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What's even crazier to me is that there are statements that aren't self-referential that are both not true and not false at the same time. For instance the cont

What's even crazier to me is that there are statements that aren't self-referential that are both not true and not false at the same time. For instance the continuum hypothesis [1], or anything to do with the axiom of choice, like the trippy Banach-Tarski paradox [2]. [1] http://en.wikipedia.org/wiki/Continuum_hypothesis http://en.wikipedia.org/wiki/Continuum_hypothesis [2] http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

jerf 14y ago
The continuum hypothesis sort of fits your description, but Banach-Tarski does not. If you take the axiom of choice in your system, Banach-Tarski is simply true (in that the described construction will create two spheres out of one). If you do not take the axiom of choice, it is simply false (in that it can not be defined). (And I'm sure somebody somewhere has worked out some sort of in-between state there, but let's keep it simple.) While it's a very interesting thing to think about, there is one sense in which the paradox is trivially resolvable; while the axiom of choice may make sense in mathematics, it almost certainly is never used by the real universe, and you certainly can not construct two real spherical shells made of atoms of the same size as one original spherical shell in the real world. (Whether the universe is continuous is an open problem, but we know matter isn't.) A great deal of the reason why the mind rebels at the Banach-Tarski paradox is that it is built on a thoroughly aphysical axiom; that the aphysical axiom permits aphysical (and therefore counterintuitive) results is not that surprising. By no means is that a criticism of BT, it is simply its nature. My personal position on constructivist vs platonic vs. blah blah blah is that in math, as long as you specify which axiom set you are using up front, there is nothing to be emotional about. (Except that I will reserve a special place of interest for whatever mathematics it turns out to be that precisely represents the real world. Alas, this is still a work in progress, though we can point to at least some characteristics of it.)
- btilly 14y ago
  But it does fit the description. There is absolutely no statement that can be proven in standard math about the integers whose truth depends on the truth or falseness of Banach-Tarski. (More technically there is a theorem that any statement that can be proven about ZFC can be proven in ZF alone - this falls out of the construction that Goedel used to prove that ZFC is consistent if ZF is. The same construction and result falls out for many other axioms that you can add to ZF, including the closure of ZF is consistent, ZF + the consistency of ZF is consistent, ZF + the consistence of ZF plus the consistency of THAT is consistent, etc.) Therefore the truth of Banach-Tarski has nothing to do with any statement you can readily make about the integers, which would include any statement that you can make about anything that can be done on a Turing machine. Incidentally the whole debate about constructivism is much more subtle than you likely appreciate. For instance it is trivial to prove that almost all real numbers that exist, cannot be written down. (Proof, I can easily write out a countably infinite set of statements that could define real numbers. So the set of real numbers that can be written down is countable. But there are an uncountable number of real numbers, so almost all cannot be written down.) In what sense do said real numbers actually exist? If our axioms say that numbers must exist, which have in a very real sense no actual existence, then are those axioms capturing what we want to mean by "exist"? If you say it does not, then you're a constructivist who hasn't thought carefully enough yet. If you say it does make sense, then you've been brainwashed by decades of standard mathematical presentations into accepting a definition of "existence" that makes no sense to the average lay person. :-P
- jules 14y ago
  > If you do not take the axiom of choice, it is simply false Nitpick: If you do not choose the axiom of choice, then Banach-Tarski is independent, not false. So you can't prove that it's true and you can't prove that it's false.
- dmvaldman 14y ago
  "If you take the axiom of choice in your system, Banach-Tarski is simply true" - The exact same is true if you take the continuum hypothesis as an axiom. Both the axiom of choice and the continuum hypothesis are independent of the other standard ZF axioms "If you do not take the axiom of choice, it is simply false" - That is incorrect. It is only independent. To say it is false, another axiom(s) would need to contradict it. "while the axiom of choice may make sense in mathematics, it almost certainly is never used by the real universe" - neither are the real numbers. would you like to claim that pi, or the square root of 2, for that matter, also don't exist? How about the number 1?
derleth 14y ago
Banach-Tarski is true if you accept the axiom of choice. Which is not especially interesting: Every theorem depends on the axioms used to prove it; axioms are, in fact, what separates math from the natural sciences, and why 'truth' is available to mathematicians as long as you stipulate that a theorem is only absolutely true if you accept the axioms as absolute. None of this involves the kind of actual (or falsidical) paradoxes the incompleteness theorems do. (Note that the Banach-Tarski Paradox is a viridical paradox, as it is a true result that is only thought paradoxical because it contradicts naïve intuition. A falsidical paradox is one that shows a true flaw or limit in a logical system, such as Russell's Paradox is to naïve set theory. This terminology is due to Quine.)
- dmvaldman 14y ago
  not all axioms are created equally. some axioms are simple, like that the empty set exists. to think that we should have all these simple axioms, and then out of left field accept the continuum hypothesis as an axiom is a bit weird don't you think? also weird is that we can build up all this mathematical machinery from the ZF axioms, but still can't answer some seemingly innocuous questions. that's a good distinction you make in the second paragraph. to me though, the viridical paradoxes are more interesting :-)
  - lmm 14y ago
    >to think that we should have all these simple axioms, and then out of left field accept the continuum hypothesis as an axiom is a bit weird don't you think? It's like the parallel postulate - famously much more awkward and complicated than the other axioms of geometry, and you can do most of geometry the same with or without it. But even so, most of us choose to use standard euclidean geometry and accept the parallel postulate. >also weird is that we can build up all this mathematical machinery from the ZF axioms, but still can't answer some seemingly innocuous questions. I think of it like this: "the language of mathematics is expressive enough that we can formulate nonsense questions". If you look at it that way the incompleteness theorem isn't terribly surprising; in natural language any five year old can ask unanswerable questions.

jerf 14y ago

The continuum hypothesis sort of fits your description, but Banach-Tarski does not. If you take the axiom of choice in your system, Banach-Tarski is simply true (in that the described construction will create two spheres out of one). If you do not take the axiom of choice, it is simply false (in that it can not be defined). (And I'm sure somebody somewhere has worked out some sort of in-between state there, but let's keep it simple.) While it's a very interesting thing to think about, there is one sense in which the paradox is trivially resolvable; while the axiom of choice may make sense in mathematics, it almost certainly is never used by the real universe, and you certainly can not construct two real spherical shells made of atoms of the same size as one original spherical shell in the real world. (Whether the universe is continuous is an open problem, but we know matter isn't.) A great deal of the reason why the mind rebels at the Banach-Tarski paradox is that it is built on a thoroughly aphysical axiom; that the aphysical axiom permits aphysical (and therefore counterintuitive) results is not that surprising. By no means is that a criticism of BT, it is simply its nature. My personal position on constructivist vs platonic vs. blah blah blah is that in math, as long as you specify which axiom set you are using up front, there is nothing to be emotional about. (Except that I will reserve a special place of interest for whatever mathematics it turns out to be that precisely represents the real world. Alas, this is still a work in progress, though we can point to at least some characteristics of it.)

btilly 14y ago
But it does fit the description. There is absolutely no statement that can be proven in standard math about the integers whose truth depends on the truth or falseness of Banach-Tarski. (More technically there is a theorem that any statement that can be proven about ZFC can be proven in ZF alone - this falls out of the construction that Goedel used to prove that ZFC is consistent if ZF is. The same construction and result falls out for many other axioms that you can add to ZF, including the closure of ZF is consistent, ZF + the consistency of ZF is consistent, ZF + the consistence of ZF plus the consistency of THAT is consistent, etc.) Therefore the truth of Banach-Tarski has nothing to do with any statement you can readily make about the integers, which would include any statement that you can make about anything that can be done on a Turing machine. Incidentally the whole debate about constructivism is much more subtle than you likely appreciate. For instance it is trivial to prove that almost all real numbers that exist, cannot be written down. (Proof, I can easily write out a countably infinite set of statements that could define real numbers. So the set of real numbers that can be written down is countable. But there are an uncountable number of real numbers, so almost all cannot be written down.) In what sense do said real numbers actually exist? If our axioms say that numbers must exist, which have in a very real sense no actual existence, then are those axioms capturing what we want to mean by "exist"? If you say it does not, then you're a constructivist who hasn't thought carefully enough yet. If you say it does make sense, then you've been brainwashed by decades of standard mathematical presentations into accepting a definition of "existence" that makes no sense to the average lay person. :-P
jules 14y ago
> If you do not take the axiom of choice, it is simply false Nitpick: If you do not choose the axiom of choice, then Banach-Tarski is independent, not false. So you can't prove that it's true and you can't prove that it's false.
dmvaldman 14y ago
"If you take the axiom of choice in your system, Banach-Tarski is simply true" - The exact same is true if you take the continuum hypothesis as an axiom. Both the axiom of choice and the continuum hypothesis are independent of the other standard ZF axioms "If you do not take the axiom of choice, it is simply false" - That is incorrect. It is only independent. To say it is false, another axiom(s) would need to contradict it. "while the axiom of choice may make sense in mathematics, it almost certainly is never used by the real universe" - neither are the real numbers. would you like to claim that pi, or the square root of 2, for that matter, also don't exist? How about the number 1?

derleth 14y ago

Banach-Tarski is true if you accept the axiom of choice. Which is not especially interesting: Every theorem depends on the axioms used to prove it; axioms are, in fact, what separates math from the natural sciences, and why 'truth' is available to mathematicians as long as you stipulate that a theorem is only absolutely true if you accept the axioms as absolute. None of this involves the kind of actual (or falsidical) paradoxes the incompleteness theorems do. (Note that the Banach-Tarski Paradox is a viridical paradox, as it is a true result that is only thought paradoxical because it contradicts naïve intuition. A falsidical paradox is one that shows a true flaw or limit in a logical system, such as Russell's Paradox is to naïve set theory. This terminology is due to Quine.)

dmvaldman 14y ago
not all axioms are created equally. some axioms are simple, like that the empty set exists. to think that we should have all these simple axioms, and then out of left field accept the continuum hypothesis as an axiom is a bit weird don't you think? also weird is that we can build up all this mathematical machinery from the ZF axioms, but still can't answer some seemingly innocuous questions. that's a good distinction you make in the second paragraph. to me though, the viridical paradoxes are more interesting :-)
- lmm 14y ago
  >to think that we should have all these simple axioms, and then out of left field accept the continuum hypothesis as an axiom is a bit weird don't you think? It's like the parallel postulate - famously much more awkward and complicated than the other axioms of geometry, and you can do most of geometry the same with or without it. But even so, most of us choose to use standard euclidean geometry and accept the parallel postulate. >also weird is that we can build up all this mathematical machinery from the ZF axioms, but still can't answer some seemingly innocuous questions. I think of it like this: "the language of mathematics is expressive enough that we can formulate nonsense questions". If you look at it that way the incompleteness theorem isn't terribly surprising; in natural language any five year old can ask unanswerable questions.