An example would be (A = {1, 2, 3, A}).
That is an example of the expression. I was asking for an actual existing set.
My position is that what you specified ( (A = {1, 2, 3, A}) ) is an impossible set - it cannot actually exist - it is purely imaginary - a set that is larger than itself.
To my understanding, there are various ways for something to qualify as being a member of itself. Consider the following:
- A thing that encompasses itself as well as other things (like a list of all lists)
- A thing that physically contains itself as well as other things (like a physical folder of all folders)
I think you are focused exclusively on 2. 1 is not controversial at all, whereas I can see how 2 is. My focus here is on 1.
Right. With this being the case:
Is it not the case that there are many sets in Existence?
Is it not the case that all existing sets are in Existence?
Therefore, is it not the case that Existence is the set of all sets?
You recognise that Existence is a set. I will try to show you that Existence is also a meaning.
The semantic of ‘triangle’ is the same for all people, yet the English call it “triangle”, whilst the Persians call it “mosallas”. We cannot deny that ‘triangle’ is a semantic. Similarly, we cannot deny that ‘Existence’ is a semantic. Thus, ‘Existence’ is a member of the set of all semantics.
Existence is a meaningful set. This means that it is an existent, a meaning, and a set.
Would you deny that ‘triangle’ is a shape? Would you deny that ‘triangle’ is a meaning? Would you deny that ‘triangles’ is a set?
100% yes. This is crystal clear.
A set that is bigger than itself is also clearly absurd. It does not matter if x is a set, or a square, or Existence. It cannot be bigger/smaller than itself at the same time.
obsrvr524,
I don’t know what you mean by “an actual existing set”. If what you mean is “physically existing set”, I am not sure why the existence of such a set is necessary given that this is a thread about mathematics (which is concerned with concepts.)
I do intuit the same as you do, namely, that a set that contains itself is an oxymorn.
I just wanted to either get something more concrete, avoiding vague abstractions, or reveal that such a set might not exist at all.
And I think maths is about quantities.
I don’t see any significant difference between those 2. A “list of all lists” is just as absurd as a folder of all folders.
- Yes
- No
All existing sets are not IN Existence - they ARE existence. They comprise what we call existence. Existence is not something that contains anything. Existence IS everything together. It is not an entity to itself but a collection.
Only in the exclusive sense (not containing itself along with other sets). You have been using Existence as an inclusive set (containing itself along with other sets).
Because Existence is everything, it cannot be contained within anything.
All of that is just improper English and word usage.
Then how can you say that a set can contain itself along with other items?
That would make the set larger than the set.
In that example - (A = {A, x, y, z} ), both A’s must be equal in every respect yet the first is set to be larger than the second by an additional “x, y, and z”. And when you say that the second A has an x, y, and z within it - that just means that the first A has 2 sets of x, y, and z.
Right, it s about quantities. But one need not show an instance of (10^{1000000}) things in physical reality in order prove that such a number conceptually exists. (Indeed, such is not even necessary to physically exist. Unicorns don’t exist physically but they do conceptually.)
On the other hand, nothing wrong with asking for clarification (:
Only Existence is exclusively a member of itself as an existent because it is not a member of anything other than itself (as an existent). All other existents are members of other than themselves (they are members of Existence, whilst they are not Existence Itself).
That would mean that I am Existence too. Whereas to my understanding, I belong to Existence. I am not Existence. If I cease to exist, Existence does not cease to exist.
I think it’s contradictory to say all existing sets are Existence. Only one existing set is Existence in my opinion. That is the set of all existents (which I have called ‘Existence’).
So you have said that a list of all lists is absurd. That would mean that if I have four lists in my room, and I wanted to make a fifth list titled “a list of all lists in my room”, this list cannot contain itself. Whilst I agree that this list cannot physically contain itself in addition to itself because it is physically itself once as opposed to twice, I do think that this list can list itself. I also think that this list is such that it lists items that are not members of themselves (because they are members of it), and it lists one item that is a member of itself (the list lists itself).
Where is the contradiction in the above? Rejecting the above is contradictory (and you have rejected a list that lists all lists).
I’ll try to present an argument in favor of the idea that a set that contains itself is a contradiction in terms.
However, in order to that, I’ll have to introduce a new term. I have to do this because I have trouble finding an existing term in mathematics that captures the concept that I want to express.
The notion that I am going to introduce will be represented by the expression “the volume of a set”.
obsrvr524 is using the word “size” to capture the same concept but that leads to a minor problem since the word “size” is already defined with respect to sets and it means something else – it refers to the number of elements. (A = {1, 2, 3, A}) has exactly (4) elements. No more and no less. The fact that (A) contains itself does not mean it has more than that.
So here we go. I’m going to kick it with a recursive definition.
Let the volume of a set be the sum of each member’s volume.
Let the volume of anything that is not a set be equal to (1).
For example, the volume of (B = {1, 2, 3}) is (3) because the set contains three elements, each one of which is not a set, which means, the volume of each element is exactly (1). Thus, the volume of (B) is (3 \times 1) which is (3).
The volume of (C = {1, 2, 3, {1, 2, 3}}), on the other hand, is equal to (6) even though its size (= cardinality) is equal to (4).
Let (n) be a number that represents the volume of (A = {1, 2, 3, A}).
What’s (n) equal to?
Let’s see. (A) contains (3) non-set elements and (1) set-element. This means the volume of (A) is (3 \times 1) plus the volume of the set-element it contains. We can simplify this to (3) plus the volume of the set-element. The set element it contains is (A) which means its volume is (n). Thus, the volume of (A) is (3 + n). This contradicts our earlier claim that the volume of (A) is (n). The volume of a set cannot be both (n) and (n + 3).
I think I get your point and I agree with it, but I don’t think it applies to what I’m saying.
Because we cannot ignore that when a list lists other lists, those items are not members of themselves (because they are members of this list whilst they are in reference to other lists), and that when a list lists itself, that item is a member of itself (because it is a member of this list whilst it is in reference to this list itself), I think it necessary for us to have a distinction between elements in a list that are not members of themselves, and an element in a list that is a member of itself (provided that a list has elements of such a nature). No list can contain more than one element as a member of itself (the same is true of sets).
So given the above, I don’t think your post highlights a contradiction in what I said.
The list of all lists is a member of itself because it is a list. Similarly, the set of all sets is a member of itself because it is a set. If you ascribe the volume x to the set of all sets, and then say the set of all sets is the volume x, plus it contains the volume x in addition to it being the volume x, then yes, I think that is contradictory. But I am not saying that the set of all sets = volume x plus volume x when I say that it is a member of itself.
No because you are NOT the entire collection. Existence is the entire collection as a whole.
The list of all lists (in your room) would - even in concept - be longer than itself - for the reason Magnus just pointed out.
- If A had a “volume” of 4 then A would have to have a volume of 7.
- And if A had a volume of 7 then A would have to have a volume 10.
- And if A had a volume of 10 then A would have to have a volume of 13.
.
.
If it is a set then it has a volume/cardinality/size.
But any statement of the volume (or “size”) of A would be false.
The only kind of set that could fully contain itself would be a null set = zero volume.
I don’t think there is such a thing as a set that contains only itself. That too is an oxymoron. It’s just that my argument does not handle that case. Change “Let the volume of a set be the sum of each member’s volume” to “Let the volume of a set be the sum of each member’s volume plus one” and the problem should be resolved.
If an empty set contains itself then it is not an empty set.
It seems to me (though I am not exactly sure) that what you’re saying is that, when you say that a set contains another set that you actually mean that it contains a reference to another set and not the set itself. If that’s the case, then I agree: what I said is unrelated to what you said. But then, I’m not sure how what you said is related to Russell’s Paradox.
A set is whatever is in the set. Whatever is in the set is the set.
An empty set is the emptiness - not a container of it.
So every set has its entire contents within it which is having the set within it. It just can’t be a separate copy of the set.
That is the difference between a set containing itself inclusively vs exclusively.
I think he keeps going back and forth - confusing a set or list with a reference to it – sometimes.
There is a difference between a set and its contents (or members.) They are not one and the same. In mathematics, two sets are identical if and only if they have exactly the same elements. (\emptyset) (which stands for an empty set) and ({\emptyset}) are not identical sets. The former has no elements, the latter has one element.
So Existence is the collection that encompasses all collections then. Existence is the collection of all collections then. All other collections are members of Existence, whereas Existence is not a member of anything other than itself.
That’s because you’re focused on a different thing to me. I said if a list lists itself, it is a member of itself. When I have only have four lists in my room and I’m making a fifth titled “the list of all lists in my room”, that list will be finite. It will consist of five items. Four of which are not members of themselves, one of which is a member of itself.
Russell’s paradox arises from the belief that you cannot have a set of all sets that are not members of themselves. To which I thought:
The set of all sets encompasses all sets that are not members of themselves, as well as itself. To which a math professor told me:
By a set of all sets that are not members of themselves, we mean a set that encompasses all sets that are not members of themselves, and no other set. To which I thought:
In that case, you cannot have a set of all sets that are members of themselves because it would amount to one set being a member of itself twice (which is contradictory. See the op for proof of this).
In truth, there is one absolute universal set (which I call Existence or Infinity or God). Everything is a member of this set/collection/existent/thing. It is a member of itself (which is the same as saying it is itself. This is not the same as saying it is itself + another of itself combined). Think of it this way: Everything is contingent on God. God is not contingent on anything other than Himself. God is the set of all contingents. God is self-contingent, whilst all other things are not self-contingent.
I disagree but it is just a matter of semantic definition.
In my language every item is a set of 1 item.
I wouldn’t say that Existence is a member of itself (that confuses the language). Existence IS itself.
What you say is only true IF your list is a list of references (or titles) to the lists including a reference to itself. Your 5th list cannot contain itself except as only a title or reference. That has been this whole argument from the beginning.
Yes, Existence is itself, but so is the collection of books I own. The collection of books I own is a member of Existence. Existence is a member of Existence. You say saying Existence is a member of itself confuses the language, whereas I think it is a necessary part of language without which the language is not as logically precise as it could be.
That is what I was saying.
I don’t think I suggested otherwise. Maybe you were under impression that I was based on our previous discussion of a folder of all folders (wherein which I did say the folder contains itself, as opposed to just referencing itself).
In any case, as far as I can see, Russell’s paradox is solved. This absurdity of rejecting an absolute universal set needs to end.
So do you agree that the set (A = {1, 2, 3, A }) cannot exist?
I think that is just a part of the cancel culture politics. They don’t care that it doesn’t make sense.