I think you have a version of Russel's Paradox here and you are fucked. I suggest that Frodo eats Sam or you ask the dragon to adhere to ZF and get back to you.

Zermelo–Fraenkel set theory

Has to do with a proposition that tries to predicate itself. The math whizzes here could shed some light.

I think Zermelo's solution was "don't do that".

Animavore wrote:I'm assured this has a correct answer.

They wouldn't lie to me?

Who told you that? Did he say "this statement is a lie"?

I need to get time to watch the whole thing.

His commentators didn't seem to have an answer. The kicker is that you set up a definition of a set with a predicate. This is called intensional. But the creation of the set changes what would be included in it and if that is included than it oscillates back again. And so on. The way around it would have to do with setting conditions in time or something.

But I am just embarrassing myself here. I'm a pup at this stuff.

To address the main question posed by the video; consciousness does at first glance appear to be a self-referential paradox. Wittgenstein remarked, “Reflection brings obscurity- which is the result of the shadow cast by the inquirer himself.” Trying to understand consciousness from within the framework of consciousness accumulates self-referential baggage although what other option do we have for understanding our respective internal and external universes?



The white cup or vase is the figure and the negative black space is the ground. The ground (consciousness) is the background or arena in which objects of perception (the figure) appear. The ground shapes our experience of the figure and vice versa; from the formless ground, the form of the figure emerges. Consciousness is the background in which the world appears to us, it gives form to particular objects and universal patterns. This relationship is recursive because although consciousness shapes the world in which we perceive; ultimately the nature of the world shapes consciousness in the way that the ground is shaped by the form of the figure.

This recursive relationship breaks down the distinctions between subject and object when we realize that we are not observers observing the universe but instead we are the universe observing itself.

Animavore wrote:I'm assured this has a correct answer.

They wouldn't lie to me?

zoon:
Does the video say there’s a correct answer? I think he avoids committing himself to that, and just says “I’ll leave that to you, my viewers, to discuss in the comments section”. I’m a non-expert too, I’m vaguely aware that variants of the liar paradox have been around at least since the Cretan Epimenides is said to have asserted that all Cretans are liars. These paradoxes are still live subjects for philosophy of mathematics. The kind of possible resolution mentioned near the end of the video, “the truth or falsity of a statement is not a property of the statement, but a judgment of the mind that evaluates the statement” looks somewhat like the levels of language resolution proposed by Tarski (I’m going by the [url=http://en.wikipedia.org/wiki/Liar_paradox#Alfred_Tarski]Wikipedia article[url] again, which is about my level of understanding.)

My own view about consciousness is that it’s primarily a judgment we make about other people, and that when we refer the judgment back to ourselves we do indeed get entangled in self-referential paradoxes. I see humans as biological organisms, complex robots, that carry around a model of the world in our brains so that we can predict the objects in the world and then decide what to do. Some of the objects out there are other people, which are especially difficult to predict – the machinery’s very fast, very complex, and hidden inside the skull. So humans have evolved a cunning indirect way of predicting other people, known as Theory of Mind. To predict another person, I set up a simulation of their model of the world in my brain, then use that simulation to guess what they will decide to do. This works pretty well because we are all the same species, with similar brains. It all happens automatically: I’m not aware of deciding to set up a copy model of the world in my brain to predict the other person, I just see that person as conscious, I reify the extra model of the world in my brain as that person’s consciousness. I classify the things in my main model of the world as real, for example I may be looking at a tree with a squirrel on my side of the tree trunk; both the tree and the squirrel which I see are straightforwardly classified as real. The things in the other person’s consciousness are classified as essentially unreal, for example, if the other person is on the far side of the tree and can’t see the squirrel and doesn’t know it’s there, I think of their model of the world as wrong, missing one squirrel.

It’s when I start trying to predict how other people will predict me that the self-referential paradox comes in. The other person will see me as conscious, with a model of the world which is now inside the model which I’ve set up of their model. If I reclassify the tree and the squirrel which I can see, and which I assumed were real, as merely part of my consciousness like another person’s consciousness, then I’ve set up a recursion in which it becomes impossible to say what’s real and what isn’t. If we philosophise on the matter we end up like computers taking pictures of their own monitors and wondering what’s gone wrong.

The original Theory of Mind trick of modelling or mirroring other people’s brains in order to guess what they will do is a straightforwardly mechanistic evolved way of improving our predictions of the world. It’s when we turn it back on ourselves that we become mired in paradoxes. Is my guess.

Pyhrro wrote:This recursive relationship breaks down the distinctions between subject and object when we realize that we are not observers observing the universe but instead we are the universe observing itself.

Yes, I think the answer has to be something to do with seeing ourselves as mechanisms, no different from the rest of the world, but I’m not at all clear how to combine this with human sociality.

The way to solve it is to get the dragon to make a commitment about the future. That would collapse the problem I think. The paradox happens when something is constructed in the absolute present from information that is not. The same thing is true of trying to introspect your consciousness ins some ways. It's like picking up a marble with a fork.

For a very long time I tried to do this introspection and kept coming up with a belief that there was some impossible thing going on. when I approached the problem from an entirely different direction the hard problem started to evaporate. All that is left of it now is a stain.

The confusion (not to be conflated with paradox) here stems from: "'If you can predict correctly what I will do with Sam... ". However, what is necessary here is that the dragon should already know what shall be done to Sam (the dragon has a preconceived idea of what it wants to do to Sam). Thus, the dragon isn't really asking Frodo to predict the future, but is asking instead for Frodo to guess what its plans are. This must be the case, since otherwise how would the dragon judge whether Frodo was wrong?

This is being viewed as a paradox because Frodo's prediction actually turns out to be correct, if Sam is eaten. However, the dragon asked Frodo to predict what it originally intended to do with Sam, not what would happen to Sam. There is no paradox, therefore, if Sam should be eaten after Frodo predicts that he would, since what Frodo predicts is still at-odds with what the dragon asked him to predict.

There's only a 50/50 chance of Sam getting out alive. Frodo must predict not what will happen to Sam, but what he thinks the dragon wants to do to him. Therefore, he must keep his prediction as general as possible, such as (You will make Sam happy/unhappy). However, since the dragon is obviously troublesome, it is almost certain that he will make Sam unhappy. I would therefore say this. Pending a further demand for detail, Sam would then be free.

Failing that, kick the dragon in the balls and do a runner.

I didn't see this before because I avoid threads with the c-word in the title.

OK, here's the solution. You have to come at this from the perspective of game theory not set theory.

As soon as the dragon finishes outlining the terms and conditions he gets a sinking feeling. What has he done? His honour and truthfulness are more important to him than anything and he's gone and put himself in a difficult position here. If Frodo predicts that the dragon will eat Sam then the dragon cannot keep his word whatever he does. That would be a disaster. The dragon really needs Frodo to predict that he will release Sam. If that happens he's free to release or eat the hobbit as he chooses without breaking the rules, so it will be his chance to devour Sam with a clean conscience.

Frodo knows that if he predicts Sam's release he's signing his death warrant. The dragon will simply declare him wrong and reach for the ketchup. If Frodo predicts that the dragon will eat Sam then there's no way of telling what will happen. It doesn't look good, but perhaps the dragon will release Sam in a fit of remorse at being forced to break his word, or maybe they'll be able to escape while the dragon has some kind of reptilian nervous breakdown. Predicting death is looking like the better option, but it's still a huge risk.

However, Frodo looks at the situation from the dragon's point of view and realises that it's in the dragon's interest for him to predict Sam's release. Perhaps he can get the dragon to offer him an incentive to do just that...

"Tell me, O Great Wyrm," our hero calls out. "What will you do if I predict that you will return my companion unharmed?"

The dragon now has three options.

1. Refuse to answer the question. After all, he's under no obligation. But he realises that as things stand Frodo's best bet is to predict death, which is his worse case scenario. This option doesn't change that situation.

2. Say that he will eat Sam. But if he says it then he'll have no choice but to do it Frodo predicts Sam's release. Frodo would then have zero incentive to predict release and would definitely predict death, which the dragon doesn't want.

3. Say that he will release Sam. This is passing up a meal because it allows Frodo a guarantee that he can correctly predict that Sam will be spared. It is, however, the only way to prevent Frodo predicting Sam's death and thereby salvage his honour.

So the dragon reluctantly says he will free Sam. Frodo makes his prediction and Sam is released. He bursts into tears on his master's shoulder and blubbers about what his old Gaffer would make of it all until suddenly the dragon swallows Frodo in one gulp because he can do that, and nobody likes a smart arse. Then he slinks into the woods, resolving to just eat people in future and not get drawn into stupid riddles.

This riddle is a paradox called the Crocodile Dilemma, see wiki and mathworld.

After a PM, here's the geek post anyway.

The point of set theory is that we want sets to be properties in extension. For every property or predicate we'd like to have a set which is the set of all things that have that property/for which that predicate is true. Russel's paradox kind of sinks that - if you try to have a set of sets which are not members of themselves paradox results.

As an aside, two other informal variants of Russel's paradox. The first is called Grelling's paradox. Some adjectives describe themselves e.g. pentasyllabic, English, short. Others don't. Does the adjective non-selfdescriptive describe itself? Then there's the barber paradox. The town barber shaves precisely those townsmen who don't shave themselves. Does the barber shave himself?

The response to the problem is to water down the idea that all properties have corresponding sets. The ZF way of doing this is based on the fact that paradoxical sets, if they could exist, would be very, very big indeed. A major rationale for ZF is avoiding creating excessively large sets. The axioms of ZF give a single infinite set and tools for building new sets from existing ones that are all about keeping the size down.

Rather than saying that for any property there is a set of all things with that property, ZF says that given a set and a property there is a subset consisting of all the members of the set that have the property. Assigning sets to properties now can't give a larger set than those we had already. Mathematicians are trained, whenever they specify a set, to specify a larger set of which it is a subset.

In particular, there are no self-membered sets in ZF and no universal set. On the other hand, there's an extensive theory of sets large enough not to exist in ZF, together with the additional axioms that would be required to bring them in to being, suggesting that a certain amount of baby got thrown out with the bath water.

An alternative response is type theory. Instead of limiting the size of sets, this approach limits what properties are admissible. Very loosely, a type one property is a property of things. A type 2 property can be a property of things and of type 1 properties, and so on. Only properties that have a type are admissible and no typed property can refer to itself. With this it's safe to say that every typed property has a corresponding set.

There are alternative set theories based on type theory, most notably Quine's New Foundations. In NF there is a universal set of all sets and there are self-membered sets. Bizarrely this include self membered singletons or Quine atoms Q = {Q}. Go figure. But there are no paradoxical sets. These systems are in some ways more natural than the slightly clunky aspects of ZF, but some of the technical workings are cumbersome. Whatever the pros and cons, ZF has become the mainstream option which maths is based on.

We've asked whether all properties have a set to go with them and found that they don't. The question also arises of whether all sets have a property that goes with them. One of the objections to the Axiom of Choice is that it violates this. The AoC says that, given any set of sets, there is a choice set which contains one member of each of them. Such a set lacks a unifying property and so, it could be argued, goes against the spirit of what set theory is supposed to be for.

logical bob wrote:The response to the problem is to water down the idea that all properties have corresponding sets. The ZF way of doing this is based on the fact that paradoxical sets, if they could exist, would be very, very big indeed. A major rationale for ZF is avoiding creating excessively large sets. The axioms of ZF give a single infinite set and tools for building new sets from existing ones that are all about keeping the size down.

Don't you need the Axiom of choice to discuss the relative sizes of the infinite sets? I'm struggling to remember, but for some reason I have my doubts that the structure of the ordering of ZF¬C can be described this neatly.

Your well made point still stands if we read ZF as ZFC, of course.