This is thetalk page for discussing improvements to theFunction (mathematics) article. This isnot a forum for general discussion of the subject of the article.

Put new text under old text.Click here to start a new topic. New to Wikipedia? Welcome!Learn to edit;get help. Assume good faith Be polite andavoid personal attacks Be welcoming to newcomers Seekdispute resolution if needed Article policies Neutral point of view No original research Verifiability

Find sources: Google (books ·news ·scholar ·free images ·WP refs) ·FENS ·JSTOR ·TWL

Archives (index):1,2,3,4,5,6,7,8,9,10,11,12,13,14,15Auto-archiving period:3 months

This level-3 vital article is ratedC-class on Wikipedia'scontent assessment scale. It is of interest to the followingWikiProjects:

MathematicsTop‑priority This article is within the scope ofWikiProject Mathematics, a collaborative effort to improve the coverage ofmathematics on Wikipedia. If you would like to participate, please visit the project page, where you can jointhe discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics Top This article has been rated asTop-priority on theproject's priority scale.

The content ofFunction (set theory) wasmerged intoFunction (mathematics). The former page'shistory now serves toprovide attribution for that content in the latter page, and it must not be deleted as long as the latter page exists. For the discussion at that location, see itstalk page.

Text and/or other creative content fromthis version ofFunction (mathematics) was copied or moved intoHistory of the function concept withthis edit. The former page'shistory now serves toprovide attribution for that content in the latter page, and it must not be deleted as long as the latter page exists.

The content ofEmpty function wasmerged intoFunction (mathematics) on 25 July 2017. The former page'shistory now serves toprovide attribution for that content in the latter page, and it must not be deleted as long as the latter page exists. For the discussion at that location, see itstalk page.

The function as relation or mapping in the single and multiple-valued and in classical functions, continuous functions, smooth functions, and about the language of functions, and domains and ranges and images and codomains, is very overloaded. This article is picking a course of opinion which does not represent its wide and varied usage, function the term. Over time, as other aspects of mathematics solidified, it's "function" the term that is most loosely thrown about, then as with regards to relations that are admitted to various sub-fields, each claiming their own definition of function has those are each distinct and different ant not compatible. This article, which could be called "mappings" instead as that's largely what it defines, does not from the outset affect to describe the development of the definition over time, nor does it very well reflect the most usual sort of arithmetic definition with which most people are familiar, or as with regards to domain and range. Mathematics is not merely differential geometry, and the definition of function is among the very most general and general throughout. So, this article should largely start explaining that "function theory" is its own sort of world, and a history and survey of "this is what is called a function historically or in these various settings", then with regards to an opinion of "this is a function today and in the most usual setting", which it is largely arguable that this article does not reflect, instead expect.

It reminds one of "graph", "chart", and "plot", about diagrams of functions, drawing a function.

Functions are modern, and Cartesian thus including the multi-valued, and not just classical functions, smooth classical continuous functions that are single-valued, and not just differential geometry's functions with neither vertical nor horizontal tangent, "functions" in mathematics are very general, and sub-fields that restrict the definition for their own purpose are presumptious that their definition is implicit, where it is not.

This article is opinionated and needs context in itself why the definition of function is so broad that it's about its own sub-field of mathematics, in matters of relation.

This article needs a brief survey of the development of the term over time, and to point to the many different intended interpretations of the term.

This article needs a thorough introduction detailing the survey of the meaning of the term "function" over time as mathematics has grown, and, specifically not removing what has become its fuller definition, in the interest of su-fields that would restrict its meaning for their own purposes in notation, where instead they should declare their own regions of syntax, because general usage does not agree.

This article has problems and hides them.97.113.179.80 (talk)15:03, 19 May 2024 (UTC)[reply]

Are you volunteering to track down a pile of sources and write that draft? –jacobolus (t)21:01, 19 May 2024 (UTC)[reply]

Keeping in mind the goal of simultaneously being readable to middle-schoolers and providing pointers to current research-level mathematics... —David Eppstein (talk)21:03, 19 May 2024 (UTC)[reply]

As noted somewhere above, the intent was to introduce something that is usable in schools mathematics courses. Ok, nice. But why not mention it explicitly? Why not say right away (I guess it was before) that this definition is specific for set theories.

Or we could add a section dedicated to the history of the term an the notion. Leibnitz, Newton, Cauchy had no clue about functions being their graphs ("pairs of values").

To me, it's a shame to promote just one specific view of things, the school-level mathematics. It's so XX century, the century were everything was "defined" as sets. These days mathematicians must be familiar with model theory, and see clearly that functions as "sets of pairs of sets" is just a model (in sets).

Vlad Patryshev (talk)00:38, 17 June 2024 (UTC)[reply]

I agree. The "Formal definition" section should be expanded to note that the graph definition is one possible formalization. We should concisely explain alternatives (e.g. other constructions, or an axiomatic approach) for which we can find citations. But keep in mindWP:UNDUE as the graph definition probably remains the most common or conventional one.

In fact, the existing text under "Formal definition" should be cut down. As written, it's not terrible, but it repeats essentially the same definition twice. Invoking the concept of a relation is unnecessary to explain the concept of the graph of a function, and its verbosity only distracts. I think it would be better to state just the graph definition directly, with a concise note that "This graph${\displaystyle R}$ can also be viewed as defining abinary relation between the domain${\displaystyle X}$ and codomain${\displaystyle Y}$".

It might also be worth connecting the graph-based definition to the "Multi-valued functions" section further down, as the graph definition is more natural for that generalization than for the basic concept of a function.73.223.72.200 (talk)02:25, 28 July 2024 (UTC)[reply]

In fact, we already have an article covering the history:History of the function concept. So perhaps we only need to explain the math of a few selected alternative formalizations without regard to their history. We can then link to the history article in lieu of putting in detailed history in this overview article.73.223.72.200 (talk)02:38, 28 July 2024 (UTC)[reply]

The text could be improved by removing the two conditions in the middle of the text and only using a worded definition followed by the set-builder form.Roryyarr (talk)12:14, 28 September 2024 (UTC)[reply]

Please, be more accurate: where the text should be modified and which conditions should be removed"D.Lazard (talk)14:05, 28 September 2024 (UTC)[reply]

Paragraph 3 of the formal definition subsection is redundant as paragraph 5 presents the same information. Below is a suggested edit.

=== Formal definition ===

The above definition of a function is essentially that of the founders ofcalculus,Leibniz,Newton andEuler. However, it cannot beformalized, since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms ofset theory. This set-theoretic definition is based on the fact that a function establishes arelation between the elements of the domain and some (possibly all) elements of the codomain. Mathematically, abinary relation between two setsX andY is asubset of the set of allordered pairs${\displaystyle (x,y)}$ such that${\displaystyle x\in X}$ and${\displaystyle y\in Y.}$ The set of all these pairs is called theCartesian product ofX andY and denoted${\displaystyle X\times Y.}$ Thus, the above definition may be formalized as follows.

A function is formed by three sets, thedomain${\displaystyle X,}$ thecodomain${\displaystyle Y,}$ and thegraph${\displaystyle R}$ that satisfy the three following conditions.

• ${\displaystyle R\subseteq \{(x,y)\mid x\in X,y\in Y\}}$
• ${\displaystyle \mathrm{\forall }x\in X,\mathrm{\exists }y\in Y,\left(x,y\right)\in R}$
• ${\displaystyle (x,y)\in R\wedge (x,z)\in R⟹y=z}$

Roryyarr (talk)11:07, 29 September 2024 (UTC)[reply]

I have edited the article for clarifying that the definitions are the same. However the two versions must been kept, because most people do not like to read a big succession of formulas without prose, while other prefer to not refer to relation theory or need help for formalizing prose.D.Lazard (talk)11:44, 29 September 2024 (UTC)[reply]

Thank you D.LazardRoryyarr (talk)14:09, 29 September 2024 (UTC)[reply]

About once a year I present corrections to this article and am disappointed to see that the indicated defects remain. Ample references showing how to do better are in the bibliography I added to the article a couple years ago, but now they appear hidden. Anyway, the current definition still states:

Inmathematics, afunction from asetX to a setY assigns to each element ofX exactly one element ofY. The setX is called thedomain of the function and the setY is called thecodomain of the function.

It is logically unacceptable because it makes both the domain and the codomain ill-defined. Indeed, let${\displaystyle f}$ be a function obeying the above definition, claiming the domain to be${\displaystyle X}$ and the codomain to be${\displaystyle Y}$. Let now${\displaystyle X^{\prime }}$ be a proper subset of${\displaystyle X}$ and let${\displaystyle Y^{\prime }}$ be a proper superset of${\displaystyle Y}$. Then, by the above definition, the same functrion${\displaystyle f}$ assigns to each element of${\displaystyle X^{\prime }}$ exactly one element of${\displaystyle Y^{\prime }}$, so its domain is${\displaystyle X^{\prime }}$ and its codomain is${\displaystyle Y^{\prime }}$, different from${\displaystyle X}$ and${\displaystyle Y}$ respectively.

The idea of making a codomain an extra attribute of a function is an unnecessary complication and also extremely limiting. One example: normally, the composition${\displaystyle g\circ f}$ is defined forany functions${\displaystyle f}$ and${\displaystyle g}$; its domain is the set of all${\displaystyle x}$ in the domain of${\displaystyle f}$ such that${\displaystyle f(x)}$ is in the domain of${\displaystyle g}$, no further conditions. Requiring (as with most definitions using codomains) that the codomain of${\displaystyle f}$ equals the domain of${\displaystyle g}$ makes composition unusable even for very basic purposes such as the chain rule in calculus.

The only way to do fully justice to the universality of the function concept is simply defining a function as being fully specified by its domain and by a unique value for every domain element, nothing more. Any basic article about functions that makes things more complicated detracts from its value to the readers. The simpler defination does not prevent defining alabeleled function as a different concept that indeed has a codomain, but that should be presented only later as a variant.

To conclude, any editors who insist on attaching a codomain to a function in the basic definition of a Wikipedia article should present very seriousarguments why they think that is necessary or even useful.Boute (talk)10:04, 8 June 2025 (UTC)[reply]

The argument for the current definition is simply that this is the one given in most reliable sources. This is the strongest possible argument since the role of an encyclopedy is not to provide new definitions. Moreover, if you remove the codomain from the definition you get "Inmathematics, afunction from asetX assigns to each element ofX exactly one element." Most people would be confused and would ask the question: what is the nature of assigned elements? The answer is: there are taken from some setY. So, we come back to the current definition.D.Lazard (talk)12:23, 8 June 2025 (UTC)[reply]

The current definition is not given in any reliable source, certainly not in Halmos as claimed. Whether or not you are a codomain-fan, the current definition is logically incorrect, as I have proven. No one is proposing any new definitions, there are plenty of acceptable ones around. Cutting the definition in half to remove the codomain would not solve the logical defect.Boute (talk)16:08, 13 June 2025 (UTC)[reply]

For context, previous discussions:Archive 6 § Codomain and mathematics,Archive 9 § Suggestions ...,Archive 10 § Slowly getting there ... and§ A Modest Proposal,Archive 11 § Letting logic prevail ...,Archive 12 § Correcting shortcomings ...,Archive 14 § Towards a coherent article ...,Archive 15 § Total reset needed .... Maybe you can summarize the arguments made by other editors at those discussions and what you think has changed in the intervening time. You may also want to ping past participants who may be interested in discussion. From a skim it looks like in past discussions people did some careful literature surveying and found that your proposed version was not as well supported by sources. I might be mischaracterizing though; I only did the briefest skim. –jacobolus (t)17:03, 8 June 2025 (UTC)[reply]

I am doubtful that covering all that old ground would be helpful. It is a matter of logic, not a political campaign. Did you not find the package of 20-odd sources I provided a couple of years ago?Boute (talk)16:16, 13 June 2025 (UTC)[reply]

@Boute: As far as I remember I followed your last-year discussion. As of today, you would have a point if we would define justa function. However, (again: if I remember right) as a result of your discussion, we now definea function from a set X to a set Y. Imo, this is the best compromise between the advocates of labeled functions and those of unlabeled functions. I wouldn't object against adding a section near the end that discusses these variants. -Jochen Burghardt (talk)17:36, 8 June 2025 (UTC)[reply]

My criticism about the current definition was certainly about "function from X to Y" (the definiendum), not just "function". In fact, my proof shows that, by the current definition, a "function from X to Y" is also a function from X' to Y' for any subset X' of X and any superset Y' of Y. That would cause no logical problem, but would result in a very uncommon definition. Before talking about repairs, and the question of whether we should make a codomain an attribute, it should be clear to everyone why the current definition is logically flawedBoute (talk)16:37, 13 June 2025 (UTC)[reply]

It's difficult to tell exactly what your objection is. The opening sentence does implicitly require that X and Y are part of the data that describe a particular function. It's maybe not immediately obvious from the way it's worded, but it's made so later on when a more formal definition is given. Composition of functions with dissimilar (co)domains is often just a minor notational abuse to avoid having to explicitly write out when codomains are extended, etc. So what's the problem?35.139.154.158 (talk)17:05, 13 June 2025 (UTC)[reply]

There are many problems, compounded by collapsing 3 definitions into one: (i) function from X to Y, (ii) domain of that function, (iii) codomain of that function. Let us disentangle.

Part (i) is implicitly quantified by "for any sets X and Y", as per common style conventions in mathematics. The opening phrase states the definiendum. In the full definition part (i) is acceptable logically, but does not lead to a very common definition (although it isnearly equivalent to Def. 2.1 in Bartle's areal analysis text). Part (ii) and (iii) are ill-defined since, by part (i), the very same function is also a function from X' to Y' (witness sets introduced in the proof). This ill-definedness is a fatal logical flaw.

As for composition of functions with a codomain f / domain g mismatch, proper definitional design means not requiring notational abuse or patching. Any necessity for abuse ("minor" being in the eye of the beholder) reveals poor design, unworthy of the elegance of mathematics. Not attaching a codomain is issue-free. Composition is just one example of the numerous issues with codomains.

A personal aside: since around 1976 I have been waiting in vain for a single well-founded argument in favor of attaching a codomain to a function. Answers have always been either based on a misunderstanding or evasive, full of red herrings etc. Here is an opportunity for giving the first sound argumentad rem!

However, since so many editors insist on codomains for unspecified reasons (fetishism?) instead of offering raders the simpler and more universal variant, they should at least ensure a logically flawless definition of the codomain variant, despite its unnecessary complications. Such flawless definitions are found in most category theory textbooks and many set theory textbooks (e.g., Bourbaki, Dasgupta): the triplet formulation with${\displaystyle ⟨f,X,Y⟩}$. This "works" because the definiendum cannot make X and Y attributes of the function; that must be done in the definiens.Boute (talk)09:23, 14 June 2025 (UTC)[reply]

If I unsterdand well your argument, you are arguing that theidentity function on the real numbers is the "very same function" as the inclusion of the positive real numbers into the real numbers, and that the identity function of the positive real numbers is the "very same function" as the inclusion of the positive real numbers into the real numbers.

Also it seems that you misunderstand the given definition by misplacing the implicit quantifiers. The definition must be interpreted as a "⁠${\displaystyle \mathrm{\forall }X\mathrm{\forall }Y}$⁠, a function from⁠${\displaystyle X}$⁠ to⁠${\displaystyle Y}$⁠ is ...". In other words, we have a parametric definition, and there is nothing flawed here.D.Lazard (talk)13:03, 14 June 2025 (UTC)[reply]

It seems you did not understand my argument at all, and leave it up to you to find the fallacy in obtaining your conclusion about the identity function.

Also, I did not misplace any quantifiers, in my text I placed them at the same spot as you did, and also observed that there is nothing wrong until that point, just an unusual definition. The flaw is in part (ii) and (iii), clearly and undeniably. Please try to read more carefully what people write and avoid red herrings that have plagued this article from the start.Boute (talk)19:22, 14 June 2025 (UTC)[reply]

I regret my impatience when writing the preceding remark. It is just frustrating to see that the single most important concept in mathematics (the function concept) cannot find proper coverage in this article. Before signing off until 2026, I need to clarify that, by "unusual definition", I do not mean the formulation of that definition, but the concept of function covered by that definition. In other words, although "part (i)" contains no logical contradiction, it does not cover the common concept of function.213.132.130.11 (talk)06:20, 15 June 2025 (UTC)[reply]

The fallacy is in using the definite article "the". My reasoning shows that,with the current Wikipedia definition, if f is a function from X to Y, it is also a function from U to V for any subset U af X and any superset V of Y. If you want to consider injection functions, which satisfy f(x) = x, then "an" injection function from X to Y is also "an" injection from U to V as defined. To make the desired injection function unique, you have to specify explicitly that the domain is X. The current Wikipedia does the opposite, by wrongly assuming that X follows from the definition.Boute (talk)03:14, 2 July 2025 (UTC)[reply]

The current Wikipedia definition does not assume "that X follows from the definition"; it includes X as a part of the definition. So, with Wikipedia definition, a function from X to Y,is not a function from U to V unless U = X and V = Y. So, you reasoning is entierely based on a fallacious interpretation of what is written in the article.D.Lazard (talk)08:33, 2 July 2025 (UTC)[reply]

Talking about the Wikipedia definition, you fail to distinguish between definiendum (function from X to Y) and the definiens (assigns to each element of X exactly one element of Y). The definiendum is a relation between a function and two sets. Only the definiens can make X and Y part of the function, but you must do that explicitly. The cause of your misinterpretation is wishful thinking that the Wiki formulation includes X a part of the definition of function, but that is not the case.

Note that even Halmos contains misleading passages inNaive Set Theory. On page 30 he defines a function f from X to Y as a functional subset of X x Y such that dom f = XThis domain part is essential. On page 31 he defines the inclusion map from X (in)to Y as usual, and correctly deduces it is the set of all pairs (x, y) in X x Y for which x = y. However, by his definition of function from X to Y, this implies that this inclusion map actually is also the identity function on X, whereas Y can be any superset of X. So for any X and any superset Y of X, the inclusion map from X (in)to Y is actually the identity map on X, which Halmos should have emphasized.Boute (talk)09:37, 2 July 2025 (UTC)[reply]

The things we are talking about in this article's lead are inhabitants of the type "functions from X to Y". There is no logical implication that they are therefore also inhabitants of a different type.

If you are insisting that a function is really some other kind of mathematical object like a set, and that the set can also be a function of a different type: how is this different than insisting that numbers are really sets and therefore can have members? —David Eppstein (talk)17:15, 2 July 2025 (UTC)[reply]

I think that the problem of Boute is not this one: He insists to use the philosophical meaning of the word "definition" without understanding that definitions are not exactly the same in modern mathematics as in classical philosophy. In particular, the classification of definitions given in the articleDefinition is totally meaningless in mathematics. On the opposite, many mathematical definitions do not fit in this classification. This is the case here where we have aparametric definition, where the object that is defined depends on two parameters (the domain and the codomain), and these two parameters must appear in both the definiendum and the definiens. By the way, the terms "definiendum" and "definiens" are terms of classical logic, and are no more in use in mathematical logic and in mathematics.D.Lazard (talk)18:28, 2 July 2025 (UTC)[reply]

David Eppstein: My criticism of the current Wikipedia definition is independent of the issues you raised. If you think the definition is OK, add a correct source. I gave a logical counterargument. BTW: did your email address change?

D. Lazard: My incidental use of Latin does not indicate any philosophical stance as you arbitrarily cooked up. It was only meant to clarify the issues for you, sorry I failed. I have been working with modern mathematics for over 50 years, so if you think the problem is my understanding, the Wikipedia definition must be very obscure indeed. In any area of mathematics, a definition mentions a concept to be defined and a characterization. You are right in saying that the two parameters must appear in both the definiendum and the definiens, but that is exactly the problem: the definiens in the Wiki example is incomplete and does not make X and Y uniquely determined by the function as defined. Look how Halmos did it correctly by adding dom f = X (not Y, which is not an attribute of f in his definition). Category theory textbooks also do it correctly via triplets. Final remark: if you really keep insisting that the current Wikipedia definition is OK, provide a valid reference, so this article can stop running in circles.Boute (talk)20:46, 2 July 2025 (UTC)[reply]

What does my email have to do with it? I have multiple email addresses that have not changed for decades, except maybe in which one I prefer for which communications. For discussion of Wikipedia content, I prefer it to proceed openly, here, rather than by email. —David Eppstein (talk)20:54, 2 July 2025 (UTC)[reply]

BTW means "by the way", unrelated to this topic. Since you bring up Wikipedia, email generates less noise and is more convenient for attachments.Boute (talk)05:31, 3 July 2025 (UTC)[reply]

A major problem in getting this article on track is that logical arguments are ignored in this discussion. Disagreement should be supported by counterarguments, not by rudely questioning someone else's understanding (in violation of Wikipedia policy). Resuming the thread, the criticism of the current Wikipedia definition still stands. Also, in the open access article[1] arguments are given why the chosen definition should reflect the use of functions in mathematics at large rather than the preferences in a restricted theoretical area.Boute (talk)16:30, 3 July 2025 (UTC)[reply]

Recently,Farkle Griffen re-added the following paragaph to the article

Strictly speaking, this does not fully describe a function in the usual sense, as there is no way toapply a function${\displaystyle f}$ to an argument${\displaystyle x}$. However, this definition is sufficient to be equivalent to a system with the ability to apply functions. Thus, one may use aconservative extension (called anextension by definition) to introduce a new primitivefunction symbol "${\displaystyle \bullet (\bullet )}$", as in "${\displaystyle f(x)}$", denotingfunction application. This function symbol takes in expressons denoting a set-theoretic function, "${\displaystyle f}$", and an arguent, "${\displaystyle x}$", then "${\displaystyle f(x)}$" denotes the unique object${\displaystyle y}$ such that${\displaystyle (x,y)\in f}$. That is, if "${\displaystyle f}$" denotes a function, and${\displaystyle x}$ is in its domain,${\displaystyle f(x)=y⟺(x,y)\in f}$. If${\displaystyle f}$ is not a function, or${\displaystyle x}$ is not in its domain, "${\displaystyle f(x)}$" denotes theempty set.

By this edit, they restart, without any discussion on the talk page, an edit war for which we were blocked fo 24 hours.

This paragraph is highly confusing by several aspects: It uses several concepts of logic that are not supposed to be known by readers of this article. Also, it start by talking of the "usual sense" of a function without saying whether this usual sense is the one used in standard set theory or the one used when a function is a primitive notion (such as inlambda calculus). It is only when a function is a pimitive concept that "application of a function" must be defined.

However, I must agree that previous version of the section lacks explanation how functional notation and usual jargon ("⁠${\displaystyle f}$⁠ maps⁠${\displaystyle x}$⁠ to⁠${\displaystyle y}$⁠", ...) are defined from the formal definition. So, I'll replace Farkle Griffen's paragaph with a short paragraph explaining this.

Please, do not change or remove it without providing strong reasons here.D.Lazard (talk)10:35, 5 September 2025 (UTC)[reply]

Hi, @D.Lazard. My biggest issue with the current version is that it doesn't really offer any explanation, and, under reasonable interpretations, is simply misleading.

if${\displaystyle x}$ and${\displaystyle y}$ are related by${\displaystyle R}$, this is commonly denoted${\displaystyle f(x)=y}$ instead of${\displaystyle (x,y)\in R}$ or${\displaystyle xRy}$.

If the whole string "${\displaystyle f(x)=y}$" is a kind of alternative notation for${\displaystyle xfy}$, as in "${\displaystyle \bullet (\bullet )=\bullet }$" is "one symbol", then the string "${\displaystyle f(x)}$" isn't defined on its own.

I agree that mathematical logic is fairly technical, but that's simply how it's done. And I don't think relevant, foundational content should be removed on the basis of being "too technical", especially content with several sources and in a section titled"Formal definition".–Farkle Griffen (talk)14:20, 5 September 2025 (UTC)[reply]

then the string "⁠${\displaystyle f(x)}$⁠" isn't defined on its own: Fixed.D.Lazard (talk)10:05, 6 September 2025 (UTC)[reply]

"[...] the unique element of the codomain that is related to${\displaystyle x}$ is denoted${\displaystyle f(x)}$."

Hmm... But how are you doing thiswithoutfunctions? Are you just introducing a whole new symbol for every possible input? As in "${\displaystyle f(2)}$" is just "one symbol"? If so, you still don't have function application; you cannot have${\displaystyle f(x)}$ for avariable x, or have an expression inside f(...). Both the input and the function need to be capable of being variables, e.g. in a differential equation, where functions can be unknowns.

Please respond to the last paragraph above in my previous reply.–Farkle Griffen (talk)13:26, 6 September 2025 (UTC)[reply]

I do not understand your last sentence, since, per the manual of style, an answermust always appearafter the answered post.

"doing thiswithout functions" is nonsensical in an article about functions in mathemtics and in a section about the formal definition of functions in terms of set theory. If you are talking of functions in other frameworks than set theory, it is in§ In the foundations of mathematics that this must be discussed.

"Both the input and the function need to be capable of being variables": No; a variable is a symbol that represents (that is, is a name for) a mathematical object. In⁠${\displaystyle f(x)}$⁠, there are two variables,⁠${\displaystyle f}$⁠ and⁠${\displaystyle x}$⁠, and the mathematical objects that they represent are defined in the preceding text. It is not a new symbol ("one symbol") that is introduced, this is a syntactic rule for forming an expression. As usual, when a variable appears in an expression, it can be substituted with any expression that can be evaluated to an acceptable value of the variable.D.Lazard (talk)14:43, 6 September 2025 (UTC)[reply]

""doing this without functions" is nonsensical in an article about functions in mathemtics"

This is a bit rude; you could simply ask for clarification. I am not talking about other foundations; I am talking about set theory. This is supported by all three sources I included in my edit.

By default, the language of ZFC only has variables, predicates, and logical connectives. There are no constant symbols, functions, or operations to create largerterms than an individual symbol. If you are asserting${\displaystyle f(x)}$ is an expression that contains two individual symbols, then you are attempting to introduce an operation, which cannot exist in the initial language of ZFC.–Farkle Griffen (talk)15:08, 6 September 2025 (UTC)[reply]

To reply to the superscripted initial line, I was asking you to respond to this paragraph:

"I agree that mathematical logic is fairly technical, but that's simply how it's done. And I don't think relevant, foundational content should be removed on the basis of being "too technical", especially content with several sources and in a section titled "Formal definition"."–Farkle Griffen (talk)15:22, 6 September 2025 (UTC)[reply]

@D.Lazard?–Farkle Griffen (talk)15:19, 7 September 2025 (UTC)[reply]

Sorry, I am unable to decipher what do you mean with "how are you doing thiswithout functions". I understand "doing this" as "defining functions", and, with this interpretation, the question is indeed nonsensical.

If you are asserting⁠${\displaystyle f(x)}$⁠ is an expression that contains two individual symbols, then you are attempting to introduce an operation, which cannot exist in the initial language of ZFC: In other words, you assert that the integers⁠${\displaystyle \mathbb{Z}}$⁠ and addition⁠${\displaystyle a+b}$⁠ cannot exist in ZFC. If you means something else, you must be clearer.

However, as this page is not for discusssing editor's posts, but for discussing the article, I give below more reasons for removing the paragraph quoted above:

• "Strictly speaking, this does not fully describe a function in the usual sense": fixed by my edit.
• "there is no way toapply a function to an argument": wrong, as the application of a function to an argument is defined as the unique element related by the function to the argument.
• "one may use aconservative extension (called anextension by definition) to introduce a new primitivefunction symbol": In other words you say that you do not work in ZFC, but in an extension of ZFC. This is wrong, since ZFC allows the use of new symbols and new notations for abbreviating complicated logical expressions.
• "primitivefunction symbol": Apparently, nobody but you consider functional notation as a primitive of ZFC

Apparently, your paragaph is an attempt for definingfunction application. Function application is abinary operation that takes a function and an element of its domain to povide an element of its codomain. It is defined this way in the linked article, where the definition uses the fact that the set of all functions from⁠${\displaystyle D}$⁠ to⁠${\displaystyle R}$⁠ form a set denoted⁠${\displaystyle R^D}$⁠. This definition does not requires any extension of set theory.

My opinion is that the view offunction application as a binary operation deserves to be mentioned in the article, but this must be the object of a separate section.D.Lazard (talk)16:28, 7 September 2025 (UTC)[reply]

"with this interpretation, the question is indeed nonsensical."

You are assuming an interpretation that is nonsensical. That does not mean what I said is nonsensical, only that your interpretation is. Again, you can simply ask for clarification.

"In other words, you assert that the integers⁠ and addition ⁠⁠cannot exist in ZFC."

Not in the usual ZFC; or not that notation, at least. The symbols${\displaystyle \mathbb{Z},0,1,-1,...,+}$ are not part of thelanguage of ZFC. The logic of the integers can bemodeled by ZFC, or a conservative extension can include them to get them explicitly.

"since ZFC allows the use of new symbols and new notations for abbreviating complicated logical expressions."

Strictly speaking, this is not correct. Introducing new symbols via a definition requires you to do anextension by definition. That is what that article is about.

There is no other way to get the function application notation to work as intended. This is all fairly standard, though Levy and Mendelson from the sources I added go into more detail if you need more convincing.–Farkle Griffen (talk)16:57, 7 September 2025 (UTC)[reply]

@D.Lazard?–Farkle Griffen (talk)18:10, 8 September 2025 (UTC)[reply]

@Farkle Griffen Can you explain a bit more about why you want to make the definition more complicated and technically detailed? To me personally, this seems like the wrong direction to move – if anything, we should be trying to make the early sections of this article about such a centrally important topic more accessible. If you think additional technical material is essential to mention somewhere, perhaps it can be moved further toward the bottom of the page, put into a footnote, or moved into a secondary article. –jacobolus (t)18:43, 8 September 2025 (UTC)[reply]

"[...] if anything, we should be trying to make the early sections of this article about such a centrally important topic more accessible."

I agree with this... The only issue is that what I'm trying to add is specific to set theory, and the only place that makes sense is right after the set-theoretic definition.

"Can you explain a bit more about why you want to make the definition more complicated and technically detailed?"

The motivation is kinda the opposite. Calling the set-theoretic definition the "formal definition" in the first section is confusing for less advanced readers, since it's not clear how toapply that function (and as it turns out, it's not super easy to get that). Honestly, I've never really come across the set-theoretic definition outside of introductory set theory books. Most authors try to immediately move to the usual notation. To your point, I wonder if the best choice is to move the set-theoretic definition to the bottom too.

I'll go ahead and make a restructuring edit so it is clear what I'm advocating for. Feel free to revert me or edit it if you disagree or think it needs more discussion.–Farkle Griffen (talk)19:05, 8 September 2025 (UTC)[reply]

If you think it's likely to be controversial, maybe this can be workshopped on the talk page instead of done as a revert war on the article? –jacobolus (t)00:01, 9 September 2025 (UTC)[reply]

More like, consensus through the talk pageand edits. I don't think any administrator would call this a revert war if we're actively talking and I'm inviting you to do so.

If you disagree with my edit and don't want to revert me for procedural reasons (I guess?) I can self-revert if you want.

In any case, do you have an opinion?–Farkle Griffen (talk)01:09, 9 September 2025 (UTC)[reply]

I don't have a particularly strong feeling about it. Just as a general matter, if you think something is more likely to be reverted than left by other editors, it can be less trouble to discuss first. YMMV. –jacobolus (t)01:43, 9 September 2025 (UTC)[reply]

I think I would only disagree with this in the case of complicated edits. Restructuring involving a lot of section moves seems like a case of "show don't tell" for getting a point across.–Farkle Griffen (talk)02:48, 9 September 2025 (UTC)[reply]

If someone filibusters a talk page and, whenever everyone else gets too tired of responding to the same points, keeps reinstating the same previously-reverted content, then I would call it an edit war. —David Eppstein (talk)02:12, 9 September 2025 (UTC)[reply]

@David Eppstein I mean, I agree, but I don't understand how that relates to the comment you're replying to. The edit mentioned is not reinstating an earlier edit. Unless this is passive-agressively hinting at some other discussion unrelated to this one, I have no idea what this reply is saying.–Farkle Griffen (talk)02:41, 9 September 2025 (UTC)[reply]

This seems a bit harsh. Farkle Griffen has only made 5 or 6 not-outrageously-long comments here, and the "everyone" seems to just be D.Lazard. You might be mixing the discussion up with one on some other page / some past discussion here atFunction (mathematics) involving different editors? –jacobolus (t)02:43, 9 September 2025 (UTC)[reply]

Unless if I missed some points, this restructuring consists mainly of:

• Renaming section "Formal definition" as "In set theory"
• Moving this section,§ Partial functions and§ Multivariate functions near to the bottom of the article
• Making section "Formal definition" a subsection of§ In foundation of mathematics
• Making the two other moved sections as subsections of a new section§ Generalizations

I stongly disagree with these changes:

• The formal definition given in the article appear in most elementary textbooks of algebra and is commonly taught to college students. So, it belongs to the beginning of the aticle, and more specifically to section§ Definition
• Partial functions are soon encountered by many college students (for example, the logaithm and the tangent function are partial function of a real variable). As partial functions are often simply called "functions", they must be defined in section§ Definition for avoiding reader confusion. Also, partial functions are not generalizations of functions; they are functions whose domain is not fully specified.
• I agree that multivalued functions are generalizations of functions. However, they are encountered soon incomplex analysis (the complexsquare root and thecomplex logarithm are multivalued functions) and are also often simply called functions. So, they must be introduced soon in the aticle. However, the place of this section deserves to be discuted, and, before a consensus, it must be kept at the previous place.

So, since no other editor support fully these changes, which are not clearly motivated, I'll restore the previous version.D.Lazard (talk)13:47, 9 September 2025 (UTC)[reply]

The points are fair and the revert is invited. This is getting a bit off topic from the main discussion topic. I think I'll be willing to make a new discussion for a better organization for this article eventually, but my motivations are tied up in other articles at the moment.

In the meantime, @D.Lazard, can you respond to my last reply about the paragraph in question?–Farkle Griffen (talk)18:07, 9 September 2025 (UTC)[reply]

I will not respond to your last post about the opening paragraph: I gave the reasons for which I think that this paragraph is not suited for Wikipedia. You answered by essentially saying that I am wrong and that I do not understand basic mathematics. I have nothing to add to my reasons, and it is to other editors to arbitrate between us.D.Lazard (talk)19:42, 9 September 2025 (UTC)[reply]

@D.Lazard, I apologise if that's how it came off, that certainly wasn't my intention. However, mathematical logic is not basic math. It's an unfortunate state of affairs that mathematicians tend to just hand-wave away foundations, but that seems like the status quo.

This doesn't need to be a black and white discussion. We can discuss how to phrase the paragraph, rather than "me vs you". I think any formulation that explains, roughly "function application is a binaryfunction symbol:${\displaystyle \bullet (\bullet )}$." would be good enough for me. Unless you're not willing to discuss at all?–Farkle Griffen (talk)20:34, 9 September 2025 (UTC)[reply]

This seems like a pedantic and confusing addition. The notation with the dots might not be understood by a significant proportion of readers, and the term "binary function symbol" is obscure and a search for «"binary function symbol" "function application"» turns up mostly non-relevant uses and uses in specialized papers about symbolic logic and lambda calculus (where it doesn't refer to the common function notation, but to some specialized new symbol used as a binary operation, so that function application is explicitly written like⁠${\displaystyle f\star x}$⁠ or the like instead of⁠${\displaystyle f(x)}$⁠). I think discussion along those lines can probably be saved for more specialized articles about those topics. The current text doesn't seem especially ambiguous or misleading. To quote:

The functional notation requires that a name is given to the function, which, in the case of a unspecified function is often the letterf. Then, the application of the function to an argument is denoted by its name followed by its argument (or, in the case of a multivariate functions, its arguments) enclosed between parentheses, such as in$${\displaystyle f(x),\sin (3),\text{or}f(x^2+1).}$$

The argument between the parentheses may be avariable, oftenx, that represents an arbitrary element of the domain of the function, a specific element of the domain (3 in the above example), or anexpression that can be evaluated to an element of the domain (${\displaystyle x^2+1}$ in the above example). The use of a unspecified variable between parentheses is useful for defining a function explicitly such as in "let${\displaystyle f(x)=\sin (x^2+1)}$".

When the symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to writesinx instead ofsin(x).

–jacobolus (t)22:39, 9 September 2025 (UTC)[reply]

[...] turns up mostly non-relevant uses and uses in specialized papers about symbolic logic and lambda calculus

Symbolic logic, specificallyfirst order logic, is indeed the correct context. It is the context which ZFC is defined.

Maybe I'm just suffering from being on the other side of it, because I honestly don't understand what's confusing. But okay, I agree it's pedantic, but there has to besome way to to make it less confusing, right?

Specifically, the audience I am trying to write this for is readers trying to understand foundations. If a reader learns that "technically, ZFC does not have functions" and wants to learn how to get back the usual notation, the first place they're likely to check isFunction (mathematics) § Formal definition. The misleading bit with what you quoted is that it gives no hint that it is not completely formal, or that there is a canonical, formal way to do it.

I think discussion along those lines can probably be saved for more specialized articles about those topics

I mean yeah, I agree, and that's fine. It is already partially there atFunction application § Set theory. I'm not asking that this article does a deep-dive into formal logic, just that it actually points to the specialized articles for interested readers.

Are you willing to at least let me workshop it?–Farkle Griffen (talk)23:44, 9 September 2025 (UTC)[reply]

Side remarks: you wrote"first-order logic, is indeed the correct context. It is the context which ZFC is defined". The lead of the linked article as well asHigher-order logic#Quantification scope imply that ZFC is not a first-order logic.

Also, the sentence"technically, ZFC does not have functions" is highly confusing, since it is true if interpretedZFC does not have functions among its primitives and it is wrong if interpreted as"ZFC does not allows a technical definition of functions".D.Lazard (talk)00:57, 10 September 2025 (UTC)[reply]

@D.Lazard, I can see how those would be very confusing, however ZFC is always first-order. The first line ofZermelo–Fraenkel set theory#Formal language says explicitly"Formally, ZFC is a one-sorted theory in first-order logic." as well as any book on axiomatic set theory. There are second- and higher-order set theories, but they are not ZFC.

Higher-order logic#Quantification scope definitely needs to be rewritten for clarity. This is confusing because in ZFC the "individuals" are also called "sets". However, the "sets" in that article refer to sets external to the theory, not internal ones. It should say something more like"Second-order logic allows for the quantification over all subsets of thedomain of discourse. Third-order logic allows for quantification over all sets of subsets of the domain of discourse, and so on."–Farkle Griffen (talk)02:07, 10 September 2025 (UTC)[reply]

@Jacobolus?–Farkle Griffen (talk)22:52, 10 September 2025 (UTC)[reply]

Personally I'd look for definitions of function found in textbooks aimed at undergraduate students (e.g. on intro analysis, abstract algebra, discrete math, or "first course in pure mathematics" kinds of courses), rather than looking for definitions from niche monographs aimed at professional logicians or whatever. YMMV. –jacobolus (t)00:08, 11 September 2025 (UTC)[reply]

We have an articleHistory of the function concept which might be more appropriate for a deep dive into the many possible alternatives. –jacobolus (t)00:10, 11 September 2025 (UTC)[reply]

@Jacobolus, you're right, I should have been clearer: the first placeon Wikipedia. But I digress. Can you respond to the last bit of my response there? To quote:

"I'm not asking that this article does a deep-dive into formal logic, just that it actually points to the specialized articles for interested readers. Are you willing to at least let me workshop it?"

I'm a bit confused by:

"[...] into the many possible alternatives"

I agree there are alternative foundations, but I don't understand how it applies to the conversation. Assuming§ Formal definition is based in ZFC, what I wrote is the canonical way to introduce function application.–Farkle Griffen (talk)00:34, 11 September 2025 (UTC)[reply]

I think "formal definition" near the top here here should be a formal definition as found in a highly cited source aimed at early undergraduates (as compared to an "informal" description aimed at an audience of 12–16 year old schoolchildren). If something more elaborate or a deeper discussion than that is required, it could be put in a footnote, further down the page, or relegated to some other article. –jacobolus (t)01:04, 11 September 2025 (UTC)[reply]

@David Eppstein, I'm using this post since it seems related enough.

What exactly is the goal ofthis revert? If the goal is to undo the typo, then why not just fix the typo? If the goal is to make sure the notation${\displaystyle f(x)}$ is mentioned, that notation is already introduced in section§ Notation. I don't understand your edit summary. The current formulation in a section titled"Formal definition" seems very misleading.–Farkle Griffen (talk)23:40, 10 September 2025 (UTC)[reply]

The goal was simply to undo a change that did not appear to me to be an improvement. It was not part of any broader agenda (such as how formalist we should be or whether function objects come equipped with domain and codomain sets) and I think my edit summary stated the same thing clearly enough. Supplying the missing verb would only have made sense if I were sure which verb you intended and if I thought that the fixed change would be an improvement, neither true in this case. —David Eppstein (talk)23:45, 10 September 2025 (UTC)[reply]

"[...] if I thought that the fixed change would be an improvement, neither true in this case."

Can you explain this a bit more? I still don't think I understand. Lets say I fix the typo. Your reply seems to imply you would still revert me. Can you explain why?–Farkle Griffen (talk)23:51, 10 September 2025 (UTC)[reply]

Ok, since you demand a longer and more in-depth explanation than the edit itself and my gut reaction to it, then the answer is, more or less,WP:USEPROSE. Your edit took a three-sentence paragraph with lots of concrete nouns and replaced it by "If $$$ is a function defined by a functional relation $$$, and $$$ or $$$, and one says that $$$ maps $$$ to $$$, or $$$ is the image by $$$ of $$$." Where are all the words? It's too formula-heavy to be readable. Now maybe what was there before had some of the same issues but your edit made it worse. I don't think it's reasonable to demand that I spend a lot of time and effort defending my choice to undo edits that in my opinion are not improvements. Put more of your energy into crafting your content better and less on arguing about it, please. —David Eppstein (talk)00:04, 11 September 2025 (UTC)[reply]

Okay, I was sloppy and should have been more careful. Point taken.

I am honestly trying to discuss. My point in moving to the talk page is not to argue to reinstate my edit, but to understand your issues with it so my next editcanbe an improvement. I apologize if that is not how it comes off. I really am trying here.

TBH, the whole paragraph as phrased seems off-topic in the section§ Formal definition, and is mostly repeated in the section§ Notation. The current paragraph seems redundant at best, and outright wrong at worst. My motivation is just to make it less misleading. Do you have an objection to simply deleting that paragraph? I can't accidentally introduce typos if I do that.–Farkle Griffen (talk)00:16, 11 September 2025 (UTC)[reply]

Yes, I object to the removal of an explanatory paragraph. It doesn't have to be that specific explanatory paragraph. But the rest of that section merely puts together some abstract mathematical objects. The paragraph is needed to explain how those objects match our less-formal intuitions and notations about what a function is supposed to do (namely, that it takes inputs and produces outputs, rather than just sitting there inertly as a set of pairs of things). —David Eppstein (talk)01:48, 11 September 2025 (UTC)[reply]

@David Eppstein, Okay, that makes sense. The paragraph reads a little better now after the most recent edit, which, given it still stands, you don't oppose?

Taking injacobolus's opinion above, would either of you object to adding just one sentence at the end of that paragraph? Roughly:

"A more formal definition of function application can be given in terms offirst-order logic (cf.Function application § Set theory)."

Satisfying jacobolus's request (hopefully?), all information is delegated to the main article, and this article would merely point there. No possibly confusing notation is added or explanation removed, hopefully satisfying your conditions. And the interested reader would have clear indication of where to find it. Does anyone oppose this?–Farkle Griffen (talk)17:58, 11 September 2025 (UTC)[reply]

Is the formalization ofsyntactic sugar important to direct readers to? Why? For the formalization of functions themselves, and the syntactic sugar that makes the formalization understandable, I can see the value. But why do readers care about formalization of syntax? —David Eppstein (talk)18:08, 11 September 2025 (UTC)[reply]

You're probably right that ~90% of readers don't care about foundations. But the last ~10% is who I'm writing for. It's kind of hard to explain concisely, but the basic gist of it is, it's not just syntax or syntactic sugar. What you're getting from first-order logic isn't just a "notation" for functions, but real, bona fide function application. And trying to formalize it any other way can result in genuine foundational issues.–Farkle Griffen (talk)18:51, 11 September 2025 (UTC)[reply]

I think you missed my point. The readers who care aboutfoundations of functions can find them in the section already existing here. The readers who care aboutfoundations of syntax can read other articles, likeformal language; the details about formalization of the syntactical sugar used for functional notation are not really specific to functions, not really about functions per se, and I think not relevant to most readers here. —David Eppstein (talk)20:12, 11 September 2025 (UTC)[reply]

If I did, I don't think I understand your point still, because I feel like my response just above mostly responds to this. In slightly stronger language (without the intention of being rude) :

The definition is not about syntax. If you treat ZFC as a set theory, and not some uninterpreted formal language, then the definitionnot just a notation butliterally function application. It is how to actually take theset-theoretic definition of "function" and introducereal function application. Without it, as you said, "functions" are just static sets of ordered pairs.

It is not about the foundations of syntax. It is exclusive to set theory, and tothisspecific set-theoretic definition of "function".–Farkle Griffen (talk)20:47, 11 September 2025 (UTC)[reply]

Who does not treat Zermelo–Fraenkelset theory (ZFC) as an uninterpreted formal language that is not a set theory? Otherwise, I cannot understand this post (what does mean "literaly" here? what is areal function application?, etc.)D.Lazard (talk)21:05, 11 September 2025 (UTC)[reply]

In mathematical logic, you work withformal theories (such as ZFC), which don't necessarily have to be given an interpretation. You would just work with the symbols and transformation rules. In this sense, none of the formulas in ZFC "mean" anything. If you give the theory an interpretation, i.e., the individuals literally represent sets (in the same way the word "egg" represents a real egg), then function application as defined is the usual function application, as you use it.

That is to say that definition is literally function application, insofar as ZFC is literally about sets.–Farkle Griffen (talk)21:20, 11 September 2025 (UTC)[reply]

If you work in a world like ZFC where everything is sets, then everything is sets: function application is just what we call certain usages of certain sets. There is no need to formalize that something is "literally" function application: that is just our interpretation of certain arrangements of set operations. To say that something is "literally" function application you need to be working in a world where there is a class of mathematical objects that are "literally" functions and their application, and not merely sets that we interpreting as those things. A world such as, say, type theory. But then if you have functions built into your foundation as first class objects, there is no longer a need to specify how to formalize them as something else. As far as I can see, the need to both have literal functions and to have a formalization of those things in sets only comes up in specialized occasions when you are showing how to interpret one formalization in another. —David Eppstein (talk)21:43, 11 September 2025 (UTC)[reply]

I don't think I understand that last sentence, but I don't really see how this objects to my previous reply to you above at all.

What I originally wrote in the article is well-sourced as being the definition of "function application" in ZFC for that particular definition of function. Unless you have a source saying that is not "real" function application, I don't see an issue with taking the authors at their word, beyond philosophical differences.–Farkle Griffen (talk)22:01, 11 September 2025 (UTC)[reply]

https://us.metamath.org/mpegif/df-fv.html includes 1 and links to 4–7 alternative definitions of function application. I think all of them are too technical for anywhere in the top several sections of this article. Maybe you can add something in§ In the foundations of mathematics (perhaps with details in a footnote) or atHistory of the function concept, or perhaps it might be relevant somewhere inZermelo–Fraenkel set theory. Edit:Function application is probably the most relevant place to delve into this. –jacobolus (t)03:49, 12 September 2025 (UTC)[reply]

Huh, interesting... following the sources, a couple link to Takeuti & Zaring, Monk, and Enderton, which useNBG and class comprehension to do it, and a couple use the foundations inPrincipia Mathematica, so definitely not ZFC.

The one using the union operator is super contrived but technically would be usable in ZFC. It doesn't link to a source though, unfortunately.

Cool link! I haven't seen that before!

I think all of them are too technical for anywhere in the top several sections of this article.

I agree. My only motivation is to mention itsomewhere in the article. I'm happy to work in§ In the foundations of mathematics if it is accepted there.

In response to your edit, I agree. I really don't want to delve into anything here, just a pointer to the main article so interested readers know where to find it.–Farkle Griffen (talk)04:23, 12 September 2025 (UTC)[reply]

I oppose addingA more formal definition of function application can be given in terms of first-order logic (cf.Function application § Set theory). for several reasons.

• This section is about the definition offunctions, not offunction application.
• "More formal" does not apply sincefunction application is not defined previously.
• The definition of functional application given in the linked article involve only set theory, not first-order logic.
• This definition uses the definition of functions and is exactly as formal as the definition of functions that is given here.
• Wikipedia is not a source, especially when the linked article has many issues (in particular here, where⁠${\displaystyle f(x)}$⁠ is defined when⁠${\displaystyle f}$⁠ is not a function or⁠${\displaystyle x}$⁠ is not an element of the domain of⁠${\displaystyle f}$⁠; applied to the square root function, it defines⁠${\displaystyle \sqrt{-1}=\mathrm{\varnothing }}$⁠).

D.Lazard (talk)20:49, 11 September 2025 (UTC)[reply]

@D.Lazard, To respond to the last point,"Wikipedia is not a source...", my original edit contained three sources, which included that fact. The choice of the empty set is by convention, and due to certain technicalities in how function symbols need to work in general. Mendelson and Levy talk about this quite explicitly, and every other source I found uses that convention.

It doesn't really cause issues further up the chain of abstraction, aside from some "technically true but weird" trivialities like that one.–Farkle Griffen (talk)20:59, 11 September 2025 (UTC)[reply]

• C-Class level-3 vital articles
• Wikipedia level-3 vital articles in Mathematics
• C-Class vital articles in Mathematics
• C-Class mathematics articles
• Top-priority mathematics articles

• All Wikipedia vital articles
• Wikipedia pages using copied template
• Wikipedia pages using merged-from template without oldid