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Why do we consider a set which is treated for all intents and purposes as a 'collection' with one element as being different from the element itself? In this 'collection' there is one element, and only one element, if we have one thing we would never draw a distinction between the one object and an imaginary 'collection' containing it? Why do we bother to make this distinction?

J D
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Confused
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    Are you asking about why this distinction is drawn in a specific set theory (such as ZFC) or asking about the intuitive interpretation of set theories in general? – Greg Nisbet May 15 '22 at 07:39
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    More intuitively, it seems strange that we make this distinction. – Confused May 15 '22 at 07:41
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    I think this question is a duplicate of this one. Mauro Allegranza's answer has a good example. The short, intuitive answer is that the identity of a set is determined by its elements. The empty set {} for example has no elements, but the set {{}} has the empty set as its only element. The sets in ZFC are all like this, if you reach down far enough you end up with the empty set. – Greg Nisbet May 15 '22 at 07:48
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    You can define a theory with some non-collection things (typically called atoms) and collections without internal structure (so a = {a}), but if you went through and formalized this idea, you'd end up with something quite different than set theory and more like a formalized mereology. I knew of a reference for such a system once, but I can't find it at the moment. – Greg Nisbet May 15 '22 at 07:53
  • @GregNisbet thanks for than answer, will read that thank you, the wording was a bit different so I didn't find it in the search. – Confused May 15 '22 at 08:00
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    Because the properties of a man are very different from the properties of a set of men. – Mauro ALLEGRANZA May 15 '22 at 09:37
  • @user1007028 We make this distinction primarily so that we can treat the empty set and sets with just one member in that same way that we treat sets with multiple members (or even potentially infinite members) without having to have exceptions to handle the first two cases. The standard concept that we use to explain/justify/rationalize this is the "set as container" metaphor/allegory. – RBarryYoung May 15 '22 at 22:30
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    A grocery bag containing an orange is not an orange. It's a grocery bag that happens to contain an orange. That's a good way to think about sets. – user4894 May 15 '22 at 23:04
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    Did you notice any set in math has an encircling bracket symbol in its explicit form which is quite like the famous Husserl's epoche bracket in phenomenology of philosophy? Once something is bracketed, its sense and reference are changed subtly and often times significantly... Thus some understanding of phenomenology hopefully might provide you with some deep and clear insight for set theory in general and this puzzle/its alternative formulations/axiomatizations in particular... – Double Knot May 16 '22 at 05:55
  • From a more philosophical point of view, the difference is that between an individual "named" by a proper name, like e,g, Napoleon: it is meaningless to use a proper name if the corresponding individual does not exists, and that of "description": "first emperor of France", that identify a class that may be empty or may have a single element.; in every case it is meaningful. – Mauro ALLEGRANZA May 16 '22 at 08:22
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    See Wlllard Van Orman Quine, Peano as logician (HPL, 1987) for a very interesting discussion about origins and philosophical meaning. – Mauro ALLEGRANZA May 16 '22 at 09:44
  • @MauroALLEGRANZA the properties of a man are exactly the same as the properties of a battalion of one man, who is that man! – user253751 May 16 '22 at 12:23
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    @user253751 - NO, a man has black hairs, a batalion thas no hairs at all. – Mauro ALLEGRANZA May 16 '22 at 12:27
  • @MauroALLEGRANZA if we talk about properties of the members of the battalion such as 'half the men have black hair' would that be a property of the battalion or the men themselves? perhaps this is where seeing a battalion and a 'set' are very different, in terms of how we treat real life collections. – Confused May 16 '22 at 14:11
  • @GregNisbet The standard reference seems to be Peter Aczel, Non-well-founded sets (1988), ISBN 0-937073-22-9 – prosfilaes May 16 '22 at 17:01
  • Could you clarify that? If not, you will face such interpretations as user4894's grocery bag containing an orange, which seems an obscure way to think about anything… – Robbie Goodwin May 18 '22 at 22:45
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    Don't you think you spin round in circles when you try to use "if we have one thing we would never draw a distinction between the one object and an imaginary 'collection' containing it?" to clarify why we consider a set which is treated for all intents and purposes as a 'collection' with one element as being different from the element itself? – Robbie Goodwin May 18 '22 at 22:51
  • Do you think a set with two, or seven, or 27 or any other number of elements would be distinct from those elements? Don't you think we need to make that distinction until you explain how the number of members in the set could make any difference? – Robbie Goodwin May 18 '22 at 23:01

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In computing, there are data models (such as the XPath data model used for XML) in which an item and a singleton collection containing that item are treated as indistinguishable. You can build a coherent and workable system on this basis. It has some advantages: most notably, you don't have to decide up-front whether properties (such as the author(s) of a paper or the email address(es) of an author) -- in mathematics, functions -- are single-valued or multi-valued; a single-valued property is a special case of a multi-valued property, not something completely different. But there are also disadvantages, notably when it comes to handling collections of collections.

So I think the answer to your question, why do we consider a singleton set as distinct from its one member, is simply because it's useful. Other models are possible and coherent, but generally less useful.

Michael Kay
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    Another example from a different domain: Set theories with "ur-elements". Ur-elements are meant to represent non-sets that could be members of sets. A popular way of modelling ur-elements within a set theory that has only sets is to use "reflexive sets". Namely, say that each ur-element is a set which contains exactly one element - itself. As Michael says, you can build a coherent and workable system on this basis if you want to. – Jirka Hanika May 16 '22 at 11:59
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    We would say that a set is not the only way to view a collection, then, and our natural way of viewing a collection where one object is just that, and where no objects cannot form a collection is just one way? The collection is rigorously defined as being almost distinct from just the items themselves, but it is also defined entirely by what those elements actually are? The set and the objects are different, but the set is one set, and another is a different set based on what elements are in each set. – Confused May 16 '22 at 17:07
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One reason why this is true is because there is such a thing as the empty set - the set with no elements at all.

Consider a set X that contains only the empty set, and nothing else. How many elements does X have? Obviously, it has just one.

But if there were no distinction between a set with one element and the element itself, then X would be the same thing as the empty set. That is, X would have zero elements. But since X has one element, and 0 is not 1, This is a contradiction.

Therefore there must be a distinction between a set with one element, and the element itself.

Dawood ibn Kareem
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You may consider a collection as a container: Apparently a thing included in a container is different from the thing without container.

Aside: Set theory provides operations to handle sets (= collections) but no operation to handle objects in isolation.

Jo Wehler
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    The first paragraph is good. The second is incorrect as written. There are set theories with urelements. Zermelo’s original set theory included urelements and there are interesting implications when adding urelements to New Foundations set theory. There is active research in set theories with urelements. – Just Some Old Man May 15 '22 at 20:20
  • @JustSomeOldMan The only "special" element you need is the empty set. And even that is simply the empty container with nothing special or "ur" about it. – cmaster - reinstate monica May 15 '22 at 20:28
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    No, he's right. Also, talking about what we "need" is misplaced, here; we need different things for different theories. An ur-element has a technical meaning: it enters into the element-of but not the set-of relation, not even over zero elements. It is a mirror of one definition of proper classes (things that enter into the set-of relation but not the element-of relation). – Kristian Berry May 15 '22 at 20:58
  • @KristianBerry I can find nothing about urelements in Zermelo's axioms (https://en.wikipedia.org/wiki/Zermelo_set_theory). And nothing that forbids things to enter the element-of or set-of relations. As such, the "ur-" stuff is some parlance that has been added on top of the theory, but it's not part of the theory itself. – cmaster - reinstate monica May 15 '22 at 21:19
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    Why in the world are you limiting your research to Zermelo's axioms? EDIT: At any rate, Zermelo's 1908 theory reportedly allowed or even had ur-elements. – Kristian Berry May 15 '22 at 21:21
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    @cmaster 'The only "special" element you need is the empty set.' – That's not true for set theories in general. In particular, NFU set theory has infinitely urelements (things which are not sets), and there is no obvious way to modify the theory to get rid of the urelements without also altering other important properties of the theory. Thus, the situation for NFU is different from the situation for ZFC, in which mathematicians have successfully eliminated all urelements, and declared that no objects exist in the theory other than sets. – Tanner Swett May 16 '22 at 12:23
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Why do we need a zero when it's conceptually the same as nothing? Because zero, as a number, has very different properties from being nothing at all.

The reasoning is similar about the empty set compared to nothing at all. Sets and their member(s) have different properties. The empty set is a set because, for example, if we have a set and remove all the members of the set, it still is a set and distinct from being nothing at all.

Niklas Rosencrantz
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    Or put another way: The number zero and the empty set both allow us to talk about having nothing. It's not useful to put a taboo on having nothing. – cmaster - reinstate monica May 16 '22 at 06:57
  • @NiklasRosencrantz "Why do we need a zero" False analogy. "Zero beans" means exactly the same as "no beans". The empty set is not a set, so it is not a set with no elements. – Speakpigeon Jun 29 '22 at 17:13
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They are distinct because a set is something different than most elements you can put into it. Sets and elements of sets usually are distinct categories or types of things (an element might be an animal, a human, a cake, or whatever abstract thing you can think of - yes, including other sets).

Think of a set like a big box where you can put stuff in.

Even if there is exactly one thing in the box, the box is still not the same as the thing.

You can do a lot with a box that you cannot do with something that is not a box, and vice versa.

Of course, when we work with sets (i.e., in maths or when writing computer programs) we could say that we identify any individual thing with the set that contains just that thing, but that would just be a convention. It might be practical when there are lots and lots of occasions of 1-element sets, or when the difference does not play that big of a role, but it would still just be a shortcut, and the one would always something different than the other.

Also, it could be a very bad idea to do this: in maths, there are some constructs where sets of sets are analyzed, and a set containing the empty set has meaning. These things would be impossible to talk about if the set which contains the empty set were identified with the empty set.

AnoE
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Try ordering water from a restaurant without a container. The container serves a purpose. In set theory, it introduces the notation of the set, and without sets, set theory wouldn't be very manageable; when we write a set, 'x and y' simply says nothing about whether or not these two identifiers are considered parts of whole; '{x,y}' does, so it has a mereological function (SEP). So, it formalizes the notation for a container, collection, system, etc., and makes it easy to determine what is in and not in a collection at a glance.

More importantly, once one introduces the idea of a container, one has also introduced the notion of a boundary which can be thought as a means of discriminating what is part of a system and what is not. One way of formalizing the notation is set-builder notation. Hence, {2,4,6,8,10} and {evens between 2 and 10 inclusive} are equivalent extensions of sets because the curly braces imply we have a collection, the first defined by a list and the second a predicate.

What happens if a thing is the same as itself and its boundary condition? Let x:={x}. now, we have an infinite recursion, and we can rewrite x={{x}} and x={{{x}}} ad infinitum by continuing to substitute the set x for itself. Since syntactically this is an infinite loop, that means every object is an infinite collection of collections of itself. And that's meaningless.

On a linguistics note, a set can be considered a linguistic artifact embodying a conceptual metaphor, and formalization of a Metaphor of Containment. Essentially, your brain may be wired to fundamentally group certain salient qualia or phenomena, such as when you subitize which may be viewed as instances of the gestalt qualities of perception.

wjandrea
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J D
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  • The unfolding of a = {a} is not at all clearly trivial. The main consequence is not that this singleton function does not exist but that it corresponds to a parafounded and more importantly unwell-ordered set, so whereas there is an obvious translation of Zermelo singletons into natural numbers, there is no such association between a Quine singleton and the numerical symbols (the ten digits or their permutations/interpositions via decimals/etc.) commonly used, which makes it seem meaningless. – Kristian Berry May 15 '22 at 21:08
  • Epsilon-delta unmasks continuity s a construction of infinite regress, and Goedelian numbers demand that truths are subject to an infinite regress of axiomatization. But what metaphysical precept is born of object circularity? I intuit that casualty is subject to itself, but I see no clear benefit to the theorizing other than some sort of taxonomy of permutations. What an I missing? – J D May 16 '22 at 00:30
  • IIRC there are possible or by now actual applications in computer science/programming/those kinds of fields. This essay's abstract mentions the computation angle (and a category-theoretic one to boot), for example. – Kristian Berry May 16 '22 at 00:47
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Plural quantification touches on the intuition you seem (in my opinion, so correct me if I'm wrong) to be having, here. E.g.:

This is that the quantifier ∃R is a plural quantifier (and would thus be better written as ∃rr) and that plural quantification is ontologically innocent. Therefore (13) does not assert the existence of any “set-like” entity over and above the sets in the range of the quantifier ∀x.

Or:

The second argument is nicely encapsulated by Boolos’s remark that “It is haywire to think that when you have some Cheerios, you are eating a set” (1984: 448–9 [1998a: 72]) ... [However] We can for instance let all predicates take plural entities as their arguments. The verb “ate” will then always receive as its interpretation the relation the-elements-of x ate-the-elements-of y, thus removing any ambiguity. Whether or not this response is ultimately acceptable, it shows that the argument in question remains inconclusive.

The problem occurs on the Benacerraf level, then. Normally, we have that {} = 0, {0} = 1, {{0}} ≈ {0, {0}} = 2, etc. If we suppose that our set a = {a} were to take the place of zero, however, both the Zermelo and von Neumann implementations of the natural numbers can be sabotaged to some extent. On the other hand, the axiom of extensionality says that set terms in the local theory are equivalent by way of indicating the same number of elements. If a is the only element of a, then a should equal 1, let us suppose. However, on its own terms a is not well-ordered (because not well-founded) so it is not believed that a is a set in the well-founded universe. A parafounded one, yes (and some theories can posit exactly one Quine atom, others class-many).

Plurality (and unity) on the one hand, and the extensionality relation on the other, though involved with each other (modulo quantifiers), do not open the same exact question in theoretical space. If we advert to plural quantifiers to eliminate discrete quantification over individual sets that 'stand in for' plurality simpliciter (as units modulo each level of plurality), we shouldn't need sets (in terms of baroque parentheses) to compile their elements into singleton markers for the discrete quantities quantified over. So the issue of saying, "A set with one element is the same thing as its one element," would not arise as such. We might identify the number 1 with a moment in plural quantification and assimilate talk of sets to some other problematique (e.g. plain order theory).

Kristian Berry
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in the same vein as some answers above:

Lets take a set theory, most set theories are endowed with an axiom of comprehension/ specification (1). Essentially, this axiom schema allows us to move between predicates and sets. If we take a strong correspondence between properties and predicates, this allows us to move easily, between, say "red" and {the set of red things}.

Now most people will accept that there is a difference between a thing and the properties it satisfies. Hence, there must be a difference between the thing and the singleton set consisting only of that thing (2).

1- actually we can use replacement as well, since this implies separation. 2- this of course relies on the correspondence between properties and sets. This is not to say that properties are in bijection with sets, and in fact, they are probably not. This is merely one such motivation.

emesupap
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    Re your "properties are in bijection with sets, and in fact, they are probably not", seems Russell already gave a definite negative answer when he was young... – Double Knot May 16 '22 at 04:58
  • Re your "most people will accept that there is a difference between a thing and the properties it satisfies", it seems there might be also quite a few other people will accept the exact opposite (no such essential ontic differences like OP), such as Leibniz's famous and perhaps widely-accepted Law of Identity of Indiscernables described in 2nd order logic... – Double Knot May 16 '22 at 05:35
  • thanks @DoubleKnot for the engagement, indeed, this is why i qualified both answers. I will say that Leibniz Law is controversial. It may hold, in, say, the caculus of inductive constructions and other type theories, but as a metaphysical principle there are some strong arguments to be had against it.. – emesupap May 16 '22 at 06:02
  • If you're in the camp of against LL as a metaphysical principle, then it should become your core argument evidence/reasoning for your above conclusion to side with OP's title, ie, a set with one element is distinct from that element. Perhaps you may explicate in your above answer "some strong arguments to be had against it" for set theory which is apparently different from other type theories that accept LL... – Double Knot May 16 '22 at 06:09
  • no need- as I note, this is just one motivation, there are other more pragmatic reasons as well. as for my comment, LL is controversial amongst philosophers, so thus should not be built into the base set theory undergirding the metalogic that such philosophers use. in other disciplines, however, LL may be the type of equality we want, and thus may be taken for granted. – emesupap May 16 '22 at 06:23
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I think this purely comes down to precision definitions of words, in the correct context, to ensure that people are discussing the exact same concept.

As a computer programmer, it's very natural for me to think of a list of one, as still a list:

List my_list_of_one = { apple };
List my_list_of_three = { apple, orange, banana };

Thus, in my mind, a list of one is still a list, and not the same as an object or item in a list.

My friend, who is a carpenter, says this is nonsense. A list of one is not (and can not be) a list. Why? Because of the definition of the word "list":

list /lɪst/

a number of connected items or names written or printed consecutively, typically one below the other.

from Old English liste ‘border‘

The definition literally says, "a number of connected items". It's plural.

Who is right? Depends on the context, the job, the task at hand.

Words are only tools, after all, and are meant to be useful.

Stewart
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    Use of a plural noun doesn't imply that there's more than one of something. If I ask you "how many children do you have?" and you say "just one", that doesn't make my question incorrect. – Dawood ibn Kareem May 17 '22 at 23:40
  • @DawoodibnKareem Correct. But using the phrase in the definition, what do you think of, if I tell you, "When I'm older I want a number of children in my family." ? – Stewart May 18 '22 at 08:09
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    Things to consider: 1) Does your friend argue that 0 and 1 are not numbers? Or are they using the "a number" colloquially to mean "some"? That is a colloquialism, and is not a good basis for their proof. 2) I just used "they" and "their" to refer to your friend as I don't know their gender (not a comment about gender identities, this is common usage of the word "they" for hundreds of years); this can't be used to prove your friend is multiple people, similar to "items", when used abstractly (example to follow) ..1/2 – Daevin May 18 '22 at 18:51
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  • Let's rephrase Dawood's example: "How many children do you have?" "I have zero children." Is the response incorrect? Or does your friend believe "I have zero child" is more correct? In English (maybe other languages, I only speak the one), using pluralized words to refer to empty collections is very much acceptable (and sometimes the only) way to build grammatically correct sentences. Your friend can't unequivocally argue that "a number of connected items" expresses that a list contains >1 item. They definitely can argue it, but there are solid and correct arguments against them. ..2/2
    • – Daevin May 18 '22 at 18:55
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    @Daevin All your points are good. The definition my friend was using, I think, is as the synonym "several", found as definition 2, 1st sub-definition at https://www.google.com/search?q=define+number You say this is a colloquialism, and therefore dismiss it. I would say colloquialism is acceptable, because we're talking about what the words "set" and "list" and "collection" mean. The OP is asking about the concept of a set of one vs the concept of the item itself. Concepts are just that. You think them into existence. A set, in this sense, is not a physical object like a bucket. (Continued ..) – Stewart May 18 '22 at 19:45
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    @Daevin In the physical universe, you can have a bucket of crabs. The bucket is an object. The crabs are objects. But, if I have a blank piece of paper, can I say I have a shopping list? A list of zero items? If I write down the word "apple", do I have an item, or a list? I'm not computer programming, I'm shopping. So it doesn't matter. Thus, as I said in my final two sentences, it all depends on context, and what a person considers the words to mean. – Stewart May 18 '22 at 19:45
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    @Stewart very valid points as well, and I do agree with context being extremely important. My comment about colloquialisms was more to address the last sentence (char limits hampered my clarity, my apologies) with regards to words being useful; they are, absolutely, but they start to become detriments when there's ambiguity, or grammatical rules that seem to violate common understanding (like the "child" vs "children" from the example). I didn't completely dismiss it, only said it wasn't a good basis for a proof based on the ambiguity of the usage. ..1/? – Daevin May 18 '22 at 20:07
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    @Stewart As for your shopping list... that is indeed a good philosophical quandary lol. What if you titled the paper with "Shopping List" but did not have any items under it; would it be a shopping list, or would the title be incorrect? Would writing "apple" under the list make the list correct, or still incorrect because there is only 1 item? Fun exercises to consider context, meanings, and concepts, for sure :D. ..2/2 – Daevin May 18 '22 at 20:14
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    @Daevin I agree with what you are saying. My first sentence starts with "precision definitions", and I guess colloquialisms, or even words with multiple definitions, are ambiguous by nature. – Stewart May 18 '22 at 22:30