Cynical Nation

Don't worry about categories; I've never had any patience for them.

But you'll still get traffic from people searching for this stuff on Google. It's been pretty cool, actually, finding out that blogging about math not only doesn't make all your readers go away, but in the long run, attracts new ones!

Go Long Tail!

I look forward to being utterly confused by the math of this paradox.

Well, on the bright side you always have the out of using alternative set theories that don't rely on the axiom of choice. :)

That's an interesting theory, though I can't see how it doesn't violate a basic Law of Thermodynamics - that matter and energy must be balanced/equal on both sides of the equation.

I don't understand how a mathmetician can postulate that you can creat a larger mass out of smaller one, without first theorizing something like, "given we can create mass/matter," in lieu of the ubiquitous a priori - "assuming a perfect vacuum," that seems to precede so many math theorems.

Are all the Laws of thermodynamics trumped by quatum mechanics?

JMK, you're confusing the model with the real world. Banach-Tarski requires that matter be infinitely divisible, which is not a property of matter in the real world. Hence speaking of it "violating" any physical law is just apples and oranges.

And no, the laws of thermodynamics are compatible with QM and hold in every context we know of -- even black holes.

I'll take your word for that Matt, but that confuses the hell out of me.

The laws of thermodynamics balance the matter/energy on both sides of every reaction/equation, so that's a tough concept for folks like me to grasp.

Matt--is there any real-world predictive power to the theory?

One of the "tricks" of the paradox is that the pieces are infinitely complex... and not even continuous, in general. In other words, they can be an arbitrary collection of discrete points.

there is no gap in reasoning with the banach-tarski paradox. some points

1:it has been proven in ZF set theory that the banach-tarski paradox is equiconsistant with the Hahn-Banach extension theorem. What this means is that if banach-tarski is in reality false so is the hahn-banach theorem. Take a look at a book on functional analysis and see what the consequences for hilbert spaces would be and by extension the mathematical model for quatum mechanics. The Hahn-Banach theorem is often describe as a conerstone of analysis. None of this involves choice so alternate set theories can't resolve the issue. my friend has a problem with ZF's power set axiom but he is to intelligent to be taken seriously. this work was done by dr. matt foreman he was at uci. if my memory serves me the proof uses forcing arguments.

2:the standard text on BT stan wagon's book presents the material in a way that renders the use of the axiom of choice mysterious. there are other proofs which use graph theory (bipartite matching theorems) that sort of clarify the issue.

3: Foreman and Doughetry poved a version of BT which runs as follows: it is possible to cut a pea into i think seven pieces and move them using rotations and translations, rigid motions, to form the sun. What's new? well the proof is in ZF using no choice of any form. the pieces form an OPEN set whose complementis nowhere dense in the pea and the same is true of the sun. What this means is that if you deleted the portions of the pea which are not used and the sun which are not filled up you couldn't see the difference. The original proof is very complicated.

4:the BT paradox is really about the uniqueness of the lebesgue integral and the possibility of extending it to an averaging functional on the entire powerset of a space loosing countable additivity but keeping invariance under some group action. The point is that this is not possible on the sphere. I wouldn't get overly worked up about geometric/mechanical metaphors like the tired pea-sun thing. physicist never tire of trying for mechanical explantations in realms where they don't apply (wave/particle duality?). I never know what it means to say that space is not "infinitely divisible" this may or may not be true i just don't know what it means. Again it is the metaphor thing. Am i to believe that the sqrt of 2 does not really exist where do you draw the line. All mathematicians mean by the continuum is something like dedikend cuts or equivalence classes of cauchy sequences. The possibility of uniqueness and extension theorems based on them.study the construction of the exponential functions to see what i mean. For sure we do not have infinite resolution in experimentation. As far as predictive power I read a paper linking hadron physics to BT but the argument was flawed because it relied on specifics of the wagon proofs which are not needed. I would venture the idea that BT like phenonmena are probably behind alot of the non-sense explanations in physics. what is so apriori about the sigma-algebra of lebesgue measurble sets anhyway?

I havn't thought about this for years so some of the specifics may be wrong.

there is no gap in reasoning with the banach-tarski paradox. some points

1:it has been proven in ZF set theory that the banach-tarski paradox is equiconsistant with the Hahn-Banach extension theorem. What this means is that if banach-tarski is in reality false so is the hahn-banach theorem. Take a look at a book on functional analysis and see what the consequences for hilbert spaces would be and by extension the mathematical model for quatum mechanics. The Hahn-Banach theorem is often describe as a conerstone of analysis. None of this involves choice so alternate set theories can't resolve the issue. my friend has a problem with ZF's power set axiom but he is to intelligent to be taken seriously. this work was done by dr. matt foreman he was at uci. if my memory serves me the proof uses forcing arguments.

2:the standard text on BT stan wagon's book presents the material in a way that renders the use of the axiom of choice mysterious. there are other proofs which use graph theory (bipartite matching theorems) that sort of clarify the issue.

3: Foreman and Doughetry poved a version of BT which runs as follows: it is possible to cut a pea into i think seven pieces and move them using rotations and translations, rigid motions, to form the sun. What's new? well the proof is in ZF using no choice of any form. the pieces form an OPEN set whose complementis nowhere dense in the pea and the same is true of the sun. What this means is that if you deleted the portions of the pea which are not used and the sun which are not filled up you couldn't see the difference. The original proof is very complicated.

4:the BT paradox is really about the uniqueness of the lebesgue integral and the possibility of extending it to an averaging functional on the entire powerset of a space loosing countable additivity but keeping invariance under some group action. The point is that this is not possible on the sphere. I wouldn't get overly worked up about geometric/mechanical metaphors like the tired pea-sun thing. physicist never tire of trying for mechanical explantations in realms where they don't apply (wave/particle duality?). I never know what it means to say that space is not "infinitely divisible" this may or may not be true i just don't know what it means. Again it is the metaphor thing. Am i to believe that the sqrt of 2 does not really exist where do you draw the line. All mathematicians mean by the continuum is something like dedikend cuts or equivalence classes of cauchy sequences. The possibility of uniqueness and extension theorems based on them.study the construction of the exponential functions to see what i mean. For sure we do not have infinite resolution in experimentation. As far as predictive power I read a paper linking hadron physics to BT but the argument was flawed because it relied on specifics of the wagon proofs which are not needed. I would venture the idea that BT like phenonmena are probably behind alot of the non-sense explanations in physics. what is so apriori about the sigma-algebra of lebesgue measurble sets anyway?

there is no gap in reasoning with the banach-tarski paradox. some points

1:it has been proven in ZF set theory that the banach-tarski paradox is equiconsistant with the Hahn-Banach extension theorem. What this means is that if banach-tarski is in reality false then so is the hahn-banach theorem. Take a look at a book on functional analysis and see what the consequences for hilbert spaces would be and by extension the mathematical model for quatum mechanics. The Hahn-Banach theorem is often describe as a conerstone of analysis. None of this involves choice so alternate set theories can't resolve the issue. my friend has a problem with ZF's power set axiom but he is to intelligent to be taken seriously. this work was done by dr. matt foreman he was at uci. if my memory serves me the proof uses forcing arguments.

2:the standard text on BT stan wagon's book presents the material in a way that renders the use of the axiom of choice mysterious. there are other proofs which use graph theory (bipartite matching theorems) that sort of clarify the issue.

3: Foreman and Doughetry poved a version of BT which runs as follows: it is possible to cut a pea into i think six pieces and move them using rotations and translations, rigid motions, to form the sun. What's new? well the proof is in ZF using no choice of any form. the pieces form an OPEN set whose complement is nowhere dense in the pea and the same is true of the sun. What this means is that if you deleted the portions of the pea which are not used and the sun which are not filled up you couldn't see the difference. The original proof is very complicated.

4:the BT paradox is really about the uniqueness of the lebesgue measure and the possibility of extending it to an averaging functional on the powers et of a space loosing countable additivity but keeping invariance under some group action. The point is that this is not possible on the sphere. I wouldn't get overly worked up about geometric/mechanical metaphors like the tired pea-sun thing. physicist never tire of trying for mechanical explantations in realms where they don't apply (wave/particle duality?). I never know what it means to say that space is not "infinitely divisible" this may or may not be true i just don't know what it means. Again it is the metaphor thing. Am i to believe that the sqrt of 2 does not really exist where do you draw the line. All mathematicians mean by the continuum is something like dedikend cuts or equivalence classes of cauchy sequences. The possibility of uniqueness and extension theorems based on them.study the construction of the exponential functions to see what i mean. For sure we do not have infinite resolution in experimentation. As far as predictive power I read a paper linking hadron physics to BT but the argument was flawed because it relied on specifics of the wagon proofs which are not needed. I would venture the idea that BT like phenonmena are probably behind alot of the non-sense explanations in physics. what is so apriori about the sigma-algebra of lebesgue measurble sets anhyway?

i am not sure i know how to post this so sorry if this is the third time

there is no gap in reasoning with the banach-tarski paradox. some points

1:it has been proven in ZF set theory that the banach-tarski paradox is equiconsistant with the Hahn-Banach extension theorem. What this means is that if banach-tarski is in reality false then so is the hahn-banach theorem. Take a look at a book on functional analysis and see what the consequences for hilbert spaces would be and by extension the mathematical model for quatum mechanics. The Hahn-Banach theorem is often describe as a conerstone of analysis. None of this involves choice so alternate set theories can't resolve the issue. my friend has a problem with ZF's power set axiom but he is to intelligent to be taken seriously. this work was done by dr. matt foreman he was at uci. if my memory serves me the proof uses forcing arguments.

2:the standard text on BT stan wagon's book presents the material in a way that renders the use of the axiom of choice mysterious. there are other proofs which use graph theory (bipartite matching theorems) that sort of clarify the issue.

3: Foreman and Doughetry poved a version of BT which runs as follows: it is possible to cut a pea into i think six pieces and move them using rotations and translations, rigid motions, to form the sun. What's new? well the proof is in ZF using no choice of any form. the pieces form an OPEN set whose complement is nowhere dense in the pea and the same is true of the sun. What this means is that if you deleted the portions of the pea which are not used and the sun which are not filled up you couldn't see the difference. The original proof is very complicated.

4:the BT paradox is really about the uniqueness of the lebesgue measure and the possibility of extending it to an averaging functional on the powers et of a space loosing countable additivity but keeping invariance under some group action. The point is that this is not possible on the sphere. I wouldn't get overly worked up about geometric/mechanical metaphors like the tired pea-sun thing. physicist never tire of trying for mechanical explantations in realms where they don't apply (wave/particle duality?). I never know what it means to say that space is not "infinitely divisible" this may or may not be true i just don't know what it means. Again it is the metaphor thing. Am i to believe that the sqrt of 2 does not really exist where do you draw the line. All mathematicians mean by the continuum is something like dedikend cuts or equivalence classes of cauchy sequences. The possibility of uniqueness and extension theorems based on them.study the construction of the exponential functions to see what i mean. For sure we do not have infinite resolution in experimentation. As far as predictive power I read a paper linking hadron physics to BT but the argument was flawed because it relied on specifics of the wagon proofs which are not needed. I would venture the idea that BT like phenonmena are probably behind alot of the non-sense explanations in physics. what is so apriori about the sigma-algebra of lebesgue measurble sets anhyway?

the above is a drunken rampage

4/1/06