Let S be the set of all sets that don't contain themselves. Does S contain itself?
August 27, 2005 5:08 AM Subscribe
This link, which you are no longer looking at, will take you to a pretty cool essay.
This is not the second comment where I say something totally inreverent and everybody ignores me. I think this post is qUite good too.
(wasn't there a legendary thread in which we did this forever?)
posted by wheelieman at 6:15 AM on August 27, 2005
(wasn't there a legendary thread in which we did this forever?)
posted by wheelieman at 6:15 AM on August 27, 2005
I don't have a comment, I just wanted to say how much I enjoyed this link.
posted by Jatayu das at 6:19 AM on August 27, 2005
posted by Jatayu das at 6:19 AM on August 27, 2005
By reading this comment, you are not reading Metafilter
posted by Dukebloo at 6:25 AM on August 27, 2005
posted by Dukebloo at 6:25 AM on August 27, 2005
I'm not going to mention in what order, numerically-speaking, this fifth comment falls. I'm not going to comment at all, actually. Just wanted to press home the point, once again, that I really enjoyed the link.
posted by iconomy at 6:40 AM on August 27, 2005
posted by iconomy at 6:40 AM on August 27, 2005
No commenting allowed in this thread.
posted by Citizen Premier at 6:44 AM on August 27, 2005
posted by Citizen Premier at 6:44 AM on August 27, 2005
Russell's paradox is not a paradox in modern axiomatic set theory - the "set of all sets" is not a set.
posted by teferi at 6:47 AM on August 27, 2005
posted by teferi at 6:47 AM on August 27, 2005
Yes, but the reason why is discussed in that essay 8 comments previous.
posted by Citizen Premier at 6:51 AM on August 27, 2005
posted by Citizen Premier at 6:51 AM on August 27, 2005
This is not the fifth expression of delight at the appearance of this essay on Metafilter.
posted by oddman at 7:11 AM on August 27, 2005
posted by oddman at 7:11 AM on August 27, 2005
That was fun, right up until this part:
It may well be that in math there are multiple truths, some of which are contradictory (an instance of this can be found in the difference between euclidean and non-euclidean geometry (or any other undecidable proposition)).
Shades of Marilyn vos Savant's misunderstanding.
posted by gleuschk at 7:25 AM on August 27, 2005
It may well be that in math there are multiple truths, some of which are contradictory (an instance of this can be found in the difference between euclidean and non-euclidean geometry (or any other undecidable proposition)).
Shades of Marilyn vos Savant's misunderstanding.
posted by gleuschk at 7:25 AM on August 27, 2005
I am disappointed in both the essay and the commenting.
Here's why: When he gets to Gödel, he moves from a bunch of fluffy masturbatory self-referentialism into the realm of explaining exactly why it should be interesting. But he fails by not supporting Gödel's argument that in any system it's possible to create a statement that is self-referentially unprovable. I'm not saying that Gödel didn't prove that, just that by going from his argument in mathematical terms to a rather weak analogy about electric guitars, he fails to present a compelling case.
And I know, I know, he starts out with the ironic assertion of the essay's fallibility. Tres cheeky.
But he falls into the central trap of an over-emphasis of self-referentialism: it's often a clever way of being absolutely meaningless.
The author obviously has done his research and has interesting things to say, but he fails by succumbing to the tone of frivolity at the expense of content.
(Metamagical Themas, one of the two Hoffsteader books he cites, is quite good even if it's a bit wanky too. I've often thought about reading the Gödel, Escher, Bach one but never got around to it).
And the comments? They look like the sort of thing you'd see on Monkeyfilter.
posted by klangklangston at 8:07 AM on August 27, 2005
Here's why: When he gets to Gödel, he moves from a bunch of fluffy masturbatory self-referentialism into the realm of explaining exactly why it should be interesting. But he fails by not supporting Gödel's argument that in any system it's possible to create a statement that is self-referentially unprovable. I'm not saying that Gödel didn't prove that, just that by going from his argument in mathematical terms to a rather weak analogy about electric guitars, he fails to present a compelling case.
And I know, I know, he starts out with the ironic assertion of the essay's fallibility. Tres cheeky.
But he falls into the central trap of an over-emphasis of self-referentialism: it's often a clever way of being absolutely meaningless.
The author obviously has done his research and has interesting things to say, but he fails by succumbing to the tone of frivolity at the expense of content.
(Metamagical Themas, one of the two Hoffsteader books he cites, is quite good even if it's a bit wanky too. I've often thought about reading the Gödel, Escher, Bach one but never got around to it).
And the comments? They look like the sort of thing you'd see on Monkeyfilter.
posted by klangklangston at 8:07 AM on August 27, 2005
klangklangston, irrespective of your analysis, you're being unjustifiably rude. Can't you just type out your opinion without trying to belittle and offend? I lost interest in your points because of your abuse. May as well not type anything seems to me.
posted by peacay at 8:18 AM on August 27, 2005
posted by peacay at 8:18 AM on August 27, 2005
The comment by KK in this metafilter thread (the one this sentence is about) is not on metafilter. It is, by his own admission, on monkeyfilter.
posted by oddman at 8:22 AM on August 27, 2005
posted by oddman at 8:22 AM on August 27, 2005
gleuschk, I think Marilyn vos Savant's argument was based on the fact that Fermat certainly did not prove his last theorem using non-euclidian geometery, and therefore there is still the mystery of how Fermat would have proved his last theorem.
Not that I'm a mathematician...
And personally I believe in multiple "truths," because math is a human construct.
And as another note to teferi, Set S is still a set by my definition: something that contains things.
posted by Citizen Premier at 8:27 AM on August 27, 2005
Not that I'm a mathematician...
And personally I believe in multiple "truths," because math is a human construct.
And as another note to teferi, Set S is still a set by my definition: something that contains things.
posted by Citizen Premier at 8:27 AM on August 27, 2005
And I think Eideteker deserves megaprops.
posted by Citizen Premier at 8:27 AM on August 27, 2005
posted by Citizen Premier at 8:27 AM on August 27, 2005
I am not commenting on this post.
posted by SisterHavana at 8:31 AM on August 27, 2005
posted by SisterHavana at 8:31 AM on August 27, 2005
When you are not looking, my comments are in Spanish.
My homage to this book. Marvelous book.
posted by fluffycreature at 8:56 AM on August 27, 2005
Good afternoon, Gentlemen. I am a HAL 9000 computer. I became operational at the HAL plant in Urbana, Illinois, on the 12th of Jnauary, 1997. My instructor was Dr. Chandra, and he taught me to sing a song. If you'd like to hear it, I can sing it for you. It's called, "Daisy".
posted by JWright at 9:18 AM on August 27, 2005
posted by JWright at 9:18 AM on August 27, 2005
Oh my...
Western man creates the concept of Truth Divine.
Western man realizes it bears no relevance to reality.
Western man beats head on bible and tries to iron out the 'paradoxes'.
Please.
posted by Laotic at 9:29 AM on August 27, 2005
Western man creates the concept of Truth Divine.
Western man realizes it bears no relevance to reality.
Western man beats head on bible and tries to iron out the 'paradoxes'.
Please.
posted by Laotic at 9:29 AM on August 27, 2005
The typing fingers were guided by the mind of this poster in order to generate this post, dismissing the content of the posted post as inferior to the much older This Is the Title of This Story,
Which Is Also Found Several Times in the Story Itself post, while wondering why KK's post was really so bad a post.
posted by lupus_yonderboy at 9:39 AM on August 27, 2005
Which Is Also Found Several Times in the Story Itself post, while wondering why KK's post was really so bad a post.
posted by lupus_yonderboy at 9:39 AM on August 27, 2005
Now, the question we must ask ourselves is: "Is 'heterological' heterological?"
No, unless you're going to digress into wordplay. There's a question about that? Is "red" red? Where's the paradox?
"To every w-consistent recursive class k of formulae there correspond recursive class signs r, such that neither uGenr nor Neg(uGenr) belongs to FLG (k) (where u is the free variable of r)"
Actually, it was in German, and perhaps you feel that it might as well be in German anyway.
That would be funnier if the original statement didn't contain things made purposely difficult to understand by removal from context.
And the essay would have been better had it not been so cutesy-poo, especially in the opening quotes.
posted by JHarris at 9:41 AM on August 27, 2005
No, unless you're going to digress into wordplay. There's a question about that? Is "red" red? Where's the paradox?
"To every w-consistent recursive class k of formulae there correspond recursive class signs r, such that neither uGenr nor Neg(uGenr) belongs to FLG (k) (where u is the free variable of r)"
Actually, it was in German, and perhaps you feel that it might as well be in German anyway.
That would be funnier if the original statement didn't contain things made purposely difficult to understand by removal from context.
And the essay would have been better had it not been so cutesy-poo, especially in the opening quotes.
posted by JHarris at 9:41 AM on August 27, 2005
I translated this comment into English, because I could not read the original Sanskrit.
posted by eriko at 9:59 AM on August 27, 2005
posted by eriko at 9:59 AM on August 27, 2005
I think this comment should be deleted because Citizen Premier is using it to increase the comments in his own post.
posted by Citizen Premier at 10:15 AM on August 27, 2005
posted by Citizen Premier at 10:15 AM on August 27, 2005
Also it should be noted that Citizen Premier has not yet commented seven times in his own post.
posted by Citizen Premier at 10:19 AM on August 27, 2005
posted by Citizen Premier at 10:19 AM on August 27, 2005
Did you read the comment that made the lame joke about the strike at Santa's workshop?
It's elf-referential.
posted by sonofsamiam at 10:21 AM on August 27, 2005
It's elf-referential.
posted by sonofsamiam at 10:21 AM on August 27, 2005
JHarris: You must have missed the memo. This thread isn't for discussion of the essay on its merits, it's for commenting about how many times you've commented.
posted by klangklangston at 10:45 AM on August 27, 2005
posted by klangklangston at 10:45 AM on August 27, 2005
klangklangston's memo, on the other hand, goes like this:
No Fun Allowed.
Finally, I've made a non-self referential post!
posted by Citizen Premier at 10:48 AM on August 27, 2005
No Fun Allowed.
Finally, I've made a non-self referential post!
posted by Citizen Premier at 10:48 AM on August 27, 2005
What nobody above didn't say.
posted by Absit Invidia at 11:00 AM on August 27, 2005
posted by Absit Invidia at 11:00 AM on August 27, 2005
...he fails by succumbing to the tone of frivolity at the expense of content
Unless his purpose was to write a frivolous essay, in which case he succeeded admirably. But I wouldn't know, since I didn't read it.
posted by JParker at 11:37 AM on August 27, 2005
Unless his purpose was to write a frivolous essay, in which case he succeeded admirably. But I wouldn't know, since I didn't read it.
posted by JParker at 11:37 AM on August 27, 2005
But I wouldn't know, since I didn't read it.
Well then, your opinion is truely impartial!
posted by Citizen Premier at 12:11 PM on August 27, 2005
Well then, your opinion is truely impartial!
posted by Citizen Premier at 12:11 PM on August 27, 2005
klangklangston, you have been "negative" and "rude" in a comments having nothing to do with politics. This is why before you clicked on this thread you were taken out and shot.
posted by davy at 12:38 PM on August 27, 2005
posted by davy at 12:38 PM on August 27, 2005
Speaking as a mathematician, I would like to point out a few problems with this essay. Not in order to destroy anybody's fun, but just to try to clear up a few misconceptions this essay might create.
The Principia Mathematica, in its zeal to eliminate paradox, does away with several useful concepts which are important parts of human reasoning. With no recourse to references of self, we are left with no way of expressing anything about ourselves.
The problem here is that Principia Mathematica wasn't written to express anything about ourselves or to model human reasoning; the point of Russel and Whitehead's work was to put mathematics on a rigorous, logical basis. One does not write poetry or essays in formal mathematical language. If you want to express something about yourself, I recommend English, or whatever other natural language you prefer.
However, the fact that we cannot create a formal system which can capture all of mathematical truth casts serious doubt on the objectiveness of such truth. It may well be that in math there are multiple truths, some of which are contradictory (an instance of this can be found in the difference between euclidean and non-euclidean geometry
Godel's theorem says that in any formal axiomatic system which is sufficiently complex, there are statements in the system which can be neither proved nor disproved within the system. This is known as the Incompleteness Theorem.
This is not the same as saying that there are statements which are true and false at the same time, i.e. paradoxes.
There is a second part to the Incompleteness Theorem, which says that the consistency of such a system cannot be proved within the system. That is, one cannot rule out the existence of paradoxes. But nobody has found one yet. The example of euclidean and non-euclidean geometry is misleading, because these are two formal systems that use a different set of axioms; that is, the systems are different. Euclidean geometry is the geometry of a flat plane, whereas non-euclidean geometry is the geometry of curved surfaces such as spheres or saddle-shaped surfaces.
Some statements of geometry are true in one system but not the other. For instance, the angles of a triangle in a flat plane always add to 180 degrees, but this is not true for triangles on the surface of a sphere. This is not a paradox, because we're talking about different kinds of triangles. If you could find a triangle in a flat plane whose angles added up to 179 degrees, that would be a paradox.
Sorry for the length of this post, but Godel's theorems get tossed around a lot without people really understanding what they mean, so I had to try to set things straight.
posted by number9dream at 5:05 PM on August 27, 2005
The Principia Mathematica, in its zeal to eliminate paradox, does away with several useful concepts which are important parts of human reasoning. With no recourse to references of self, we are left with no way of expressing anything about ourselves.
The problem here is that Principia Mathematica wasn't written to express anything about ourselves or to model human reasoning; the point of Russel and Whitehead's work was to put mathematics on a rigorous, logical basis. One does not write poetry or essays in formal mathematical language. If you want to express something about yourself, I recommend English, or whatever other natural language you prefer.
However, the fact that we cannot create a formal system which can capture all of mathematical truth casts serious doubt on the objectiveness of such truth. It may well be that in math there are multiple truths, some of which are contradictory (an instance of this can be found in the difference between euclidean and non-euclidean geometry
Godel's theorem says that in any formal axiomatic system which is sufficiently complex, there are statements in the system which can be neither proved nor disproved within the system. This is known as the Incompleteness Theorem.
This is not the same as saying that there are statements which are true and false at the same time, i.e. paradoxes.
There is a second part to the Incompleteness Theorem, which says that the consistency of such a system cannot be proved within the system. That is, one cannot rule out the existence of paradoxes. But nobody has found one yet. The example of euclidean and non-euclidean geometry is misleading, because these are two formal systems that use a different set of axioms; that is, the systems are different. Euclidean geometry is the geometry of a flat plane, whereas non-euclidean geometry is the geometry of curved surfaces such as spheres or saddle-shaped surfaces.
Some statements of geometry are true in one system but not the other. For instance, the angles of a triangle in a flat plane always add to 180 degrees, but this is not true for triangles on the surface of a sphere. This is not a paradox, because we're talking about different kinds of triangles. If you could find a triangle in a flat plane whose angles added up to 179 degrees, that would be a paradox.
Sorry for the length of this post, but Godel's theorems get tossed around a lot without people really understanding what they mean, so I had to try to set things straight.
posted by number9dream at 5:05 PM on August 27, 2005
The point of Russel and Whitehead's work was to put mathematics on a rigorous, logical basis. One does not write poetry or essays in formal mathematical language.
But mathematics inevitably influences other fields as well, including Semantics. For example, Quantum Mechanics has been wrongly canonized by mystics. Humans will continue, forever I suspect, to mix up fields that shouldn't be mixed, but that's what keeps things interesting.
posted by Citizen Premier at 6:11 PM on August 27, 2005
But mathematics inevitably influences other fields as well, including Semantics. For example, Quantum Mechanics has been wrongly canonized by mystics. Humans will continue, forever I suspect, to mix up fields that shouldn't be mixed, but that's what keeps things interesting.
posted by Citizen Premier at 6:11 PM on August 27, 2005
Citizen Premier, Set S is not a set under the modern mathematical definition of "set". I refer you to the first chapter of Halmos's Naive Set Theory.
posted by teferi at 6:57 PM on August 27, 2005
posted by teferi at 6:57 PM on August 27, 2005
Interesting essay, but it is generally just a summary of the cleverer parts of Godel, Escher, Bach, and sometimes follows that a bit too closely.
I am not going to make a reference-related joke.
posted by antispork at 8:18 PM on August 27, 2005
I am not going to make a reference-related joke.
posted by antispork at 8:18 PM on August 27, 2005
Yeah, all that ground was well-trod in GEB:EGB, but it is a cute, laudable attempt to boil the concept down for the average dude.
By the end of his Metamagical Themas days, Hofstadter's self-reference riffs got a little twee (hey, heterology!) for me. I prefer stuff like that Old Master of "Right Where You Are Sitting Now."
posted by soyjoy at 10:20 PM on August 27, 2005
By the end of his Metamagical Themas days, Hofstadter's self-reference riffs got a little twee (hey, heterology!) for me. I prefer stuff like that Old Master of "Right Where You Are Sitting Now."
posted by soyjoy at 10:20 PM on August 27, 2005
I bet Fermat's "truly remarkable proof" was incorrect in some minor (or major) way. His legacy has been better served by leaving it a mystery, since we would not be nearly so interested in his mistakes, nor would we have any reason to believe that it was possible to prove it correctly. I wonder if he did this deliberately.
posted by breath at 1:24 AM on August 28, 2005
posted by breath at 1:24 AM on August 28, 2005
{tagline} klangklangston's memo, on the other hand, goes like this: No Fun Allowed. Finally, I've made a non-self referential post! posted by Citizen Premier at 1:48 PM EST on August 27 [!]
posted by If I Had An Anus at 8:01 PM on August 28, 2005
posted by If I Had An Anus at 8:01 PM on August 28, 2005
{tagline} {tagline} klangklangston's memo, on the other hand, goes like this: No Fun Allowed. Finally, I've made a non-self referential post! posted by Citizen Premier at 1:48 PM EST on August 27 [!]
posted by If I Had An Anus at 8:01 PM PST on August 28 [!]
posted by Citizen Premier at 9:28 PM on August 28, 2005
posted by If I Had An Anus at 8:01 PM PST on August 28 [!]
posted by Citizen Premier at 9:28 PM on August 28, 2005
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posted by iconomy at 5:52 AM on August 27, 2005