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Thread: Applications of Russell's Paradox

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    chriscase's Avatar Chairman Miao
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    Default Applications of Russell's Paradox

    I promised an explanation of Russell's paradox and how I believe it can be used outside of a purely mathematical context.

    Interested parties should of course read the explanation on wiki or your favorite reference, but I'll give my informal (simplified) synopsis with what I see as the key features.

    Naive set theory, as proposed in informal language, asserts Cantor's definition of set as "a gathering together into a whole of definite, distinct objects of our perception or of our thought." Although the logical language of set theory is first-order predicate logic, at this level of description we can use the simpler form of propositional logic: a set is well defined if we can formulate a propositional formula that describes its elements. That is, because the logical definition of a propositional formula P(x) is that for every x we can say (in principle) that P(x) is true or false but not both.

    It follows from the assertion of this "definition" of set that the propositional formula P(x) iff x is a set is a well defined proposition - otherwise the notion of set would be poorly defined. But because P(x) iff x is a set is assumed to be a well formed proposition, by the definition of set it must define a set, i.e., the set of all things that are sets. At this point I'll note that we aren't really using many of the properties of sets. What we've really done is piggyback on the logical foundation to assert the existence of some collections of things that correspond with the action of propositions on them.

    However, now that we have the set of all sets, we do need to assert properties of membership. Still we do not need the full machinery of Cantor's set theory to generate a paradox, only some rather intuitive rules about things that are members of collections. Specifically, if a collection of things or a set is well defined, and I take some of the members of that collection, the collection I have taken inherits the well-definition of the original collection. In talking about sets, a subset of a set is also a set.

    And at this point we can also assert that the proposition of self-membership must be well defined, since the members of a set are definite. So whether a set has a particular member must be propositionally true or false, and since the set of all sets is itself a set, it follows that it is a member of itself.

    Given rules of propositional logic, the negation of a proposition is also a proposition, so we can form a proposition S(x) iff x is not a member of itself, which derives from the negation of the proposition P(x) iff x is a member of itself - a proposition already established as well defined. Now we get another subset of the set of all sets, the set of all sets that are not members of themselves. This must be a set since it is a subset of the collection of all sets.

    Note that the key move here is this assertion that the aggregate of all objects of a given type is itself an object of the same type.

    At this point, the collection of all sets that are not members of themselves - In mathematical language, S = { X | X is not an element of X } - generates a paradox when we ask, is S a member of itself?

    1. Start by assuming S is not a member of itself. In that case, it satisfies the definition of membership in S which is defined as the set of all sets that do not belong to themselves. That means S must be a member of S, i.e. it does belong to itself, which contradicts the assumption we started out with.
    2. Suppose S is a member of itself. Then it must satisfy the conditions for membership in S which is defined as the set of all sets that do not belong to themselves. Thus we conclude that S must not be a member of itself, which again contradicts the assumption we started out with.
    ______

    Although much deeper predicate logic can be used to describe this, for the purposes of general discussion, this paradox does not really require much beyond:
    1. Naive use of propositional logic in a free universe of consideration (notice we never scoped our universe of consideration to objects of a particular type - we were free to consider anything that might be encoded in a propositional formula).
    2. A rudimentary notion of membership directly tied to propositional formulas.

    Even the notion of self-membership (which is where things seem to start going off the rails) is necessitated by (1) and (2), which give us the assumption that the "collection of all collections" or set of all sets is well defined.

    _______

    So when I encounter mystical notions such as a God that is the "transcendent summation of all that exists", I see a very similar construction to naive set theory:
    1. Is P(x) iff x exists a valid propositional formula? If not, we can stop right here and say our logical foundations are simply unusable. In this case, the statement that the God under discussion "exists" or perhaps does not is itself incoherent.
    2. On the other hand, if we do (as most would) assume that existence is a well defined propositional property, then this God that contains the transcendent culmination of all that exists has properties very much like the set of all sets, and allows us to construct an analog of Russell's paradox.

    To me, it's also a giveaway that type has become mixed here. If we postulate a thing that subsumes the entirety of existence, it seems suspiciously odd that this monstrous aggregation would have any properties in common with its lowly members. This is where the transcendent God appears to contradict a loving personal God. Suppose we define some other property such as P(x) iff x is capable of feeling love. Is the set or collection of all x that are capable of feeling love itself capable of feeling love? If we think it might, we're back in a set of all sets situation, and can very likely construct a paradox.

    I suspect this prohibition on free mixing of type is a general principle. Can the free aggregation of all objects with a particular property itself be an object with that property? If so, it seems we'd be able to construct another Russell-type paradox.

    What do you think? Is this paradox of use in general discourse? Or does it belong solely in Set Theory class?
    Last edited by chriscase; September 17, 2017 at 12:12 AM.

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    basics's Avatar Vicarius Provinciae
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    Default Re: Applications of Russell's Paradox

    chriscase,

    Perhaps the question should be, is the paradox worth talking about at all since very few will ever be able to understand any application for it outside of the simple mind? I'd love to be a fly on the wall in your house just to find out if the wife and kids get the same treatment. And here was me thinking that you were a simple dad but I bow to you and thank God that He made life really simple for ordinary folks.

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    chriscase's Avatar Chairman Miao
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    Default Re: Applications of Russell's Paradox

    Hey basics, anyone who studies set theory in college/university will spend time with this sort of construction. Not sure if either of my kids will be so inclined, but that's really up to them. In any case, when we deal in logic, we generally inherit an aversion to paradox, so this sort of thing comes with the territory.

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    Default Re: Applications of Russell's Paradox

    Quote Originally Posted by chriscase View Post
    Hey basics, anyone who studies set theory in college/university will spend time with this sort of construction. Not sure if either of my kids will be so inclined, but that's really up to them. In any case, when we deal in logic, we generally inherit an aversion to paradox, so this sort of thing comes with the territory.
    chriscase,

    You know that i'm having a little fun at your expense simply because all this is way above my meagre brain. Sign of a mispent youth. God bless you.

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    Default Re: Applications of Russell's Paradox



    Kinda topical

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    Sir Adrian's Avatar the Imperishable
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    Default Re: Applications of Russell's Paradox

    Quote Originally Posted by chriscase View Post
    So when I encounter mystical notions such as a God that is the "transcendent summation of all that exists", I see a very similar construction to naive set theory:
    1. Is P(x) iff x exists a valid propositional formula? If not, we can stop right here and say our logical foundations are simply unusable. In this case, the statement that the God under discussion "exists" or perhaps does not is itself incoherent.
    2. On the other hand, if we do (as most would) assume that existence is a well defined propositional property, then this God that contains the transcendent culmination of all that exists has properties very much like the set of all sets, and allows us to construct an analog of Russell's paradox.

    To me, it's also a giveaway that type has become mixed here. If we postulate a thing that subsumes the entirety of existence, it seems suspiciously odd that this monstrous aggregation would have any properties in common with its lowly members. This is where the transcendent God appears to contradict a loving personal God. Suppose we define some other property such as P(x) iff x is capable of feeling love. Is the set or collection of all x that are capable of feeling love itself capable of feeling love? If we think it might, we're back in a set of all sets situation, and can very likely construct a paradox.

    This is the most roundabout way of saying I don't believe in God I have ever seen. Also you have to prove that all creation does indeed fulfill the characteristics and requirements of a set in order for the paradox to be applicable.
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    Default Re: Applications of Russell's Paradox

    Well that's one of the points I am making here - the basic structure of the paradox isn't only applicable to sets. All we really need is propositional logic, and the ability to define collections based on propositional formulas. In fact, as set theory is formulated now the paradox does not apply - that's kind of the point of axiomatic set theory. So the association with sets is more of a historical curiosity.

    As for God, the paradox only applies in the case where we can claim that God lies entirely within the scope of logic (i.e. we can apply the principles of propositional logic to the entirety of God). For many this is a non-starter.

    Why is it that mysteries are always about something bad? You never hear there's a mystery, and then it's like, "Who made cookies?"
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    Default Re: Applications of Russell's Paradox

    chriscase,

    The thing is one cannot tie God down to anything that might tick certain boxes just because to us that might be logical. Put it this way, as John tells us, " there is nothing made that was not made by Him " and so even our logic is disciplined to what He has endowed to us. I think I can therefore say that them that now know Him do so only because He makes it so by doing the illogical yet supernatural thing on their behalf each time someone is drawn to rebirth. They believe what our laws don't allow for because it is outside the boxes we live by.

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    Default Re: Applications of Russell's Paradox

    Yeah, the original context for this was an assertion that a transcendent notion of God and propositional logic are entirely compatible.

    Why is it that mysteries are always about something bad? You never hear there's a mystery, and then it's like, "Who made cookies?"
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    Default Re: Applications of Russell's Paradox

    Quote Originally Posted by chriscase View Post
    Yeah, the original context for this was an assertion that a transcendent notion of God and propositional logic are entirely compatible.
    chriscase,

    Yet John tells us quite clearly that the prayers of unbelievers God doesn't listen to, but on the other hand He does have appeal for those that do cry out to Him with a broken and contrite heart. But then again looking deeper that only comes by the Spirit divulging the heavy weight sin has on that person. To them the reward is beyond all reason but to the world it is totally illogical.

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    Default Re: Applications of Russell's Paradox

    Yeah but that's just, like... a book, man.

    Why is it that mysteries are always about something bad? You never hear there's a mystery, and then it's like, "Who made cookies?"
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    Default Re: Applications of Russell's Paradox

    Quote Originally Posted by chriscase View Post
    Yeah but that's just, like... a book, man.
    chriscase,

    There's only one book that I know of that has the power to change lives not only for this world but also that to come and it's called the Bible. People all across the world that were never allowed it are crying out for it, why? Because within it there is something quite supernatural in its promises. Yes to many it contains nothing but the dead letter but to others it opens out a new exciting conclusion to a life that may not always be burdened by poverty, oppression, injustice and yes guilt to heavy to bear anymore.

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    Default Re: Applications of Russell's Paradox

    Dune. That's the one. Seriously though, that's not what the thread is about.

    Why is it that mysteries are always about something bad? You never hear there's a mystery, and then it's like, "Who made cookies?"
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    Default Re: Applications of Russell's Paradox

    Quote Originally Posted by chriscase View Post
    Yeah, the original context for this was an assertion that a transcendent notion of God and propositional logic are entirely compatible.
    chriscase,

    But the basis for that is surely the knowledge that we have come to know about God which can both be textual or experimental experienced through others?

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    Default Re: Applications of Russell's Paradox

    On the other hand, I'm sure you'll agree that it's quite easy for us, as humans, to fall into error. Further, I also assume you'd agree that some errors are the result of simple lack of information, whereas other errors are due to misinterpretation of information that's available. Given that we have to make interpretations in order to reach any sort of conclusion at all, it's crucial that we avoid pitfalls in how we proceed from observations or established knowledge to conclusions about new knowledge. It's equally important that we are able to revisit earlier conclusions that may have been erroneous, given new information. In this process, logic plays the role of watchdog and guide, warning us about known mistakes of reasoning even as it illuminates the way with valid methods for making inferences.
    Last edited by chriscase; September 24, 2017 at 12:14 PM.

    Why is it that mysteries are always about something bad? You never hear there's a mystery, and then it's like, "Who made cookies?"
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    Default Re: Applications of Russell's Paradox

    chriscase,

    That's quite true, us humans probably make more mistakes than anything else in creation. For the simple like me it's a way of learning not to repeat mistakes although I have to say that history does sometimes repeat itself before the penny drops. Of course the more information one can absorb one would think errors in humanity might have been eliminated by now especially if one goes along the Darwinistic line but again history proves that not to be the case. So, when the Bible tells us that there is nothing new on the earth, that seems to be the case in general. Of course being a people of inquisitive natures we will utilise information to develope many things that are both advantageous as well as disadvantageous thinking that mankind will benefit from them and now all we seem to get are the cries that we are destroying the planet meaning that those who invent them whilst being superior to the rest of us are actually the ones who are the most guilty. It makes knowledge and information in their hands more dangerous than the poor ole Joe Bloggs who make up the bulk of the population.

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    chriscase's Avatar Chairman Miao
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    Default Re: Applications of Russell's Paradox

    OK, but let's just consider maternal death due to complications during childbirth:



    What do you think? Is this a good development? Or should we stop tinkering with medicine?

    Why is it that mysteries are always about something bad? You never hear there's a mystery, and then it's like, "Who made cookies?"
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    Default Re: Applications of Russell's Paradox

    chriscase,

    OK, these are fair figures you present and if continued they surely reflect our ability to make pregnancy safer but and there's always a but about these things and what comes to mind is our Western culture's low birth rate compared to other nations as well as the rise in abortion. In other words Western women are not prepared to bring life to the fore for many personal reasons regardless of medicine's ability to protect a fetus not just outside the womb but also inside and that can only be a good thing in itself. However abortions are one horrific stain on our society and if libs get their way it will become even worse. So, in this we see the good and bad side of medicine where doctors and abortionists are defying the very oaths they take to preserve life. Is this not the very tinkering with all that medicine should on oath stand for?

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    Default Re: Applications of Russell's Paradox

    Knowledge is morally neutral: it can be used for good or bad purposes. I would propose, however, that the desire for knowledge is inescapable. For example, how to deliver a baby safely will inevitably come to the fore for every person who witnesses a loved one die on the birthing-bed. Remaining in ignorance goes against our most noble impulses. The question I am posing is: how can we acquire that knowledge we all crave in order to do good?

    I am willingly tabling the question of how to teach the moral use of knowledge, since logic - like other types of knowledge - is morally neutral. It's a worthy question, but since the acquisition of knowledge is so central to our nature, it's clear that we must seek knowledge and learn how to use it morally. In this discussion, however, I am focusing on the fundamental tools that enable us to acquire knowledge with reliability, and logic is part of the toolkit. Obviously, if the knowledge we are talking about is so unreliable it's apt to backfire in any case, the ethical route is to hold off until our knowledge is good enough for us to reliably obtain the intended outcome.

    Interestingly, though logic is morally neutral, it can help us recognize false reasons for actions, and in this way can help comprise a moral framework. So we might make a moral statement that false logic is immoral, but it's easy enough to construct a thought experiment that contradicts this. I think it's better to say that unsound reasoning can't be part of a good rationale for doing something, though the objective itself might otherwise be laudable.

    Going back to the contrast between the doctors of 1900 (with their abysmal obstetric record) and today, what's made the difference? I'm going to go out on a limb and say that modern doctors' reading more of the Bible isn't going to be high on the list. Solid research in medicine - and, of course, sound reasoning - are much more plausible explanations.
    Last edited by chriscase; September 25, 2017 at 02:11 PM.

    Why is it that mysteries are always about something bad? You never hear there's a mystery, and then it's like, "Who made cookies?"
    - Demetri Martin

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    Default Re: Applications of Russell's Paradox

    chriscase,

    If I may I'll begin with your last statement first. When I had my heart valve replaced the replacement part was a pig's valve, inserted by a Muslim surgeon into a born again Christian and I never batted an eyelid about that. The Bible may not have played a part as far as the pig was concerned or the surgeon but it certainly put me at ease throughout the experience, why? Because I knew that God was in control and I was in a win win situation. Therefore I do not claim that a doctor who does read the Bible is any greater than the doctor who doesn't. It certainly plays a part where abortion is concerned because it brings into question when life actually begins. If it doesn't begin at conception and build on that why is it that many operations are now done inside the mother's womb to save a baby and yet it is OK to tear/disect a fetus by the million every year not necessarily for medical purposes, rather personal intention? Where stands the Hypocratic oath then?

    You began with knowledge which of course is a good thing provided it is used for the benefit of man. Whether all knowledge is morally right is quite another question. One man's meat is another man's poison comes to mind. Jesus said that He is the truth, the way and the life and I now believe that whence once I didn't. Therefore if you can believe that He died on a cross in your place to get rid of all your imperfections then you are well on the way to never having to experience immorality or imperfection ever again. Of course we retain our memories even when born again and it is these that spring up all the time to make one doubt, stumble and fall even so one has to fight that by not looking back just as Jesus logically said. To God there is nothing morally neutral, one is either on His side or on the other. So, there are two knowledges, the worldly and the Spiritual, having both outweighs anything.

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