Banach-Tarski: Given the axiom of choice (!), there exists (!) a partition of the unit ball into five disjoint subsets such that the whole unit ball may be recovered by taking the union of only 3 members of that partition after rotating each in a certain way - as well as by joining just the (appropriately rotated) other two. This merely tells us that our notion of volume does not consistently extend to ALL subsets of space (and in fact those subsets defying it are not explicitly constructable).
okzoia. I agree with your comments. There are an almost infinite number of different objects we could define.In this way, we could define an almost infinite number of university maths courses. In the same way, we could write an almost infinite number of books, or an almost infinite number of grammars. But how many such objects is it healthy to define and think about?
Mathematical "objects" don't have any properties. There are no such things: it's all in the mind - which means we imagine them along with round squares, Pegasus, and hypercubes. "They" are what we make them.
so how is this video about model theory? i mean, you talk a little about tarski, the founding father of "model theory" (though his form of model theory is different from most modern presentations), and then you talk a bunch of shit for awhile, and then what else? you get off on pretending to explain something complicated, i take it?
@nortexoid You could be right. I saw books on "Model Theory" and could not understand them at all. Then a few years later I noticed that the term "model theory" is used to describe the approach exemplified by Birkhoff and Maclane in their freshperson book "A Survey of Modern Algebra". I found B&M easy to understand, so I suspect that the more complex maths books are failing to present material in a way that is intelligible to the beginner. In 1950 Turing predicted this...
@nortexoid In 1950 Turing predicted this - he was only talking about people who write computer code. But I since begin to think that this disease of obfuscation exists also in many other areas. Perhaps Maths and Philosophy is where it started. It has now spread to many areas such as "business management".
Thanks for your message. (I liked watching your videos too). I haven't studied maths for 40 years. Without funding, I lost interest, or perhaps did not have time.
I think the Banach-Tarski paradox says you could cut up a ball and make from the pieces, two balls of the same size.
This suggests that there is something wrong with the theory. It does not sound like a very useful theory for describing objects in our world.
Argh - I hate how youtube totally just ate my comment. So no, I'm not a math major - I'm the daughter of a Chemist/Math Enthusiast (check out his videos - AncalagonxThexBlack) and actually a Journalism Major myself. I'm just a masochist in that I enjoy reading about and researching mind blowing paradoxes in the company of my roommates.
Fun stuff. As for the theory being "useful" - I couldn't agree more. I'm all about useful beliefs and theories - and the application thereof. I think I'll be okay letting go of this paradox; that is, not needing to understand it. :)
Banach-Tarski is not about objects in the real world. It is about mathematical objects, which sometimes have paradoxical properties. Sometimes you can modify the properties of a mathematical object to remove a paradox, and sometimes you can't. If you want to keep the axiom of choice, you're stuck with Banach-Tarski.
Banach-Tarski: Given the axiom of choice (!), there exists (!) a partition of the unit ball into five disjoint subsets such that the whole unit ball may be recovered by taking the union of only 3 members of that partition after rotating each in a certain way - as well as by joining just the (appropriately rotated) other two. This merely tells us that our notion of volume does not consistently extend to ALL subsets of space (and in fact those subsets defying it are not explicitly constructable).
Maxmuetze 3 months ago
okzoia. I agree with your comments. There are an almost infinite number of different objects we could define.In this way, we could define an almost infinite number of university maths courses. In the same way, we could write an almost infinite number of books, or an almost infinite number of grammars. But how many such objects is it healthy to define and think about?
richardmullins44 11 months ago
@richardmullins44
"[...] [H]ow many such objects is it healthy to define and think about?"
Ask you're doctor how many model-theoretic objects are right for you.
Side-effects could include: A fundamental feeling of incompleteness.
The99Metric 1 week ago
Thanks okzoia.
richardmullins44 11 months ago
Mathematical "objects" don't have any properties. There are no such things: it's all in the mind - which means we imagine them along with round squares, Pegasus, and hypercubes. "They" are what we make them.
okzoia 11 months ago
so how is this video about model theory? i mean, you talk a little about tarski, the founding father of "model theory" (though his form of model theory is different from most modern presentations), and then you talk a bunch of shit for awhile, and then what else? you get off on pretending to explain something complicated, i take it?
nortexoid 4 years ago
@nortexoid You could be right. I saw books on "Model Theory" and could not understand them at all. Then a few years later I noticed that the term "model theory" is used to describe the approach exemplified by Birkhoff and Maclane in their freshperson book "A Survey of Modern Algebra". I found B&M easy to understand, so I suspect that the more complex maths books are failing to present material in a way that is intelligible to the beginner. In 1950 Turing predicted this...
richardmullins44 11 months ago
@nortexoid In 1950 Turing predicted this - he was only talking about people who write computer code. But I since begin to think that this disease of obfuscation exists also in many other areas. Perhaps Maths and Philosophy is where it started. It has now spread to many areas such as "business management".
richardmullins44 11 months ago
I'm asking what the Banach--Tarski paradox is - and why can't I wrap my head around it?
xmarlasingerx 4 years ago
Thanks for your message. (I liked watching your videos too). I haven't studied maths for 40 years. Without funding, I lost interest, or perhaps did not have time.
I think the Banach-Tarski paradox says you could cut up a ball and make from the pieces, two balls of the same size.
This suggests that there is something wrong with the theory. It does not sound like a very useful theory for describing objects in our world.
Are you studying math?
richardmullins44 4 years ago
Argh - I hate how youtube totally just ate my comment. So no, I'm not a math major - I'm the daughter of a Chemist/Math Enthusiast (check out his videos - AncalagonxThexBlack) and actually a Journalism Major myself. I'm just a masochist in that I enjoy reading about and researching mind blowing paradoxes in the company of my roommates.
xmarlasingerx 4 years ago
Fun stuff. As for the theory being "useful" - I couldn't agree more. I'm all about useful beliefs and theories - and the application thereof. I think I'll be okay letting go of this paradox; that is, not needing to understand it. :)
xmarlasingerx 4 years ago
Banach-Tarski is not about objects in the real world. It is about mathematical objects, which sometimes have paradoxical properties. Sometimes you can modify the properties of a mathematical object to remove a paradox, and sometimes you can't. If you want to keep the axiom of choice, you're stuck with Banach-Tarski.
jstarret 2 years ago