Russell's Paradox: The Achille's Heel of Solipsism?

Mentat · Nov 24, 2003

I had always taken it for granted that nothing could disprove Solipsism, but now I think there may be an actual logical problem with it!

I understand that I could easily be wrong, and that's why I'm posting it: for constructive criticism.

Alright, now, the first think I might have gotten wrong is the name...Russell's paradox is the paradox that states that no set can contain itself, isn't it?

If so, then isn't this a huge (possibly fatal) blow to Solipsism (which dictates that there is nothing that exists, except for what exists in my mind)?

You see, if the Mind &equiv; the Set of all things that exist, then how can the Mind itself exist at all?

Any comments, constructive critiques, or corrections are appreciated.

phoenixthoth · Nov 24, 2003

first of all, that something contradicts an arbitrary contrivance such as logic means nothing. but i don't think solipsism leads to russell's paradox so that issue is moot.

here's russell's paradox. one can view it as a theorem and not a paradox. it's based on the tautology
[p→(q↔¬q)]→¬p.

p says ∃U∀x(x∈U). in this context, this means U is a set containing all other sets.

by the subsets axiom, S={x∈U:x!∈x} is a set.

q will be the statement S∈S.

note that for all (sets) z, z∈S↔(z∈U&and;z!∈z). one can show that since z∈U is true for all z, this is equivalent to z∈S↔z!∈z. now taking z=S, we have that S∈S↔S!∈S, a statement of the form q↔¬q which was a consequence of p. conclusion? ¬p. in other words, it is not the case that there exists a set U such that for all (sets) x, x∈U. in other words, for all sets U there is a set x such that x!∈U.

there is, however, a class of all sets and a category of all sets and a category of all categories (in that situation the difficulty is relieved by the absence of an equivalent "subsets axiom" but I'm no category theory person).

when one moves to multi-valued logic such as one in which truth values aren't just 0 (F) or 1 (T) but anything in [0,1], the truth value of the statement S∈S under basic assumptions is 0.5; S is considered a fuzzy set.

in solipsism, one hardly claims that their mind contains all sets; just perhaps sets related to the universe. but, then again, if the mind is better modeled by a category than a set, then the whole question of whether it is somehow equivalent to the set of sets is moot.

Mr. Robin Parsons · Nov 24, 2003

Originally posted by Mentat
(SNIP) You see, if the Mind &equiv; the Set of all things that exist, then how can the Mind itself exist at all? (SNoP)

The 'mind' can simply exist as the (small) reflector of 'all things' in the greater mind.

Just like you have memories, in you, that are not in your 'presently conscious mind', but you can re-call them. Your mind equvalently works out to be a "subset access system" of your own, greater, mind.

Put it this way, for the totality of 'mind' you still cannot concieve of the actuality of an Avogadro's number, as that number exceeds all of the Neurons, and neuronal connections, (even neurotransmitters rates, as individualized counts) that exist within a humans mind.

Your mind, like everyones mind, has limitations upon it, hence it would be difficult, at best, to prove (or disprove?) solipsism, by that methodology.

phoenixthoth · Nov 24, 2003

oh yeah, and about that snip/snop quote, it can exist it just wouldn't be a set in two valued logic.

Mentat · Nov 25, 2003

Originally posted by phoenixthoth
first of all, that something contradicts an arbitrary contrivance such as logic means nothing. but i don't think solipsism leads to russell's paradox so that issue is moot.

here's russell's paradox. one can view it as a theorem and not a paradox. it's based on the tautology
[p→(q↔¬q)]→¬p.

p says ∃U∀x(x∈U). in this context, this means U is a set containing all other sets.

by the subsets axiom, S={x∈U:x!∈x} is a set.

q will be the statement S∈S.

note that for all (sets) z, z∈S↔(z∈U&and;z!∈z). one can show that since z∈U is true for all z, this is equivalent to z∈S↔z!∈z. now taking z=S, we have that S∈S↔S!∈S, a statement of the form q↔¬q which was a consequence of p. conclusion? ¬p. in other words, it is not the case that there exists a set U such that for all (sets) x, x∈U. in other words, for all sets U there is a set x such that x!∈U.

there is, however, a class of all sets and a category of all sets and a category of all categories (in that situation the difficulty is relieved by the absence of an equivalent "subsets axiom" but I'm no category theory person).

when one moves to multi-valued logic such as one in which truth values aren't just 0 (F) or 1 (T) but anything in [0,1], the truth value of the statement S∈S under basic assumptions is 0.5; S is considered a fuzzy set.

Ok...I'm not that good at symbolic logic, but I understood most of that . Anyway, isn't the end result that there can be no set of all sets?

in solipsism, one hardly claims that their mind contains all sets; just perhaps sets related to the universe.

That's bad semantics, since the term "Universe" means "everything".

but, then again, if the mind is better modeled by a category than a set, then the whole question of whether it is somehow equivalent to the set of sets is moot.

I didn't understand the "category of sets" thing, could you please clarify?

Mentat · Nov 25, 2003

Originally posted by Mr. Robin Parsons
The 'mind' can simply exist as the (small) reflector of 'all things' in the greater mind.

That doesn't help at all, since the "greater mind" would have to be a set of all sets", and would still fall into Russell's paradox...besides, Solipsism makes all things the function of my own mind, thus the thing that I'm supposed to be "reflecting" should also be a function of my mind...IOW, there is no objective reality, nothing outside my own mind.

Mentat · Nov 25, 2003

Originally posted by phoenixthoth
oh yeah, and about that snip/snop quote, it can exist it just wouldn't be a set in two valued logic.

How else would you have me establish validity? I'm not trying to discover "truths", just validity, and this seemed like a valid argument. But, is there some other way to check its validity by another method completely (besides logic)?

Mr. Robin Parsons · Nov 25, 2003

Originally posted by Mentat
That doesn't help at all, since the "greater mind" would have to be a set of all sets", and would still fall into Russell's paradox...besides, Solipsism makes all things the function of my own mind, thus the thing that I'm supposed to be "reflecting" should also be a function of my mind...IOW, there is no objective reality, nothing outside my own mind.

But you have missed my simple point of the fact that not all things exist in your mind simultaneously, (consciously) ergo you MUST be dealing with a 'subset' (Conscious) of the 'greater set' (Sub-Conscious) even if the entire 'greater set' is in your mind only!

phoenixthoth · Nov 25, 2003

the *blank* of all things just isn't a set unless you allow fuzzy sets to exist. the mind would just not be a set.

You see, if the Mind ß the Set of all things that exist, then how can the Mind itself exist at all?

could you substitute the word "universe" for "mind" in a nonsolipsistic theory? in essence, whatever you call the thing that contains (or is equivalent to) the thing of all things has the same problem the mind does in solipsism.

either way, the problem is resolved by realizing that the *blank* of all things that exist just may not be a set.

http://mathworld.wolfram.com/Category.html

anyway, if a theory (science, set theory, logic) can't make sense of the truth, it doesn't make the truth nontruth, it just points out limitations in the theory.

Mr. Robin Parsons · Nov 26, 2003

From the Solipsist's viewpoint I would like to know how your mind goes about generating information that you, in your mind, are completely unaware of, prior to it's presentation to you, by another!

How does that work??

phoenixthoth · Nov 26, 2003

a modified solipsism is that reality is like a dvd and you are the laser beam that reads the disc. you're the only one that exists; everyone else is just a bunch of 1's and 0's on some disc (perhaps a hologram). the whole of reality is in some sense only present in one moment but the illusion of time is created as you (the laser) shifts awareness from one point on the "disc" to another. there are no other minds, just illusions of minds. perhaps the mind created this hologram and perhaps it didn't. maybe it wasn't created but has always existed out of time. generally speaking, I'm not sure if it is thought that the mind isn't generating stuff prior to your experience of it and there would be no others to present it to you. I'm not sure if it is thought that the mind is what creates things. perhaps it is the creator (the laser beam and the disc) and perhaps it is just the only perceiver (just the laser beam).

Mentat · Nov 26, 2003

Originally posted by Mr. Robin Parsons
But you have missed my simple point of the fact that not all things exist in your mind simultaneously, (consciously) ergo you MUST be dealing with a 'subset' (Conscious) of the 'greater set' (Sub-Conscious) even if the entire 'greater set' is in your mind only!

I'm sorry, I don't understand what you're getting at. If we are always dealing with subsets of a greater set, that set would still have to hold all of reality at once, wouldn't it?

Mentat · Nov 26, 2003

Originally posted by phoenixthoth
the *blank* of all things just isn't a set unless you allow fuzzy sets to exist. the mind would just not be a set.

What's a "fuzzy set"? Besides, the mind isn't really the set; all things that exist are "mindful" things, according to Solipsism, but then the mind itself must also be a "mindful" thing. That's my point.

could you substitute the word "universe" for "mind" in a nonsolipsistic theory?

No, the Universe is not an entity. The word "Universe" just refer to everything.

Mr. Robin Parsons · Nov 26, 2003

Originally posted by Mentat
I'm sorry, I don't understand what you're getting at. If we are always dealing with subsets of a greater set, that set would still have to hold all of reality at once, wouldn't it?

The 'subset' presently being dealt with, is the activity currently going on in your mind, but that is not the entirety of the 'greater set' (in your mind) that contains all of the other things you can remember, even though that 'greater set' exists, presently, (dormant? or just quiet?) within you.

But the Solipsis determines that that greater set carries all of the knowledge that Exists, and then what?? presents that through others, to itself?? Why the need of the redundancy?

Mentat · Nov 26, 2003

Originally posted by Mr. Robin Parsons
The 'subset' presently being dealt with, is the activity currently going on in your mind, but that is not the entirety of the 'greater set' (in your mind) that contains all of the other things you can remember, even though that 'greater set' exists, presently, (dormant? or just quiet?) within you.

But the Solipsis determines that that greater set carries all of the knowledge that Exists, and then what?? presents that through others, to itself?? Why the need of the redundancy? [/B]

Uh...you goin' somewhere with this? You're making good points, but they're scattered, so...what are you trying to say?

phoenixthoth · Nov 26, 2003

some people think the universe is an entity. not necessarily a conscious entity though some think it is but its consciousness doesn't resemble ours.

a fuzzy set x, to me, is a set such that there is another set y such that the statement y∈x is neither true nor false but some other truth value. for example, in russell's "paradox," you have S and the statement S∈S has truth value not equal to 0 or 1 and so if a set of all sets exists, you have to accept the existence of fuzzy sets. fuzzy sets and a universal set or crisp sets and no universal set. one can also show that the powerset of the universal set would equal the universal set and the cantor diagonal argument which usually proves that there is no function from a set to its powerset (which would show that the identity map from U to P(U) isn't onto which would contradict their equality) actually reduces to russell's theorem and the fuzzy set S; again, no contradiction if more than two truth values are allowed. i wonder if P(x)=x implies x is the universal set... the universal set itself, btw, is crisp though it has fuzzy subsets.

Mr. Robin Parsons · Nov 26, 2003

Originally posted by Mentat
Uh...you goin' somewhere with this? You're making good points, but they're scattered, so...what are you trying to say?

Scattered?, Humm, you mean like seeds??

Solipsism requires that the universe act in a redundant fashion with respect to "education of self", and requires that your mind contain the either the entirety of the Universe, or the knowledge of the entirety of the Universe which, on simple numerical values alone, it cannot.

As for the rest, simply put in your language of 'set theory', your mind and its functions, your consious mind, (as I previously said) "the subset" that is active in "the greater set" (subconscious) that is the 'entirety of your mind' which includes all of your memories etc. etc.

Does that help?
(perhaps what you need is really some time to think about it...sleep on it, see what develops tomorrow, or later still...)

phoenixthoth · Nov 26, 2003

Solipsism requires that the universe act in a redundant fashion with respect to "education of self", and requires that your mind contain the either the entirety of the Universe, or the knowledge of the entirety of the Universe which, on simple numerical values alone, it cannot.

or that your mind is the entirety of the universe.

Mr. Robin Parsons · Nov 26, 2003

Originally posted by phoenixthoth
or that your mind is the entirety of the universe.

And how then does it go about hidding 'itself' from 'itself'? as clearly we do not know the entirety of the Universe, right?
(or are you perhaps laying the claim??)

phoenixthoth · Nov 26, 2003

whatever appears hidden doesn't exist.
either that or perhaps its a kind of dissociative amnesia.
we are the entirety of the universe but are not aware of it (unless we are). i can control the sun no more than i can control my toenail growth. a mind doesn't have to completely be aware of itself, know itself, or control all of its parts.

Mr. Robin Parsons · Nov 27, 2003

Originally posted by phoenixthoth
whatever appears hidden doesn't exist.
either that or perhaps its a kind of dissociative amnesia.
we are the entirety of the universe but are not aware of it (unless we are). i can control the sun no more than i can control my toenail growth. a mind doesn't have to completely be aware of itself, know itself, or control all of its parts.

Begs the question of "why" would we, the entirety of the Universe, do this to ourselves?

Mentat · Nov 28, 2003

Originally posted by phoenixthoth
some people think the universe is an entity. not necessarily a conscious entity though some think it is but its consciousness doesn't resemble ours.

a fuzzy set x, to me, is a set such that there is another set y such that the statement y∈x is neither true nor false but some other truth value. for example, in russell's "paradox," you have S and the statement S∈S has truth value not equal to 0 or 1 and so if a set of all sets exists, you have to accept the existence of fuzzy sets. fuzzy sets and a universal set or crisp sets and no universal set. one can also show that the powerset of the universal set would equal the universal set and the cantor diagonal argument which usually proves that there is no function from a set to its powerset (which would show that the identity map from U to P(U) isn't onto which would contradict their equality) actually reduces to russell's theorem and the fuzzy set S; again, no contradiction if more than two truth values are allowed. i wonder if P(x)=x implies x is the universal set... the universal set itself, btw, is crisp though it has fuzzy subsets.

So the allowance of these fuzzy (universal) sets allows for the Solipsistic approach, while the alternative doesn't?

Mentat · Nov 28, 2003

Originally posted by Mr. Robin Parsons
Solipsism requires that the universe act in a redundant fashion with respect to "education of self", and requires that your mind contain the either the entirety of the Universe, or the knowledge of the entirety of the Universe which, on simple numerical values alone, it cannot.

As for the rest, simply put in your language of 'set theory', your mind and its functions, your consious mind, (as I previously said) "the subset" that is active in "the greater set" (subconscious) that is the 'entirety of your mind' which includes all of your memories etc. etc.

Does that help?
(perhaps what you need is really some time to think about it...sleep on it, see what develops tomorrow, or later still...)

I've been "sleeping on it", and I still don't get exactly what you are trying to say.

Jeebus · Nov 28, 2003

The paradox stems from the acceptance of the following axiom: If P(x) is a property then:

{x : P}

is a set. This is the Axiom of Comprehension (actually an axiom schema). By applying it in the case where P is the property "x is not an element of x", we generate the paradox, i.e. something clearly false. Thus any theory built on this axiom must be inconsistent.
In lambda-calculus Russell's Paradox can be formulated by representing each set by its characteristic function - the property which is true for members and false for non-members. The set R becomes a function r which is the negation of its argument applied to itself:

r = \ x . not (x x)

If we now apply r to itself,

r r = (\ x . not (x x)) (\ x . not (x x))
= not ((\ x . not (x x))(\ x . not (x x)))
= not (r r)

So if (r r) is true then it is false and vice versa.
An alternative formulation is: "if the barber of Seville is a man who shaves all men in Seville who don't shave themselves, and only those men, who shaves the barber?" This can be taken simply as a proof that no such barber can exist whereas seemingly obvious axioms of set theory suggest the existence of the paradoxical set R.

Now if we applied this to the Mind we would say:

"The mind is false."

This four word sentence is confusing because it appears to be a single, solidified argument that contradicts itself. It is only one sentence. There is only one predicate and object. There is only one speaker of the sentence, yet something is seriously wrong with the statement.

Now, the thing I have noticed about all self-referential paradoxes is that they all seem to contain a strange thing called a self-reference. Notice in the example above that the words "This Sentence" is not actually a thing, it is a reference to a thing. Replace the pointer "This sentence" with the full sentence "This sentence is false," and you can see that there is more than one argument involved in the statement:
"The mind is false" is false.
Each time you evaluate the reference "The mind" you expand the argument and can see how it toggles back and forth: """The mind is False" is false." is false" is false.

By definition, all self referential paradox contain a reference back to itself. Although it appears that a given self-referential paradoxes is a single logical entity that contradicts itself, I am of the opinion that self referential paradoxes actually contain two logical steps: the resolution of the reference and the contradiction. A self-referential argument is just like the playground incident where two children have differing opinion.

Sentence A: "Sentence B is false."
Sentence B: "Sentence A is false."

Self-referential paradoxes appear as a different class of argument simply because we don't think of resolving the reference as a full logical step.

Thus this just shows that we cannont accordinly concieve and compare mixing Solipsism with the Russell Paradox. It just doesn't work.

Mentat · Nov 28, 2003

Jeebus, you didn't lose me until near the end. My question is: Why was your statement, "The mind is false"; that was not my statement.

Jeebus · Nov 28, 2003

I know it wasn't your statement, and it wasn't directed at you personally.

It meant that "The mind is false" statement is directly parallel to Solipsism. Since solipsism is defined as: "belief in self as only reality: the belief that the only thing somebody can be sure of is that he or she exists, and that true knowledge of anything else is impossible" -- You can relate that between the Russell Paradox sets to the solipsism logic.

That's why I said "The mind is false" is false because if you use the Russell paradox system to the Solipsism standpoint you can conclude that the belief in yourself is only reality is not really completely true using the Russell paradox.

You dig?

Mr. Robin Parsons · Nov 30, 2003

If you were to discover a "previously unrecognized-self evident truth", something new and groundmaking/changing in physics, or some other area of science/politics/law/art/cinema/etc. you would be demonstrating that "the Universe, had kept hidden, from itself, something" such that only the one "true" 'Solipsis' could enjoy, what?? the Game??

God's Grace in my life I have discovered one of those, ("previously unrecognized-self evident truth") and I can assure you, from what little(?) I know of reality, I am not a (or 'the') solipsis, cause I have clearly seen that I still need be;
"A student, who teaches(?)"

Mumeishi · Nov 30, 2003

You see, if the Mind = the Set of all things that exist, then how can the Mind itself exist at all?

Its an interesting thought but we could use the same argument to prove that the universe cannot exist. And then where would we b...

Mentat · Dec 1, 2003

Originally posted by Jeebus
I know it wasn't your statement, and it wasn't directed at you personally.

It meant that "The mind is false" statement is directly parallel to Solipsism. Since solipsism is defined as: "belief in self as only reality: the belief that the only thing somebody can be sure of is that he or she exists, and that true knowledge of anything else is impossible" -- You can relate that between the Russell Paradox sets to the solipsism logic.

That's why I said "The mind is false" is false because if you use the Russell paradox system to the Solipsism standpoint you can conclude that the belief in yourself is only reality is not really completely true using the Russell paradox.

You dig?

Glad to see we're on the same frequency now.

Mentat · Dec 1, 2003

Originally posted by Mumeishi
Its an interesting thought but we could use the same argument to prove that the universe cannot exist. And then where would we b...

Actually, I dealt with this before. The Universe really doesn't exist as a coherent entity. There are things that exist, and there is a limit to the amount of things that exist. The sum total of these things is called "the Universe", but the Universe itself doesn't exist.

Mumeishi · Dec 1, 2003

Wow. You've thought of everything haven't you?

The universe doesn't exist... have to remember that one.

Mentat · Dec 1, 2003

Originally posted by Mumeishi
Wow. You've thought of everything haven't you?

The universe doesn't exist... have to remember that one.

Ha ha...no, the Universe doesn't exist, not as an entity in itself, and there's nothing (that I can think of) that's logically wrong with that. However, because of this you cannot say that "everything exists inside the Universe", since there both is no "outside the Universe" and no "Universe", you have to say "the sum total of everything that exists, is called 'Universe'".

Mumeishi · Dec 1, 2003

Oh I wasn't mocking - I genuinely like your argument. It makes a lot of sense. But still - it was a bit of a shock to hear that the universe does not exist...

phoenixthoth · Dec 1, 2003

Begs the question of "why" would we, the entirety of the Universe, do this to ourselves?

which begs the questions (1) why ask why and (2) why did the chicken cross the road and (3) why does there have to be a reason?

So the allowance of these fuzzy (universal) sets allows for the Solipsistic approach, while the alternative doesn't?

it doesn't either allow it or not allow it. "it" can do whatever "it" "wants." it's not clear to me that the two have much to do with each other. but for how I'm percieving your take on solipsism, yeah, having a universal set that contains fuzzy subsets gels with solipsism.

the mind doesn't exist, not as an entity in itself, and there's nothing (that I can think of) that's logically wrong with that. However, because of this you cannot say that "everything exists inside the mind", since there both is no "outside the mind" and no "mind", you have to say "the sum total of everything that exists, is called 'mind'".

Royce · Dec 1, 2003

Okay, the universe does not exist as an entity in itself but is simply a term that we use to mean everything that does exist. Right so far? Then what about the idea, or hypothesis, that the universe is a singularity, all that exists, exists within the singularity that is the universe.
When you are speaking of all that exists, are you speaking materialisticly or all inclusively, assuming that there is that which is not material yet exists?

Russell's Paradox: The Achille's Heel of Solipsism?

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