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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?

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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? May 15 at 7:39
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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. May 15 at 7:48
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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. May 15 at 7:53
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    Because the properties of a man are very different from the properties of a set of men. May 15 at 9:37
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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 at 23:04

11 Answers 11

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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.

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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. May 16 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. May 16 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.

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    Comments are not for extended discussion; this conversation has been moved to chat.
    – Philip Klöcking
    May 17 at 5:45
  • So, if A={A}, then count(A)=count({A}). For example, if A={1,2,3}, then count(A) is both 3 (as normal) and also 1.
    – Nat
    May 18 at 14:50
  • @Nat I think you and I may have both assumed that count(...) is always somehow defined. Along with 1 and 3. If the empty set doesn't exist, then probably 1 and 3 don't exist either. May 19 at 11:06
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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.

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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. May 16 at 6:57
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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.

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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. May 15 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. May 15 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). May 15 at 20:58
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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. May 15 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. May 16 at 12:23
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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.

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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.

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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. May 15 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 at 0:30
  • 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. May 16 at 0:47
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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).

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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.

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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... May 16 at 4:58
  • 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... May 16 at 5:35
  • 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..
    – Papuseme
    May 16 at 6:02
  • 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... May 16 at 6:09
  • 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.
    – Papuseme
    May 16 at 6:23
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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.

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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. May 17 at 23:40
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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 at 18:51
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    3) 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 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 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 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 at 19:45
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There are various rigorous "mathematical" systems that involve sets. In some of these, the contents of a set can not change. That [the latter] is not definitive of "set".

A "set" is a conceptual grouping. If I create a set whose content is, for example, one particular horse (this horse is named "Philip")... the horse does not change. Particularly, the fact that I generated this set -- this conceptual construct -- is not now an attribute or property of this horse.

The corollary is that... if I were to create a set with two horses [glossing over whether or not the same horse is involved] and a duck in it... the concept of a set here is exactly the same as the concept of the other set. It is the fact of my creating a conceptual grouping.


[ Further discussion, if useful.

When we are discussing sets, there is usually an unconscious "universe of consideration". For example, if I were to mention the set {1,2,4}, my audience would automatically assume that we are talking about numbers (and probably just integers); it would be rather odd for (for instance) a sloth to get involved. That is because we use sets towards conceptually sophisticated ends, and it is far from apparent what purpose a set with numbers and wild animals and who-knows-what in it... might have.

I suggest that perhaps what is confusing about the issue raised is that, when we are talking about, particularly, the concept of a set containing only one element -- say the horse "Philip" -- we have three conceptual elements in play (not two). The obvious two are * Philip and * the "set" construct under discussion. However, there is also the fact that we have picked out this one horse for the purpose of this discussion about sets with only one element. We might have used numbers, or farm animals, or emotions, or... anything. Conversely, then, what is special about the mentioned entity [here Philip] is that the universe of consideration is, so to speak, the fact of there being just the one entity. If it happens to be a horse, for instance... we are not entertaining the thought of any other horses... and conversely the issue never arises that it might have been a number, or a patient with (particularly) cancer, or a class of technology, or what-have-you.

In other words... we have unconsciously attached the idea to this horse. This is a one-set horse.

A set with only one element in it is not a special case. If one was working with sets, towards some real-world end, one would not think twice about it (unless it was a startling result in real life). It appears to be a special case particularly within a discussion about sets with only one element... but that is merely an artefact of (the fact of there being) that discussion.
]

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Why is a set with one element distinct from the element itself?

A set with one element is not distinct from the element itself.

The notion of set is an abstraction, an idea in the mind of the observer. There is no difference between a set and its elements taken together. However, we make the distinction ourselves by either considering each element for itself or the set of all elements taken together.

This is also why the notion of empty set is nonsense. A set is the collection of elements. No element, no set.

Question from Nat:

What's the intersection of {1,2} and {3,4}? – Nat, May 2022

{1, 2} is a set of two numbers, viz., 1 and 2...

{3, 4} is also a set of two numbers, viz., 3 and 4.

So, what is the intersection of {1, 2} and {3, 4}?

The two sets have no element in common... so we can say that the two sets have no element in common. Or that they have no part in common; there is no intersection; there is no set which is the intersection; their intersection is nothing, or refers to nothing; there is no element which is member of the two sets; there is no solution to the intersection etc. etc.

But it is misleading to say that the intersection is empty. The intersection of two sets is neither a box nor a crossroad. It cannot be empty.

The question of how to talk about the intersection of two disjoint sets was never the reason for using the expression "empty set". Mathematicians themselves don't refer to the intersection of two disjoint planes as the "empty plane". The reason was to accommodate the consequences of mathematical logic into mathematics.

FOUR, NO, FIVE VOTES DOWN!

All by people who cannot argue their dogma.

It is really not difficult to understand first that the notion of empty set is nonsense: no element, no set. What is there not to understand? This is totally intuitive. All children understand that. Mathematicians themselves, before Bertrand Russell, understood that. Gottlob Frege himself explained a set couldn't be empty.

It is also clear that the notion of empty set came to be used to accommodate the use of mathematical logic (FOL), because it is FOL that necessitates the notion of empty set.

Here are some of the comments demonstrating the vacuity of the arguments of the people who voted down this answer:

The intuition behind Cantor’s concept of a set is to form ‚a collection of definite, distinguishable objects of perception or thought conceived as a whole‘. In general one identifies the elements by a condition they have to satisfy. E.g., all numbers x with square x**2=-1. Sometimes one does not know a priori whether there exists any object at all which satifies the condition. Hence the set defined by the condition can be empty. In the example above it depends on the decision which numbers are admissible whether the set, defined by the condition, is empty or not.

How does that justify that there are empty sets? Suppose the police put out the description of a murderer and then it turns out nobody fits the description. Will you say the police has the murderer? Of course not. Nobody would.

But there is a collection of elements with no set, so a set can’t be the same as its elements. So there is a difference. The set of all sets is not a set. Or at least it isn’t clear how you’d handle this.

The sentence "The set of all sets is not a set" is nonsense, although according to Bertrand Russell, it would be false. What you mean is that there is no set of all sets, or that sets don't make up a set. Whether or not there is a set of all sets is irrelevant. To have a set, you need to have elements. No element, no set. I didn't claim that when there is no set, then there is nothing.

Here is another comment on my answer, by a regular critique of my answers:

Frege thought logic and by extension, mathematics should basically be the language of metaphysics. As 0 and an empty set basically are negations and literally correspond to "not anything (contained)" they are meaningless in his theory. But logic, algebra, and set theory are incomplete and important operations cannot be defined without them, so your argument is a) moot and b) a mere opinionated rant. Why should everything that only exists in abstract automatically be nonsense? That is the question you should answer in your post before presenting questionable conclusions. – Philip Klöcking

This comment is a bit more substantial than the others, so I'll reply to every of the allegations in Philip Klöcking's comment...

Frege thought logic and by extension, mathematics should basically be the language of metaphysics. As 0 and an empty set basically are negations and literally correspond to "not anything (contained)" they are meaningless in his theory. – Philip Klöcking

I did mention Gottlob Frege in my answer. I say that Gottlob Frege himself explained a set couldn't be empty. However, contrary to what is suggested by Philip Klöcking's comment, Gottlob Frege's position on this was motivated by the intuitive notion of set (what he called "a class"). Here is the relevant passage:

A class, in the sense in which we have so far used the word, consists of objects; it is an aggregate, a collective unity, of them; if so, it must vanish when these objects vanish. If we burn down all the trees of a wood, we thereby burn down the wood. Thus there can be no empty class. — Gottlob Frege, A critical elucidation of some points in E. Schroeder’s Vorlesungen ueber die algebra delogik (1895)

We can see that his argument is based on our intuitive notion of set because he offers the analogy of a wood made up of trees, and that in the same sense as a set is made up of its elements. No tree, no wood; no element, no set.

So we don't need to convoque the entirety of Frege's work to decide what his position was on the notion of empty set.

Elsewhere, he also says that "a name must be a name for something". However, the name "empty set" is really a name for nothing. And this is why it is nonsense. This is the same egregious nonsense as the many question whether nothing exists, or whether the universe could have come out of nothing, as if, and here is the problem, nothing was something, just as the empty set is presented as something, namely, the set with no element.

But logic, algebra, and set theory are incomplete and important operations cannot be defined without them. – Philip Klöcking

This is fallacious. We all know what logic is and there is no reason to suspect that it is somehow incomplete. Philip Klöcking is talking about something else. The problem is that he uses the word "logic" to refer to something entirely different, namely, what mathematicians have decided to call "mathematical logic". However, mathematical logic is not logic. It is literally not logic. So the mathematical "fact" that FOL is "incomplete" is totally irrelevant to logic.

Second, the idea presented as fact by Philip Klöcking, viz., that "important operations cannot be defined without" the notion of empty set is simply false. The situation is very different. The situation is that mathematicians have been unable to find a way to apply mathematical logic to mathematics without introducing the nonsensical notion of empty set. And then this only demonstrates the failure of mathematical logic, not the necessity of the notion of empty set. Nobody has demonstrated that it is impossible to do mathematics without the notion of empty set.

Why should everything that only exists in abstract automatically be nonsense? That is the question you should answer in your post before presenting questionable conclusions. – Philip Klöcking

This is also fallacious. I already explained at the beginning of my answer what there was to explain. Here it is again:

This is also why the notion of empty set is nonsense. A set is the collection of elements. No element, no set. - Speakpigeon

And I also never claimed that "everything that only exists in abstract" is nonsense. I only discussed the specific concept of empty set and I fully explained why it is nonsense: a set is a collection of things, so, if there no things, there is no set. As Frege explained, no trees, no wood. There is nothing more to explain. This is self-evident.

The notion of the empty set is very interesting. First, it is clearly nonsense. It flies in the face of everything mathematicians themselves had done until then. The question is why make a nonsensical notion fundamental to set theory? The answer is that it became unavoidable once mathematicians decided to adopt so-called "mathematical logic" as the logic of mathematics. There was nothing necessary in that. This was entirely motivated by expediency. Mathematicians have been unable to produce of better model of human logic and they had to make do with whatever they had.

No wonder mathematicians and would-be mathematicians have routinely downvoted most of my answers on the subject of logic or have even removed my questions on logic. If someone has to explain themselves, it is not me.

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