The overlooked part of Russell's paradox

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philosopher19
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The overlooked part of Russell's paradox

Post by philosopher19 »

If x, y, and z are sets that are not members of themselves, and I form a set of these three sets, to represent this, I can write something like: p = {x, y, z}. I cannot write x = {x, y, z} as that would amount to x being a member of itself with y and z not being members of themselves.

If x, y, and z are sets that are members of themselves, and I form a set of these three sets, to represent this, I can write something like: p = {x, y, z}. I cannot write x = {x, y, z} as that would either amount to x being a member of itself with y and z not being members of themselves (consistent with above in red), or it would amount to x being a member of itself twice (which is contradictory as nothing can be a member of itself twice, or be itself twice), with y and z being members of themselves once. This shows the following:

You cannot have a set of ALL sets that are not members of themselves because it will result in at least one set not being included in the set. In other words, some set x will have to be included in x, but it can't.

You cannot have a set of ALL sets that are members of themselves because it will result in at least one set being a member of itself twice. In other words, some set x will have to not be included in x, but it can't.

For a detailed solution to Russell's paradox:

http://philosophyneedsgods.com/2021/05/ ... -infinity/
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Steve3007
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Re: The overlooked part of Russell's paradox

Post by Steve3007 »

philosopher19 wrote:If x, y, and z are sets that are members of themselves, and I form a set of these three sets, to represent this, I can write something like: p = {x, y, z}...
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.
...I cannot write x = {x, y, z} as that would either amount to x being a member of itself with y and z not being members of themselves (consistent with above in red)...
The part I've highlighted in bold is incorrect. x = {x, y, z} does not amount to y and z not being members of themselves. It says nothing about the members of y and z. It simply says that x is a member of itself and y and z are also members of it.
...or it would amount to x being a member of itself twice (which is contradictory as nothing can be a member of itself twice, or be itself twice), with y and z being members of themselves once.
Nothing you've written says anything equivalent to "x = {x, x}". You can, of course, have an infinite hierarchy of set membership. A set that is a member of itself will always form a hierarchy like that. "x = {x, ...}" can obviously be expanded ad infinitum. "x = {{x, ...}, ...}" etc. Nothing wrong with that so long as you don't expect it to represent physical objects. (Sets are abstract concepts. They may or may not be useful for representing groupings of physical objects.)
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Terrapin Station
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Re: The overlooked part of Russell's paradox

Post by Terrapin Station »

Aside from Steve3007's comment above, I didn't catch what the thread title promised: what's the "overlooked" part of Russell's paradox?
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RJG
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Re: The overlooked part of Russell's paradox

Post by RJG »

philosopher19 wrote:If x, y, and z are sets that are members of themselves
Phil, this is not possible. X (or Y, Z etc) cannot logically be a "set that is a member of itself". To help understand, imagine a box (to represent a "set"), now put this box inside itself. It can't be done.

"Sets that are members of themselves" are logical impossibilities (form X<X).
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Re: The overlooked part of Russell's paradox

Post by Terrapin Station »

RJG wrote: June 8th, 2021, 8:46 am
philosopher19 wrote:If x, y, and z are sets that are members of themselves
Phil, this is not possible. X (or Y, Z etc) cannot logically be a "set that is a member of itself". To help understand, imagine a box (to represent a "set"), now put this box inside itself. It can't be done.

"Sets that are members of themselves" are logical impossibilities (form X<X).
Sets aren't boxes. They're just abstract ideas. To have a set that's a member of itself, all you have to do is define it, and say, for example, that set x = {x, 1, 2}. Thus x is a member of itself.
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Steve3007
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Re: The overlooked part of Russell's paradox

Post by Steve3007 »

philosopher19 wrote:You cannot have a set of ALL sets that are not members of themselves because it will result in at least one set not being included in the set. In other words, some set x will have to be included in x, but it can't.
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.
You cannot have a set of ALL sets that are members of themselves because it will result in at least one set being a member of itself twice. In other words, some set x will have to not be included in x, but it can't.
You can have a set of all sets that are members of themselves. In your scenario of 3 sets that are members of themselves, if we call it x. So: x = {x, y, z}; y = {y, ...}; z = {z, ...}. That just means that there's an infinite hierarchy, or nesting. In the abstract world of sets there's nothing wrong with that.

Just as in mathematics there's nothing wrong with such concepts as infinite series. There is a whole class of numbers (transcendental numbers) that are only definable using such things as sums of infinite series, as opposed to finite length algebraic equations.

The concept of sets that are members of themselves may not have much use as references to physical objects, but I use them in my job all the time. They are undoubtedly useful logical concepts.
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RJG
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Re: The overlooked part of Russell's paradox

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Steve3007 wrote:x = {x, 1, 2}. Thus x is a member of itself.
Not possible. ...as this now give the variable "x" infinite different assignments, thereby defeating the purpose (uniqueness) of a variable. This creates an impossible infinite regress (a self looping) scenario.

For example, if x = {x, 1, 2}, then using substitution, x= {{x, 1, 2}, 1, 2} and x= {{{x, 1, 2}, 1, 2}, 1, 2}, and x= {{{{{{{{{{{{{{{{{{{{...to infinity and beyond, ...meaning "x" has no meaning!
Last edited by RJG on June 8th, 2021, 9:34 am, edited 1 time in total.
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Steve3007
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Re: The overlooked part of Russell's paradox

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RJG wrote:imagine a box (to represent a "set")
That's a bit like saying "imagine a cat to represent the word 'cat'". The direction of representation is the wrong way around.

Abstract concepts can (but don't have to) be used to describe or represent physical phenomena. Not the other way around. Sure, we can think of, or even construct, material models if it helps to visualize or illustrate an abstraction, but that doesn't mean that the abstraction is constrained by the physics of the material model that we've made. It's constrained by logic, which is not the same as saying that it's constrained by the physics of material models we might use to illustrate it. I think that's a mistake that's often made in all kinds of contexts where physical/material models and graphics are used to illustrate abstract idea. It's a dilemma for the people trying to explain those ideas.
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Steve3007
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Re: The overlooked part of Russell's paradox

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That one was TS.
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Re: The overlooked part of Russell's paradox

Post by Terrapin Station »

RJG wrote: June 8th, 2021, 9:32 am Not possible. ...as this now give the variable "x" infinite different assignments, thereby defeating the purpose (uniqueness) of a variable. This creates an impossible infinite regress (a self looping) scenario.
It's not something we're programming into a computer or anything real. It's an abstract idea. It creates an infinite regress insofar as you bother thinking of that.
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RJG
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Re: The overlooked part of Russell's paradox

Post by RJG »

Steve and TS, my point is that a given variable (abstract or real) is unique, it cannot simultaneously have 2 or more different meanings.
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Steve3007
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Re: The overlooked part of Russell's paradox

Post by Steve3007 »

RJG wrote:Steve and TS, my point is that a given variable (abstract or real) is unique, it cannot simultaneously have 2 or more different meanings.
Since we're now talking about variables: In my usage variables have values. So if x = 2, then the value of x is 2. If y = 2 then x = y is equivalent to saying x = 2. There's nothing wrong with x simultaneously being equal to 2 and to y, and to an arbitrarily large number of other possible expressions.

If we go back to considering sets:

If x = {x, y, z} then, by substituting for x, that's equivalent to x = {{x, y, z}, y, z}, and to an infinite number of other possible substitutions. Nothing wrong with that. Nothing wrong with substituting an expression with an equivalent expression. And, as I said earlier, there's nothing logically wrong with infinite sequences. They can be very useful. We wouldn't, for example, have definitions of pi or e without them. Pi and e are very useful.
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RJG
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Re: The overlooked part of Russell's paradox

Post by RJG »

Steve3007 wrote:Since we're now talking about variables: In my usage variables have values. So if x = 2, then the value of x is 2. If y = 2 then x = y is equivalent to saying x = 2. There's nothing wrong with x simultaneously being equal to 2 and to y, and to an arbitrarily large number of other possible expressions.
I agree with this because there are multiple ways to say "2". So the meaning is the same in all cases.

But we can’t simultaneously mean "2" and also mean "3". We can't have multiple simultaneous meanings for a single variable.

Steve3007 wrote:If we go back to considering sets:

If x = {x, y, z} then, by substituting for x, that's equivalent to x = {{x, y, z}, y, z}, and to an infinite number of other possible substitutions. Nothing wrong with that. Nothing wrong with substituting an expression with an equivalent expression. And, as I said earlier, there's nothing logically wrong with infinite sequences. They can be very useful. We wouldn't, for example, have definitions of pi or e without them. Pi and e are very useful.
There is a difference here. Pi (as with any variable) we can get to stand by itself, but x = {x, 1, 2} or X<X cannot be resolved; we can't get X to stand by itself so we can know what it is (what it means).

If you believe X<X is logically impossible, then why don't you believe x={x, 1, 2} is likewise impossible?

What's the difference? In both cases, x can never be resolved, and therefore impossible.
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Re: The overlooked part of Russell's paradox

Post by Terrapin Station »

RJG wrote: June 8th, 2021, 9:39 am Steve and TS, my point is that a given variable (abstract or real) is unique, it cannot simultaneously have 2 or more different meanings.
First off, when we're talking about how people are thinking about things--which is what we're talking about when we're talking about sets, what would make it true that a given variable is unique?
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RJG
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Re: The overlooked part of Russell's paradox

Post by RJG »

Terrapin Station wrote:First off, when we're talking about how people are thinking about things--which is what we're talking about when we're talking about sets, what would make it true that a given variable is unique?
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.
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