Griffen (talk) 00:01, 1 June 2026 (UTC)

It adds correctness. The unequivocal statement that sets can be totally ordered by cardinality, without the qualification that this is under choice, is incorrect. It also adds a natural way of briefly and not very technically referring to the later "without the axiom of choice" section, fulfilling the expectation that lead sections provide brief summaries of later material.

I think calling this "alternative foundations" and mentioning the law of the excluded middle is misleading. There is no mention of the excluded middle in our article, and ZF without assuming C is not particularly alternative. —David Eppstein (talk) 00:34, 1 June 2026 (UTC)

But it also requires the axiom of replacement, and axiom of pairing. About half the article (and half of every mathematics article) would need to be rewritten to qualify which axioms of ZFC each proof assumes.

Underlying assumptions about the foundations of mathematics seems tangential to the subject, rather than foundational. I think it's reasonable to say anything is unequivocally true without qualification if it assumes standard foundations of mathematics, which seems like a low enough bar given >99% of mathematicians will never deal with anything else. – Farkle Griffen (talk) 00:56, 1 June 2026 (UTC)

The reasoning in your edit: "adds a natural way of briefly and not very technically referring to the later "without the axiom of choice" section" seems reasonable enough. – Farkle Griffen (talk) 01:09, 1 June 2026 (UTC)

GA review

This review is transcluded from Talk:Cardinality/GA1. The edit link for this section can be used to add comments to the review.

Nominator: Farkle Griffen (talk · contribs) 03:54, 15 May 2026 (UTC)

Reviewer: Carabinieri (talk · contribs) 23:48, 31 May 2026 (UTC)

This looks like an interesting article. I'm looking forward to reviewing it. It may take me a few days to get my first comments up.Carabinieri (talk) 23:48, 31 May 2026 (UTC)

Hi Carabinieri, thanks for taking on this review! I'm looking forward to reading your comments. – Farkle Griffen (talk) 00:06, 1 June 2026 (UTC)

Hey Carabinieri, just wondering if you had time for some initial comments? – Farkle Griffen (talk) 03:16, 14 June 2026 (UTC)

Apologies for how long this has taken. I was actually almost two weeks ago, but then I lost most of my notes when my computer shut down because I hadn't saved them. I then had a hard time motivating myself to start over. I'm on it now and will be done within the next day.--Carabinieri (talk) 09:15, 18 June 2026 (UTC)

No worries on time, there is no deadline, I'm just glad to hear back from you. And I'm really sorry you had to restart, I know how discouraging that can be. If there's anything I can do to make the review easier, let me know! – Farkle Griffen (talk) 15:41, 18 June 2026 (UTC)

I think the article looks to be in pretty good shape overall and I think this could be passed rather quickly. Don't worry about the length of the list of my comments. That may just be me being overly nitpicky. Many of these comments are just suggestions. At times I found it a little jarring the way the article moves from detailed explanations of very basic concepts to briefly skimming over rather advanced ones. I suppose that's unavoidable with this topic, though. Here are my detailed comments:

• I'd consider reorganizing the first two subsections of the "Basics" section. I found it a little strange that cardinality is defined using bijections before bijections are introduced.
  • I added a brief summary of bijection in the definition section. If we can help it, I'd generally prefer to have "definition" be the first section so readers don't think they need two paragraphs of prereqs to get the basic idea. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "which are somewhat more abstract than their counterparts" would suggest removing "somewhat"
  • Removed. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "A function, or correspondence, maps members of one set to the members of another" changing this to "maps each member of one set to a unique member of another" would be more precise
  • Changed. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• That is, for each possible input, one can determine the output" Would suggest something like "That is, for each possible input, the output is defined", because this might otherwise be confused with computability
  • Added. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "it seems reasonable to say there are the same number of objects in each set" would suggest dropping "it seems reasonable to say"
  • Removed. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• I found the proof sketch of the Schröder–Bernstein theorem rather difficult to understand. Could you make it a bit clearer what the function being constructed maps to what in the end? And maybe also hint at why it is bijective?
  • It's not super clear which part makes it confusing, but I think I figured it out. Let me know if this helps. If not, I'm happy to work on this section more. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "the function ⁠$f(n)=2n$⁠ is a bijection from ⁠$\mathbb {N}$⁠ to ⁠$E$⁠." this way of referring to a function is not introduced in the "sets and functions" subsection. Maybe add a sentence there?
  • Changed the notation used in the example in "Sets and functions". – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "This is written ⁠$A\preceq B$⁠, ⁠$A\lesssim B$⁠ and eventually ⁠$\vert A\vert \leq \vert B\vert$⁠" I'm not sure what eventually means here.
  • An artifact from an early version that included "once the |A| notation has been defined". I just removed it to not overcomplicate things. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "These technically over cover the rationals" strike "technically", "over cover"->"over-cover". Since the term "injective" has been introduced, why not use it here?
  • Corrected, and I like to use simpler terms when I can if possible, especially in the earlier sections. – Farkle Griffen (talk) 21:21, 19 June 2026 (UTC)
• "the rational number ⁠$\textstyle {\frac {1}{2}}$⁠ gets mapped to by all the fractions ⁠$\textstyle {\frac {2}{4}},\,{\frac {3}{6}},\,{\frac {4}{8}},\,\dots$⁠," the following is a little easier to parse in my opinion: "the fractions ⁠$\textstyle {\frac {2}{4}},\,{\frac {3}{6}},\,{\frac {4}{8}},\,\dots$⁠ are all mapped to ⁠$\textstyle {\frac {1}{2}}$⁠"
  • Fixed. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "So this function shows ⁠ | Q | ≤ | N | ⁠ not ⁠ | Q | = | N |" This might be a little confusing since "≤" was defined based on the existence of an injective function.
  • Added a brief note that surjection is equivalent to injection in the inequality section with a footnote linking to the section on AC. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "A number is called algebraic if it is a solution of some polynomial equation (with integer coefficients). " I would drop the parentheses since that part is indispensable to the definition.
  • Fixed. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "Since the set of algebraic numbers is countable while the real numbers are uncountable (shown in the following section)" it is shown in the same section, but a later subsection.
  • Changed to "subsection" and added a link.– Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "Considering these real numbers in a column, it is always possible to create a new number such that the first digit of the new number is different from that of the first number in the column, the second digit is different from the second number in the column, and so on" Saying that it's a column seems to imply that there should be multiple columns. I would consider changing it to list. You actually switch from column to list halfway through the next proof.
  • Changed.– Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "For example, if the digit isn't 2," I think the MOS tends to avoid contractions
  • Removed as many contractions as I could find.– Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• The proofs in the fourth and sixth paragraph of the "Uncountable sets" subsections are essentially the same. I think it might be worth condensing this.
  • I think it's worth keeping them as explicit possible. Diagonal arguments tend to be a real snag point for novice readers. It's only two sentences in the last paragraph. I'm not sure how much more that could be condensed without hurting the average reader. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "Cardinal numbers are not usually thought of in terms of their formal definition, but immaterially in terms of their arithmetic/algebraic properties" I'm not sure I fully grasp what the word "immaterially" means here
  • Meaning without regard to what the actual object is. For example, does |{1,2}| represent the Frege-Russel cardinal number, or von Neumann? Mathematicians don't care, and use it the same either way. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "if a relation satisfies the properties of an equivalence relation, the objects used to materialize this relation are equivalence classes, which groups all the objects equivalent to one another" -> "each of which group objects equivalent to one another"
  • Added. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "These called the Frege–Russell cardinal numbers" Missing "are". Also, what exactly is "these" referring to? Presumably, equivalence classes. Would suggest replacing "these" with somehting along the lines of "equivalence classes under equinumerosity relation".
  • Added "equivalence classes"; I think "under the equinumerosity relation" is implied.– Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• In the initial part of "Cardinal numbers" section I was a little confused about the following: Do Frege-Russel cardinals rely in any way on Hume's principle? Or are these just different approaches to defining cardinals? If the latter is true, does focus on the algebraic properties of cardinals, rather than their definition, only apply to the Hume approach or is this true of both paradigms? Would suggest clarifying this or reorganizing the information a little here.
  • Hume's principle just says "two sets are equinumerous exactly when they map to the same cardinal number". Both definitions have Hume's principle as a theorem, but you can just take it as a kind of axiom without regard to what |{1,2}| is made of and be just fine. I tried to clarify this a bit. Let me know if this area is still confusing. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "In formal contexts, the natural numbers can be understood as some construction of objects satisfying the Peano axioms—a list of properties, such that any system satisfying these properties is, in a certain sense, just like the natural numbers." Is this really relevant here? Wouldn't using the axiom of infinity instead of the Peano axioms lead to the same properties of finite cardinals?
  • The axiom of infinity just says that some infinite set exists. The Peano axioms are more abstract: they tell you what counts as "the natural numbers". In that section we want to recreate the notion of, for example, the number 7, from the concept of cardinality. We could just point to to a collection of objects and say "obviously this is the same as my concept of '7'" but that's philosophically circular, especially if we are trying to show how to recover that concept. What we want to is point to how mathematicians actually think about the natural numbers (through the Peano axioms) and show that using it, we can recover the cardinal intuition of natural numbers "two apples plus two more apples is four apples" rather than abstract notion of "two" and "four". – Farkle Griffen (talk) 21:21, 19 June 2026 (UTC)
• "For example, a set is called Dedekind-finite if it cannot be put in one-to-one correspondence with a proper subset of itself, though this definition requires the axiom of choice." Is it really the definition that requires the axiom of choice? Isn't it more that showing the equivalence of those two definitions requires AC?
  • Clarified. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "Since ⁠ ⁠$\omega \sim \mathbb {N}$⁠ by the natural correspondence" Isn't ω = N?
  • Depends. You could define N:=ω, but some mathematicians have a disposition to saying "2={0,1}". There are other possible constructions of N too, such as the Zermelo ordinals. ω is a specific, well-defined set. ⁠$\mathbb {N}$⁠ is a more abstract, immaterial set, which could be constructed in many different ways. – Farkle Griffen (talk) 21:21, 19 June 2026 (UTC)
• "Proving that such a set always exists is known as Hartogs' theorem, wherein the smallest ordinal not less than or equal to than a set" strike "than"
  • Stricken. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "The intuitive principle that is ⁠ A" "is" -> "if"
  • Corrected. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "It is hardly controversial to modern mathematicians, however, because of its unique historical controversy it is often given special treatment not given to other axioms that basic proofs which use it ought to call it out." -> "It is hardly controversial to modern mathematicians. However, because of its unique historical controversy it is often given special treatment not given to other axioms in that basic proofs which use it often call it out."
  • Fixed. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "Specifically, there exists sets, such that there is a surjection from ⁠ A onto ⁠ B⁠, but no injection from ⁠ B into ⁠ A , thus injection is a strictly stronger notion" New sentence starting with "thus". It might make sense to explicitly state that an injection from A into B implies a surjection from B onto A.
  • Added and mentioned explicitly. I think the expected reader of this section can put together why, but let me know if this needs to be clarified further. – Farkle Griffen (talk) 21:21, 19 June 2026 (UTC)
• Why are universe and absolute capitalized in the "Proper classes" section?
  • No idea why "Universe" was, and "Absolute" was because Ferreirós capitalized it. Looking at other sources, it looks like both capitalized and uncapitalized are common. I defaulted it to uncapitalized. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• In the "Skolem's paradox" section, isn't it necessary to require that the theory in question itself be countable? Otherwise, a language with an uncountable number of symbols and a theory stating that these symbols be interpreted pairwise differently would not have a countable model.
  • Mentioned "countable" – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "From the 6th century BCE, the writings of Greek philosophers, such as Anaximander, discuss infinite sets or objects, however, it was generally viewed as paradoxical and imperfect (cf. Zeno's paradoxes)." What does "it" refer to?
  • Clarified. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "In A Treatise of Human Nature (1739), David Hume is quoted for saying "When two numbers are so combined, as that the one has always a unit answering to every unit of the other, we pronounce them equal"" Why not just "said" instead of "is quoted for saying"? Also, quotes shouldn't be italicized.
  • Fixed. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "This created a new area of mathematical analysis studying what is now called space-filling curves." is->are
  • Corrected. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• "Though Russell initially had difficulties accepting Cantor's and Frege’s intuitions of cardinality" Not a complete sentence.
  • Fixed and moved to footnote. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)
• The "History" section is inconsistent in what tense it uses when reporting what someone said or wrote: "the writings of Greek philosophers, such as Anaximander, discuss", "Aristotle distinguished", "Galileo Galilei presented". I would avoid "would be codified" and "would be the first" as that makes the chronology hard to follow.

--Carabinieri (talk) 17:16, 19 June 2026 (UTC)

I really appreciate the detailed notes. I think I managed to fix everything, save for the stuff on cardinality/natural numbers being "immaterial". See my notes above. I'm happy to discuss any of the changes. – Farkle Griffen (talk) 21:17, 19 June 2026 (UTC)

Hey Carabinieri, just wondering if you have any further comments? – Farkle Griffen (talk) 12:03, 27 June 2026 (UTC)

@Carabinieri, It's been a little over a month since the review started. How should we proceed? – Farkle Griffen (talk) 18:04, 8 July 2026 (UTC)

You're right. I'm really sorry about my inactivity. Based on your responses, I'm happy to pass this now. I'll have a few comments within the next few days that I hope might be helpful to you, but irrespective of those comments I'm satisfied that the article meets the good article criteria. Great work here.--Carabinieri (talk) 05:03, 17 July 2026 (UTC)

Squaring cardinals to get choice?

In the last paragraph of Cardinality#Without the axiom of choice, it says that $A\times A\sim B\times B\implies A\sim B$ (the function which squares cardinals is injective) implies the axiom of choice. That it follows from the axiom is apparent from Talk:Tarski's theorem about choice#Simplified proof because $A\sim A\times A\sim B\times B\sim B$ if they are infinite and it is easily proved by mathematical induction if they are finite. However, I am not clear on how to derive the axiom from it. Perhaps we could try using A = S×H and B = S∪H if S≠{}? JRSpriggs (talk) 13:17, 10 July 2026 (UTC)

It's theorem 11.8 in Jech's The Axiom of Choice pp.157-158. Here's a version available online. You need all of section 11.3 though, since it builds off a few lemmas and he introduces some uncommon notation like ⁠$\aleph ({\mathfrak {q}})$⁠ for the Hartogs number of ⁠${\mathfrak {q}}$⁠. – Farkle Griffen (talk) 14:42, 10 July 2026 (UTC)

Thank you for the link. That is a very interesting paper by Jech. JRSpriggs (talk) 19:41, 10 July 2026 (UTC)

Did you know nomination

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• Comment or view
• Article history

)

• ... that in certain theories of cardinality, not only are there infinitely many sizes of infinity, ⁠$\aleph _{0},\aleph _{1}...\aleph _{\omega }...$⁠, but there is a largest one?

• Source: Ferreirós, José (2007). Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Science Networks - Historical Studies (2nd ed.). Basel: Birkhäuser. p. 379. doi:10.1007/978-3-7643-8350-3. ISBN 978-3-7643-8349-7. LCCN 2007931860. Archived from the original on 2019-07-09.

• • Reviewed:
  • Comment: Apologies for the slightly late entry (one day outside the limit), I was attempting to improve the article Absolute infinite since it is the subject of this hook, and then I went on vacation for a few days. The hook is a summary of Cardinality § Proper classes.

Improved to Good Article status by Farkle Griffen (talk). Number of QPQs required: 0. Nominator has fewer than 5 past nominations.

– Farkle Griffen (talk) 16:42, 25 July 2026 (UTC).