The overlooked part of Russell's paradox

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Steve3007
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Re: The overlooked part of Russell's paradox

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RJG wrote:Yes, a list can contain a reference-to-itself, ...but NEVER, ever, ever, ever itself.
Yes, and you understand that, by their nature, lists and sets contain references not referents? My shopping list doesn't have actual apples glued to it. Yes?
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Re: The overlooked part of Russell's paradox

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Terrapin Station wrote:unrestricted comprehension principle
Interesting.
https://en.wikipedia.org/wiki/Axiom_sch ... cification
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RJG
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Re: The overlooked part of Russell's paradox

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**********

Steve it seems you are avoiding my questions. Do you agree with the below or not?

1. So then you agree that the x's represented in the set x={x, 1, 2), are two different x's? (...i.e. one x is a list (object) and the other x is a reference to the list; two different x's)? ...YES/NO?

2. And so then you further agree that it is logically impossible for listX to contain listX? (...i.e. listX can only contain a reference-to-listX, but never listX itself). ...YES/NO?

...agreed?
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Re: The overlooked part of Russell's paradox

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RJG wrote:1. So then you agree that the x's represented in the set x={x, 1, 2), are two different x's? (i.e. one x is a list (object) and the other x is a reference to the list; two different x's!)? ...YES/NO?
No. Disagree. In both cases 'x' is a reference to a list. '{x, 1, 2}' is the list, containing further references, as lists do.
2. And so then you further agree that it is logically impossible for listX to contain listX? (i.e. it can only contain a reference-to-listX, but never itself). ...YES/NO?
It wouldn't then be a list. It would be a physical object. Obviously, as I've said many, many, many times before, a real physical box can't contain itself. But we're not talking about boxes are we? As I said, it is in the nature of abstract constructs like lists and sets that they refer to things. As I said, I could glue all the items on my shopping list to the list but then it wouldn't be my shopping list. It would be my shopping bag. So if your question amounts to: "Is my shopping list actually my shopping bag?" then obviously the answer is no.
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RJG
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Re: The overlooked part of Russell's paradox

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...you're killing me smalls. ...have a good day.
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Steve3007
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Re: The overlooked part of Russell's paradox

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Can we take that to mean we now both agree that lists and sets reference things and that sets aren't boxes? I have to ask because you've been round this loop a few times before with both me, TS and others and you just keep coming back and repeating the same irrelevant stuff about boxes.
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Re: The overlooked part of Russell's paradox

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RJG wrote: June 9th, 2021, 8:56 am **********

Steve it seems you are avoiding my questions. Do you agree with the below or not?

1. So then you agree that the x's represented in the set x={x, 1, 2), are two different x's? (...i.e. one x is a list (object) and the other x is a reference to the list; two different x's)? ...YES/NO?

2. And so then you further agree that it is logically impossible for listX to contain listX? (...i.e. listX can only contain a reference-to-listX, but never listX itself). ...YES/NO?

...agreed?
Again, sets are NOT objects in that sense. They're abstract ideas. Re the list analogy, x isn't the list itself as an object. It's an abstract idea--a way of referencing the list itself.
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Re: The overlooked part of Russell's paradox

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philosopher19 wrote: June 8th, 2021, 10:19 pm
Thomyum2 wrote: June 8th, 2021, 11:43 am Didn't Kurt Gödel resolve this dispute about 90 years ago?
Not to my knowledge.
Do you mean that you're not aware of his work, or that you are aware of it but think it did not resolve the dispute?

Without considering Gödel's later insights into these issues, I think that any discussion of Russell's paradox kind of turns into a fulfillment of the adage 'those who do not learn history are doomed to repeat it', since this topic was so thoroughly debated in the first 30 years of the 20th Century.
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Re: The overlooked part of Russell's paradox

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RJG wrote:1. So then you agree that the x's represented in the set x={x, 1, 2), are two different x's? (...i.e. one x is [represents] a list (object) and the other x is [represents] a reference to the list; two different x's)? ...YES/NO?

2. And so then you further agree that it is logically impossible for listX to contain listX? (...i.e. listX can only contain a reference-to-listX, but never listX itself). ...YES/NO?

...agreed?
Terrapin Station wrote:Again, sets are NOT objects in that sense. They're abstract ideas. Re the list analogy, x isn't the list itself as an object. It's an abstract idea--a way of referencing the list itself.
Yes, but I'm not really talking about "objects" per se, I'm talking about "math/logic" here. The questions above still hold if you wish to answer them. In other words, in set x={x, 1, 2}, do the x's (x and x) represent the SAME "abstract idea", or DIFFERENT "abstract ideas"?

If one is a referent to the other, then these two x's are DIFFERENT "abstract ideas".

..agreed?


**************
RJG wrote:If we want to make sense then we first have to know what we are talking about. So my question is - What is X?

Does X=X? or
Does X={X, 1, 2}

Which of these two different (unique) meanings/solutions are we talking about? In other words, which X are we talking about? We can't have two different meanings/solutions to the same variable and still make sense.

X={X, 1, 2} logically contradicts X=X. Therefore (and assuming we hold X=X as true, then), X={X, 1, 2} is as logically impossible as X<X.
Terrapin Station wrote:??? If X = {X, 1, 2} than X = X is (X = {X, 1, 2}) = (X = {X, 1, 2})
You are missing my point altogether. Let me try again, -- what does x equal in the set x={x, 1, 2}? In other words, there are 2 x's in this statement. So what is x? (...is "x" the red x or the blue x?)

Does x = {x, 1, 2}? or
Does x = x (itself)?

x cannot be both. Since "x" and "{x, 1, 2}" are NOT the same, they CANNOT logically, nor mathematically, equal each other, ...hence the CONTRADICTION, and LOGICAL IMPOSSIBILITY of set x={x, 1, 2}.
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Re: The overlooked part of Russell's paradox

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RJG wrote: June 9th, 2021, 11:41 am Yes, but I'm not really talking about "objects" per se, I'm talking about "math/logic" here. The questions above still hold if you wish to answer them. In other words, in set x={x, 1, 2}, do the x's (x and x) represent the SAME "abstract idea", or DIFFERENT "abstract ideas"?
X is the set {x, 1, 2}

So that works out to {{x, 1, 2}, 1, 2} and {{{x, 1, 2}, 1, 2}, 1, 2} and so on--in other words, for every occurrence of x we can substitute {x, 1, 2}, because that's what x represents, and that is only an issue insofar as anyone thinks about that.
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Re: The overlooked part of Russell's paradox

Post by Terrapin Station »

RJG wrote: June 9th, 2021, 11:41 am x cannot be both.

There's no "both." X is the set {x, 1, 2}.

So when "x" occurs in the set, {x, 1, 2}, "x" is again the set {x, 1, 2}
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Re: The overlooked part of Russell's paradox

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RJG wrote:In other words, in set x={x, 1, 2}, do the x's (x and x) represent the SAME "abstract idea", or DIFFERENT "abstract ideas"?
Terrapin Station wrote:So when "x" occurs in the set, {x, 1, 2}, "x" is again the set {x, 1, 2}
So I take it that you believe "both x's are the SAME, and one x is NOT different nor does NOT refer to the other x"?

Terrapin Station wrote:So that works out to {{x, 1, 2}, 1, 2} and {{{x, 1, 2}, 1, 2}, 1, 2} and so on...
So, from your belief, the set "x" is a never-ending infinite regress? ...that thusly contains an infinite number of members?
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Re: The overlooked part of Russell's paradox

Post by philosopher19 »

Steve3007 wrote: June 9th, 2021, 4:49 am
philosopher19 wrote:
Steve3007 wrote:You're wrong to state above that "p = {x, y, z}" represents x, y and z being members of themselves. It doesn't. The statement "x is a member of itself", using similar notation, could be represented as "x = {x, ...}". Likewise for y and z.
So assume I have three sets that are not members of themselves: x, y, and z. I then form a set of these three sets. I write this in the following manner: p = {x y z}. No problems here. If I write x = {x y z} then problems occur because it implies x is a member of itself.
Why is that a problem? You've simply changed your mind and decided to define x as being a member of itself. In the above, you've effectively written: "x is not a member of itself. No, actually, let's make x a member of itself after all."
So assume you have three sets that are not members of themselves: x, y, and z. You are then told to form a set of these three sets. How would you write this in a manner that is consistent with the above? And by this I mean the letters outside { } represents the set, and the letters inside the { } represent the elements of that set.
p = {x, y, z}
philosopher19 wrote:
Steve3007 wrote:You can't have a set of all sets that are not members of themselves because then it could neither be a member of itself nor not a member of itself without contradiction. That's the point of the paradox.
Take V to be the set of all sets. V encompasses all sets that are not members of themselves and it is a member of itself because it is a set. Where is the contradiction in what I've just written?
As I said, the contradiction in the concept of a set of all sets, including sets that are not members of themselves, is that it cannot with logical consistency either be a member of itself nor not be a member of itself, even though those are the only two logically possible options. As I said, that's the whole point of the paradox.

But remember, this is just a logical game. Sets aren't real. They're abstract concepts that we create either for fun or because they happen to have some use in describing some aspects of the world. Or both. (Frequently they start as the former and end up, perhaps unexpectedly, as the latter). If we discover a paradox like this in set theory then all we have to do is note that the self-contradictory concept "the set of all sets that are not members of themselves" is not much use for anything except philosophical discussions.
Whilst it is clearly contradictory to have a set of all sets that are not members of themselves that is itself not a member of itself,...
The point is that the term "the set of all sets..." implies self-membership. It's self-referential. It doesn't say "the set of all sets except this one..." or similar.
...it is necessarily the case that the set of all sets encompasses all sets that are not members of themselves as well as itself (because it too is a set).
Yes, that's the whole point of the paradox! That's why it's called a paradox.
Rejecting the set of all sets is blatantly contradictory. Yet, from what I've heard, this is what's ongoing in mainstream maths, philosophy, and logic.
I've no idea what you mean by "rejecting the set of all sets". As I said, this is a game of logic. If anybody wants to consider the concept of "the set of all sets" and its logical implications they're free to do so. Doing so does not, in itself, say anything about the nature of the real world. Applicability of abstract concepts as descriptions of aspects of the real world is handy, but it's not necessary.
I think the only thing worth adding to what I've already said is that if you can have more than one thing, you can have a set of all of that thing. If you can have more than one set, you can have a set of all sets. Rejection of this is clearly contradictory, and I've discussed the universal set as well as Russell's misunderstanding in the link provided in the OP. I don't think any further discussion between us will bear any fruit. I think we'll have to agree to disagree on this topic as well. If it is truth and goodness you are sincerely in pursuit of, then I wish you the best of luck.
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Re: The overlooked part of Russell's paradox

Post by philosopher19 »

Terrapin Station wrote: June 9th, 2021, 7:36 am
philosopher19 wrote: June 8th, 2021, 9:59 pm
Terrapin Station wrote: June 8th, 2021, 7:23 am Aside from Steve3007's comment above, I didn't catch what the thread title promised: what's the "overlooked" part of Russell's paradox?
Everyone recognises that you cannot have a set of all sets that are not members of themselves. The overlooked part is that you cannot have a set of all sets that are members of themselves.
The point of the paradox isn't to list every contradiction that results from the unrestricted comprehension principle.

At any rate, when you say, " nothing can be a member of itself twice," how would you formalize (or at least semi-formalize) what you're claiming the contradiction is there?
I think the OP shows the problem of having a set of all sets that are members of themselves with sufficient clarity. If not, then the link provided in the OP sheds further light on this issue. Given all previous discussions between us (both on Russell's paradox, and on the nature of Existence) I don't think any further discussion between us will bear any fruit. Again, I think the OP is clear enough in highlighting what has been overlooked, and the link provided provides a clear solution to the problem at hand. I don't feel the obligation or need to add any thing else (precisely because I feel like what I've provided is already clear and sufficiently detailed enough).
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Re: The overlooked part of Russell's paradox

Post by philosopher19 »

Thomyum2 wrote: June 9th, 2021, 10:04 am
philosopher19 wrote: June 8th, 2021, 10:19 pm
Thomyum2 wrote: June 8th, 2021, 11:43 am Didn't Kurt Gödel resolve this dispute about 90 years ago?
Not to my knowledge.
Do you mean that you're not aware of his work, or that you are aware of it but think it did not resolve the dispute?

Without considering Gödel's later insights into these issues, I think that any discussion of Russell's paradox kind of turns into a fulfillment of the adage 'those who do not learn history are doomed to repeat it', since this topic was so thoroughly debated in the first 30 years of the 20th Century.
I'm not aware of his work. But to my knowledge, mainstream maths, philosophy, and logic, all reject an absolute universal set. To me, this is completely unacceptable as it is blatantly contradictory. Rejecting the set of all sets is the very last thing logicians should be doing.
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