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May. 1st, 2007 03:55 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stormdogAfter reading the first chapter of Gödel, Escher, Bach: An Eternal Golden Braid on the way home from work yesterday, I've been stuck on the concept of strange loops. The example that's most intriguing me right now is this one.
Say there are two different types into which all sets of things will fit: sets that contain themselves and sets that do not. An example of a set that contains itself as a member might be the set of all sets. Because it is a set, it is itself a member of the set. An example of a set that does not contain itself might be the set of all computers that I own. The set itself is not a computer and I do not own it, therefore, it is not a member of the set.
Now suppose you have a set that consists of all sets that do not contain themselves as a member of the set. Which type of set is it? This causes a strange loop in reasoning. If it contains itself then it cannot be a member of the set but if it is not a member of the set then it does not contain itself and therefore it must be a member of the set, but if it is a member of the set, then it can't contain itself. It's a circular path that can be followed to infinity.
One way to resolve this might be to posit that the initial assumption is flawed; there must be more than two kinds of sets. But if that's the case, how can that third type be defined? Sets that are both? Sets that are neither? Each of those definitions has its own logical problems. Can it be proved that there are more than these two kinds of sets?
That night, I was reading the copy of Chemistry for Dummies that I put in the bathroom to go through while I'm, well, doing what it is that people do in the bathroom (No, not that you sicko.). I read about electron orbitals and the quantum properties of electrons. I read about how an orbital, defined by an electron's energy level (principle quantum number) and it's pattern of orbit (it's angular momentum quantum number) are really just a description of where an electron is most likely to be. And it occurred to me that the logical strange loop posed by the conflicted set above is just as uncertain as the attributes of a particle.
That set, and it's type (either a set that contains itself as a member or one that doesn't), exist as a sort of quantum superposition of it's two possible states, just as Schröedinger's cat, until you open the box, is a quantum superposistion of a live cat and a dead cat. You have to look at it to resolve it definitively into one or the other. When you analyze it logically, it flips back and forth from one to the other like an air bubble that slips back and forth under your finger as you try to squeeze it out from under a window cling. First it's in one set, then the other. First it's in one set, then the other. As long as you're actively trying to analyze it, you can keep pinning it down to one thing or the other. But when you're not looking at it, when you're not thinking about it, when it's free to be what it will, then it's both. And neither.
If it weren't for the fact that the first chapter of GEB is this amazing meshing of Bach's canons with Escher's infinite loops with Gödel's ideas on number theory, I very much doubt that the idea would have occurred to me. But it is, and it did, and because of the oddly smooth way ideas in GEB meshed together, I've been in a frame of mind wherein I've been trying to mesh other things together too.
Does the idea of a logical construct within a formal system existing as an unresolvable quantum uncertainty make sense to anyone? Is it a load of horse shit? Is it addressed further into the book? I guess we shall see....
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infintysquared, thanks so much for the loan of this book! I must own a copy.)
Say there are two different types into which all sets of things will fit: sets that contain themselves and sets that do not. An example of a set that contains itself as a member might be the set of all sets. Because it is a set, it is itself a member of the set. An example of a set that does not contain itself might be the set of all computers that I own. The set itself is not a computer and I do not own it, therefore, it is not a member of the set.
Now suppose you have a set that consists of all sets that do not contain themselves as a member of the set. Which type of set is it? This causes a strange loop in reasoning. If it contains itself then it cannot be a member of the set but if it is not a member of the set then it does not contain itself and therefore it must be a member of the set, but if it is a member of the set, then it can't contain itself. It's a circular path that can be followed to infinity.
One way to resolve this might be to posit that the initial assumption is flawed; there must be more than two kinds of sets. But if that's the case, how can that third type be defined? Sets that are both? Sets that are neither? Each of those definitions has its own logical problems. Can it be proved that there are more than these two kinds of sets?
That night, I was reading the copy of Chemistry for Dummies that I put in the bathroom to go through while I'm, well, doing what it is that people do in the bathroom (No, not that you sicko.). I read about electron orbitals and the quantum properties of electrons. I read about how an orbital, defined by an electron's energy level (principle quantum number) and it's pattern of orbit (it's angular momentum quantum number) are really just a description of where an electron is most likely to be. And it occurred to me that the logical strange loop posed by the conflicted set above is just as uncertain as the attributes of a particle.
That set, and it's type (either a set that contains itself as a member or one that doesn't), exist as a sort of quantum superposition of it's two possible states, just as Schröedinger's cat, until you open the box, is a quantum superposistion of a live cat and a dead cat. You have to look at it to resolve it definitively into one or the other. When you analyze it logically, it flips back and forth from one to the other like an air bubble that slips back and forth under your finger as you try to squeeze it out from under a window cling. First it's in one set, then the other. First it's in one set, then the other. As long as you're actively trying to analyze it, you can keep pinning it down to one thing or the other. But when you're not looking at it, when you're not thinking about it, when it's free to be what it will, then it's both. And neither.
If it weren't for the fact that the first chapter of GEB is this amazing meshing of Bach's canons with Escher's infinite loops with Gödel's ideas on number theory, I very much doubt that the idea would have occurred to me. But it is, and it did, and because of the oddly smooth way ideas in GEB meshed together, I've been in a frame of mind wherein I've been trying to mesh other things together too.
Does the idea of a logical construct within a formal system existing as an unresolvable quantum uncertainty make sense to anyone? Is it a load of horse shit? Is it addressed further into the book? I guess we shall see....
(
infintysquared, thanks so much for the loan of this book! I must own a copy.)
