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.

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.

Under Zermelo-Fraenkel set theory, there can be no infinitely deep or cyclic hierarchies of sets, nor any elements which are not sets, so if you recursively replace each set with its elements, you eventually get to the empty set. I'm interested to hear which of those axioms you replace in your conception of set theory.

Try explaining that to a mathematician. And not be seen as not knowing what you are talking about afterwards. The notion of the one and only empty set is at the core of set theory.

@JoWehler And? 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.

@Speakpigeon Set theory is a creation of the human mind. To ban the empty set from set theory is like banning zero from natural numbers: You can no longer subtract 5-5 etc.

@JKusin 1. "The set of all sets is not a set." This is typical nonsense According to Bertrand Russell, your sentence is false! What you mean is that there is no set of all sets, or that sets don't make up a set. - 2. Whether or not there is a set of all set 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.

@Kevin 1 "Under Zermelo-Fraenkel set theory, there can be no infinitely deep or cyclic hierarchies of sets" We don't need this axiom. It is a direct consequence of the assumptions on which FOL is based and these assumptions are false. 2. "there can be no (...) elements which are not sets" Elements are not sets anyway. Sets are collections. Whether a particular element can form a set depends on the element, not on some unjustified axiom.

@cmaster-reinstatemonica 1. "The notion of the one and only empty set is at the core of set theory." So set theory is wrong. 2. "*Try explaining that to a mathematician. And not be seen as not knowing what you are talking about afterwards. *" So, mathematicians know set theory. So, they know a theory which is wrong. So, they don't know that it is wrong. So I know more than they do.

@JoWehler 1. "Set theory is a creation of the human mind" Don't be ridiculous, there is no such a thing as "the human mind". Set theory was invented by some human minds. - 2. "To ban the empty set from set theory is like banning zero from natural numbers: You can no longer subtract 5-5". Sorry, analogies always make for bad logic. A set is a collection of elements. No elements, no set. Is that really so difficult to understand?

@Speakpigeon Well, the empty set is the zero-element of set theory, you can hardly avoid calling that "being at its core". You can just as well say that zero is not a number, and I'd simply write you off as not knowing what you are talking about. Of course, mathematicians know set theory, and of course they talk about the empty set a lot. Just like you talk about zero a lot when you learn about arithmetic. It's arguably the most important number of them all.

@cmaster-reinstatemonica 1. "the empty set is the zero-element of set theory" Reasoning by analogy? Bad logic. 2. "at its core "I know the notion of empty set is fundamental to set theory, which is why set theory is wrong. 3. "they talk about the empty set a lot." Truth by repetition?! 4. You don't seem to have any reasonable argument.

@cmaster-reinstatemonica "I don't think that it's any use to continue arguing with someone who thinks "set theory is wrong" So you only argue with people who agree with you? No, I don't think so. So, it must be that you have no argument.

Frege thought logic and by extension, mathematics should basically be the language of metaphysics. As 0 and an empty set 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.

What's the intersection of {1,2} and {3,4}?

@Nat Hello, Nat. The intersection of {1, 2} and {3, 4}? Is this all you have to motivate the notion of empty set? Let me see... Well, {1, 2} I guess is a set of two numbers, namely 1 and 2, while {3, 4} is a set of two numbers, namely 3 and 4. So we can say that the two sets have no element in common; they have no part in common; there is no 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.

@Speakpigeon: It'd seem that you'd agree that, if there were a solution, it wouldn't contain any members -- so then I guess it's an issue of how to describe that: do we say that it's a set without members, or do we say that there's no solution? I kinda like the empty-set interpretation because we can continue doing logic.

I mean, I'm big into constructivism: if I'm figuring something out, I'll tend to pick concepts conducive to the overall process. For example, if I'm doing physics with a numeric-variable that can't be negative (say, mass in classical-physics), then.. can that variable be negative, in a mathematical sense? Meh, who cares, right? -- this is, it'd seem unnecessary to make assumptions either way -- though I could later add in such logic if there should be a call for it.

Then, I'd tend to feel the same way about an empty-set: not really too concerned if it doesn't matter. But when it does, hey, sure, let's do it, unless there's cause to avoid it.

@Nat "It'd seem that you'd agree that, if there were a solution, it wouldn't contain any members" No, I certainly do not agree with that, nor did what I said suggests I do. I said "*there is no solution to the intersection. *" So, if there were a solution, it would have elements. This is what a solution is. The solution is the set of elements that are members of the intersection.

@Nat "I kinda like the empty-set interpretation because we can continue doing logic." No, you cannot, and this is the point. The notion of empty set is illogical. . And mathematical logic is definitely not logic. It is totally wrong.

@Speakpigeon: What do you mean by the notion of an empty-set being "illogical"? For example, do you believe that, if someone were to make an algorithm that made predictions about real-life events based on the presumption of the empty-set being a valid logic, then that algorithm would malfunction and produce invalid results?

@Speakpigeon Since [your] conscious mind is the only thing [you] know and that the rest is beliefs, then you cannot state it as fact. It's I think, therefore I am, not I think, therefore it is, so maybe stop arguing with other peoples' beliefs so assuredly and pig-headedly. Also, can you elaborate on your assertions that A set with one element is not distinct from the element itself and Elements are not sets anyway. Sets are collections.?

@Speakpigeon can you also explain how 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. is valid when you've stated Sorry, analogies always make for bad logic and Reasoning by analogy? Bad logic.? Perhaps temper your confidence, since you complained you were being downvoted All by people who cannot argue their dogma when your own dogmatic argument is riddled with contradictions.