The other side of Russell's paradox

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The other side of Russell's paradox

Postby Certainly real » Sat May 08, 2021 8:36 am

There can be no set of all sets that are members of themselves that is itself a member of itself (just as there can be no set of all sets that are not members of themselves that is itself not a member of itself).

Proof of the above:

When I write x (x, y, z), I mean to say set x contains elements x, y, and z. With this in mind, consider the following:

x (x, y, z): Here, if I say x is a set that is not a member of itself, we get a contradiction. (because x is in x)
x (x, y, z): Here, if I say x is a set that is a member of itself, we get no contradiction (because x is in x)
x (x, y, z): Here, if I say x is a set of three sets that are members of themselves, we get a contradiction (it amounts to x being a member of itself twice because x is in x. Either y and z are not members of themselves (making x the only member of itself), or x is a member of itself twice whilst y and z are members of themselves once (which is contradictory). It cannot be the case that x is a set of three sets that are members of themselves).

x (p, q, y): Here, x may be a set of three sets that are members of themselves precisely because it does not include itself. This shows that a set of all sets that are members of themselves, that is itself a member of itself, is impossible (the notion of any set containing more than one set as a member of itself is contradictory).

Russell's paradox arises from the belief that you cannot have a set of all sets that are not members of themselves that is itself not a member of itself, whilst you can have a set of all sets that are members of themselves that is itself a member of itself. The above shows that this belief is contradictory (semantically inconsistent). Thus, paradox resolved.
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Re: The other side of Russell's paradox

Postby obsrvr524 » Sun May 09, 2021 2:26 am

Can you give an example of a set that contains itself (as a whole) as a member?
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              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just same Satanism as always -
    • separate the bottom from the top,
    • the left from the right,
    • the light from the dark, and
    • blame each for the sins of the other
    • - until they beg you to take charge.
    • -- but "you" have been observed --
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Re: The other side of Russell's paradox

Postby Meno_ » Sun May 09, 2021 3:52 pm

Art for art's sake.
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Re: The other side of Russell's paradox

Postby Certainly real » Sun May 09, 2021 4:10 pm

obsrvr524 wrote:Can you give an example of a set that contains itself (as a whole) as a member?


What I mean by a member of itself, does not necessarily mean physically containing itself. For example:

The list of all lists, lists itself, therefore, it is a member of itself. The list of all animals does not list itself, therefore, it is not a member of itself.

The set of all sets, encompasses all sets, including itself, therefore, it is a member of itself.
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Re: The other side of Russell's paradox

Postby obsrvr524 » Sun May 09, 2021 5:09 pm

Certainly real wrote:
obsrvr524 wrote:Can you give an example of a set that contains itself (as a whole) as a member?


What I mean by a member of itself, does not necessarily mean physically containing itself. For example:

The list of all lists, lists itself, therefore, it is a member of itself. The list of all animals does not list itself, therefore, it is not a member of itself.

The set of all sets, encompasses all sets, including itself, therefore, it is a member of itself.

But do you have a real example.

I ask because I believe such a list or set type is pure irrational fantasy - and that is why there seems to be so much confusion about it. It is a senseless entity that cannot exist at all.

A paradox is a logical consequence that SEEMS contradictory. In this case I think Russel's paradox seems to lead to a logical contradiction because it is counter-intuitive to recognize that a "set of all sets" is an impossible oxymoron so the normal mind tries to resolve a strawman logic problem.

It is just a trick of words.
Member of The Coalition of Truth - member #1

              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just same Satanism as always -
    • separate the bottom from the top,
    • the left from the right,
    • the light from the dark, and
    • blame each for the sins of the other
    • - until they beg you to take charge.
    • -- but "you" have been observed --
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Re: The other side of Russell's paradox

Postby Certainly real » Sun May 09, 2021 8:10 pm

obsrvr524 wrote:But do you have a real example.

I ask because I believe such a list or set type is pure irrational fantasy - and that is why there seems to be so much confusion about it. It is a senseless entity that cannot exist at all.

To me, irrationality is where you have something that is semantically inconsistent or contradictory (like a round square). Saying triangles are irrational, is irrational (precisely because triangles is not an absurd concept like round-square). Similarly rejecting a list of all lists, or a set of all sets, is irrational in my opinion because it implies that you think such a thing to be semantically inconsistent or contradictory (when it is not).

But do you have a real example.


Call absolutely any thing (number, shape, tree, human, dream, colour) an ‘existent’. Call the set of all existents, ‘Existence’. Note that I am not referring to how real something is/exists, just that it is an existent (a member of Existence). Numbers are numbers (which is the same as saying numbers exist as numbers in Existence). The alternative is to say numbers don’t exist in Existence, or that there is/exists no such thing or existent as numbers, or that numbers are in non-Existence (like round squares and other absurdities).

All existents (including Existence) are a member of Existence (because they are all existents). Only Existence is a member of itself as an existent. Since no other existent is a member of itself as an existent, the set of all existents that are not members of themselves, is Existence.

Existence is a meaning, so it is a member of the set of all meanings. But then again, Existence IS the set of all meanings because there is no other thing, existent, set, or meaning that existentially contains all meanings, sets, or existents. The set of all ducks is not some existing animal or shape. The set of all ducks is Existence Itself (which is an existing meaning/set/existent/truth).
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Re: The other side of Russell's paradox

Postby obsrvr524 » Sun May 09, 2021 9:27 pm

Certainly real wrote:you think such a thing to be semantically inconsistent or contradictory (when it is not).

Yes I think it is contradictory (when it is).

Certainly real wrote:
But do you have a real example.

Call absolutely any thing (number, shape, tree, human, dream, colour) an ‘existent’. Call the set of all existents, ‘Existence’. Note that I am not referring to how real something is/exists, just that it is an existent (a member of Existence). Numbers are numbers (which is the same as saying numbers exist as numbers in Existence). The alternative is to say numbers don’t exist in Existence, or that there is/exists no such thing or existent as numbers, or that numbers are in non-Existence (like round squares and other absurdities).

None of those are examples of a set that is inside itself. The set of trees is not a tree. The set of numbers is not a number. It is only when you speak of "the set of all sets" that you get confused.

Certainly real wrote:All existents (including Existence) are a member of Existence (because they are all existents). Only Existence is a member of itself as an existent. Since no other existent is a member of itself as an existent, the set of all existents that are not members of themselves, is Existence.

Existence is a set - the set of all that exists.

Certainly real wrote:Existence is a meaning, so it is a member of the set of all meanings.

No I don't agree. "Existence" as a word has a meaning. The word is not a meaning itself.

And Existence as the set of all that exists might be said to have a meaning - but is not a meaning itself. It is just a set.

Certainly real wrote: The set of all ducks is Existence Itself (which is an existing meaning/set/existent/truth).

Not quite - but ---

  • Do you believe it is irrational to say that you have a square that is bigger than itself?
  • How is that any different than saying that you have a set bigger than itself?
Member of The Coalition of Truth - member #1

              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just same Satanism as always -
    • separate the bottom from the top,
    • the left from the right,
    • the light from the dark, and
    • blame each for the sins of the other
    • - until they beg you to take charge.
    • -- but "you" have been observed --
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Re: The other side of Russell's paradox

Postby Magnus Anderson » Mon May 10, 2021 12:58 am

obsrvr524 wrote:Can you give an example of a set that contains itself (as a whole) as a member?


An example would be \(A = \{1, 2, 3, A\}\).
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Re: The other side of Russell's paradox

Postby obsrvr524 » Mon May 10, 2021 1:41 am

Magnus Anderson wrote:
obsrvr524 wrote:Can you give an example of a set that contains itself (as a whole) as a member?


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.
Member of The Coalition of Truth - member #1

              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just same Satanism as always -
    • separate the bottom from the top,
    • the left from the right,
    • the light from the dark, and
    • blame each for the sins of the other
    • - until they beg you to take charge.
    • -- but "you" have been observed --
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Re: The other side of Russell's paradox

Postby Certainly real » Mon May 10, 2021 9:07 am

obsrvr524 wrote:None of those are examples of a set that is inside itself.


To my understanding, there are various ways for something to qualify as being a member of itself. Consider the following:

1) A thing that encompasses itself as well as other things (like a list of all lists)
2) 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.

Existence is a set - the set of all that exists.


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?

No I don't agree. "Existence" as a word has a meaning.


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.

And Existence as the set of all that exists might be said to have a meaning - but is not a meaning itself. It is just a set.


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?

Do you believe it is irrational to say that you have a square that is bigger than itself?


100% yes. This is crystal clear.

How is that any different than saying that you have a set bigger than itself?


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.
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Re: The other side of Russell's paradox

Postby Magnus Anderson » Mon May 10, 2021 9:15 am

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.
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Re: The other side of Russell's paradox

Postby obsrvr524 » Mon May 10, 2021 12:56 pm

Magnus Anderson wrote: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.

Certainly real wrote:1) A thing that encompasses itself as well as other things (like a list of all lists)
2) 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.

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.

Certainly real wrote:1) Is it not the case that there are many sets in Existence?
2) Is it not the case that all existing sets are in Existence?

1) Yes
2) 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.

Certainly real wrote: is it not the case that Existence is the set of all sets?

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.

Certainly real wrote: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.

All of that is just improper English and word usage.

Certainly real wrote: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.

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.
Member of The Coalition of Truth - member #1

              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just same Satanism as always -
    • separate the bottom from the top,
    • the left from the right,
    • the light from the dark, and
    • blame each for the sins of the other
    • - until they beg you to take charge.
    • -- but "you" have been observed --
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Re: The other side of Russell's paradox

Postby Magnus Anderson » Mon May 10, 2021 2:31 pm

obsrvrs524 wrote: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.


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 (:
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Re: The other side of Russell's paradox

Postby Certainly real » Mon May 10, 2021 5:32 pm

obsrvr524 wrote: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.


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).

obsrvr524 wrote: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


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').

Then how can you say that a set can contain itself along with other items?
That would make the set larger than the set.


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).
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Re: The other side of Russell's paradox

Postby Magnus Anderson » Mon May 10, 2021 9:08 pm

Certainly real wrote:Where is the contradiction in the above?


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 \(n + 3\).
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Re: The other side of Russell's paradox

Postby Certainly real » Mon May 10, 2021 11:14 pm

Magnus Anderson wrote:
Certainly real wrote:Where is the contradiction in the above?


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.
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Re: The other side of Russell's paradox

Postby obsrvr524 » Mon May 10, 2021 11:48 pm

Certainly real wrote:That would mean that I am Existence too.

No because you are NOT the entire collection. Existence is the entire collection as a whole.

Certainly real wrote: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).

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.
    - 16.
    - 19.
    .
    .
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.
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              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just same Satanism as always -
    • separate the bottom from the top,
    • the left from the right,
    • the light from the dark, and
    • blame each for the sins of the other
    • - until they beg you to take charge.
    • -- but "you" have been observed --
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Re: The other side of Russell's paradox

Postby Magnus Anderson » Tue May 11, 2021 1:34 am

obsrvr524 wrote: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.
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Re: The other side of Russell's paradox

Postby Magnus Anderson » Tue May 11, 2021 1:47 am

Certainly real wrote: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),


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.
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Re: The other side of Russell's paradox

Postby obsrvr524 » Tue May 11, 2021 5:12 am

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.

Magnus Anderson wrote: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.

I think he keeps going back and forth - confusing a set or list with a reference to it -- sometimes.
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              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just same Satanism as always -
    • separate the bottom from the top,
    • the left from the right,
    • the light from the dark, and
    • blame each for the sins of the other
    • - until they beg you to take charge.
    • -- but "you" have been observed --
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Re: The other side of Russell's paradox

Postby Magnus Anderson » Tue May 11, 2021 1:13 pm

obsrvr524 wrote: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.


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.
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Re: The other side of Russell's paradox

Postby Certainly real » Tue May 11, 2021 2:11 pm

obsrvr524 wrote:No because you are NOT the entire collection. Existence is the entire collection as a whole.


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.

The list of all lists (in your room) would - even in concept - be longer than itself - for the reason Magnus just pointed out.


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.
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Re: The other side of Russell's paradox

Postby Certainly real » Tue May 11, 2021 2:22 pm

Magnus Anderson wrote:I'm not sure how what you said is related to Russell's Paradox.


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).

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.


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.
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Re: The other side of Russell's paradox

Postby obsrvr524 » Tue May 11, 2021 4:38 pm

Magnus Anderson wrote:
obsrvr524 wrote: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.


There is a difference between a set and its contents (or members.) They are not one and the same.

I disagree but it is just a matter of semantic definition.

Magnus Anderson wrote: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.

In my language every item is a set of 1 item.
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              You have been observed.
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    • the left from the right,
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Re: The other side of Russell's paradox

Postby obsrvr524 » Tue May 11, 2021 4:47 pm

Certainly real wrote:
obsrvr524 wrote:No because you are NOT the entire collection. Existence is the entire collection as a whole.


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.

I wouldn't say that Existence is a member of itself (that confuses the language). Existence IS itself.

Certainly real wrote:
The list of all lists (in your room) would - even in concept - be longer than itself - for the reason Magnus just pointed out.


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.

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.
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              You have been observed.
    Though often tempted to encourage a dog to distinguish color I refuse to argue with him about it
    It's just same Satanism as always -
    • separate the bottom from the top,
    • the left from the right,
    • the light from the dark, and
    • blame each for the sins of the other
    • - until they beg you to take charge.
    • -- but "you" have been observed --
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