# Banach-Tarski Paradox: Exploring the Ultimate Magic Trick

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• thinkandmull
In summary: The circumference of a circle is a finite number, and the ratio of the circumference to the diameter is ##\pi##, which is a number that happens to be infinitely long when expressed in decimal notation. In summary, The Banach-Tarski paradox is a mathematical paradox that involves dividing a measurable set into a finite number of non-measurable sets and then putting them back together differently. It is possible to avoid this paradox by giving up on certain mathematical concepts, such as the existence of non-measurable sets, but this can limit other important mathematical theorems. The concept of a point is also counterintuitive in this paradox, as it can add up to a line despite being
thinkandmull
Hi! Does anyone think Banach-Tarski's paradox needs reworking? I first came across it in a video by Vsauce. I've been told that things might be reworked as to avoid the paradox, just as set theory was fixed so as to avoid Russell's paradox. How to make sense of a smallest unit of space is what it's about for me. If you take the smallest length and make it the two shorter sizes of a right triangle, than half the hypotenuse would be the true "smallest", and the process goes on unto infinity. The ultimate mathematical magic trick would if someone used the Banach-Tarski paradox to mathematically get a bigger object out of one with smaller measure.

thinkandmull said:
Hi! Does anyone think Banach-Tarski's paradox needs reworking?
No.
The ultimate mathematical magic trick would if someone used the Banach-Tarski paradox to mathematically
get a bigger object out of one with smaller measure.
You don't need Banach-Tarski for that. It can always easily be done in topology. "bigger" requires a volume, topology does not.

I once listened to a lecture about it where the professor emphasized his view, that our understanding of the concept "point" is to blame here rather than AC. We have similar paradoxa, e.g. the Hilbert curve which also indicate that topological dimensions don't quite match intuition. However, this does not need fixing, or as another professor of mine used to say: The real world is discrete. And despite of it, we calculate well with continuous or even smooth functions and our bridges and buildings don't collapse.

dextercioby, Klystron and jedishrfu
My understanding of B/T was that it worked through the uncountable infinity of points within the original object. This is reconcilable with the object being discrete?

thinkandmull said:
My understanding of B/T was that it worked through the uncountable infinity of points within the original object. This is reconcilable with the object being discrete?
Yes, and that's why the dimensions are crucial, in this case zero dimensional points. A real life object would have finitely many molecules, a mathematical ball has uncountably many points. This is why it would never work in reality. That's what I meant: we calculate with a continuum but we can only ever align finitely many things.

Mathematical concepts and real life objects are two entirely different things. We cannot even "produce" a circle! Not if we allow an electron microscope to investigate them. Therefore there are some mathematical concepts which are counterintuitive. A point is one of them.

B/T is based upon dividing a measurable set into a finite number of non-measurable sets and then putting them together differently.

S.G. Janssens
thinkandmull said:
I've been told that things might be reworked as to avoid the paradox, just as set theory was fixed so as to avoid Russell's paradox.

You are right that they might be "reworked", but by "reworking" (= avoiding the existence of non-measurable sets) you may lose more than you will gain. Perhaps most notably, it is possible to prove that you will have to give up on the Hahn-Banach extension theorem. That theorem is central in analysis.

So, you could indeed lay the foundations such that you avoid the paradox (which is - by nature of every paradox - only apparent anyway), but you will be short on bricks to build the rest of the house.

A necessary assumption is that you have infinitely-many atoms to do these many partitions. In actuality, the number of atoms in the universe is bounded/finite.

Is a non-measurable part the same as a "simple substance", to use an old scholastic way to say it? This solution seemed to me to be saying that 0 and 0 can equal 1 (or a size). It seems the more I look, the more counter-intuitive is truth

thinkandmull said:
Is a non-measurable part the same as a "simple substance", to use an old scholastic way to say it? This solution seemed to me to be saying that 0 and 0 can equal 1 (or a size). It seems the more I look, the more counter-intuitive is truth
No, ##0+0\neq 1##. But uncountably many ##0## can add up to ##1##: ##[0,1] = \cup \{\,\{\,\frac{r}{s}\}\,|\,r,s\in \mathbb{Z}, r\leq s\,\}##. And again, it is the concept of a point which is counterintuitive, since they can add up to a line although they are dimensionless.

Finding where exactly something becomes that zero tangles me into knots, probably because I struggle with infinity. Aristotle said something can potentially be divided into an infinity, but not actually. I think that the object has the same measure whether divided or not. Math then seems to be getting something from infinite nothingness, or having something both finite and infinite in cardinality. The circumference of a circle is a factor of endless pie, as pointed out in the Vsauce video

thinkandmull said:
Finding where exactly something becomes that zero tangles me into knots, probably because I struggle with infinity. Aristotle said something can potentially be divided into an infinity, but not actually.
It's probably better to leave Aristotle out of the discussion, since mathematics has come a long way since the geometry of the Greeks.
A concrete thing cannot be divided into an infinite number of parts, since you would eventually get down to individual atoms, which can't be divided much further, practically speaking.
thinkandmull said:
I think that the object has the same measure whether divided or not.
Math then seems to be getting something from infinite nothingness, or having something both finite and infinite in cardinality. A finite number is one that is strictly less than infinity and strictly greater than negative infinity. Having a decimal representation that is infinitely long has nothing to do with the number being infinite.
No, and this doesn't make any sense. A set is either of finite cardinality (such as the set {1, 2, 3}) or is of infinite cardinality, where "infinity" here is further categorized as countably infinite (e.g., the set of integers) or uncountably infinite (e.g., the set of real numbers in the interval [0, 1])
thinkandmull said:
The circumference of a circle is a factor of endless pie, as pointed out in the Vsauce video
This also doesn't make much sense. The decimal representation of ##\pi## (spelled pi, not pie) is endless, but the number itself is very much finite, and is just one of many numbers that lie between 3.1 and 3.2.
The circumference isn't a factor of ##\pi## -- both the radius and ##\pi## are factors of the circumference.

What you seem to be struggling with are ideas from measure theory, a fairly advanced topic in mathematics. If you start with the real interval [0, 1], its measure is 1, which agrees with its length. You can remove an infinite number of points of the form ##\frac a b##, where ##0 \le \frac a b \le 1##, and still end up with a set whose measure is 1. What we have removed are the the rational numbers in this interval. Although there are an infinite number of rational numbers in the interval [0, 1], they aren't packed in there as tightly as the real numbers that aren't rational (the irrationals).

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dextercioby
I've imagined Thompson's lamp in physical representation: half an object followed by it's quarter, ect. In arithmetic it would equal one, but in physical space the line of ever smaller objects can't go on forever, for then when put together the object wouldn't be perfectly finite anymore. An answer I've heard is that standard analysis doesn't bother about a final term, but peering at the end of the line makes one curious how it would end. HOWEVER, what I've gather here is that I am on the wrong track in understanding Cantors theory of uncountable points in an object by reducing everything to cardinality, instead of two other concepts of measure and density. I don't know if Banach-Tarski aids in this or makes it harder.

thinkandmull said:
I've imagined Thompson's lamp in physical representation: half an object followed by it's quarter, ect. In arithmetic it would equal one, but in physical space the line of ever smaller objects can't go on forever, for then when put together the object wouldn't be perfectly finite anymore.
Well, there's physical reality, and then there's mathematics, and the two don't necessarily coincide in all matters. In physical reality, we can't take some thing and successively divide it an infinite number of times, and then add all the pieces together.

What you're talking about here is an infinite series - a sum of infinitely many terms that are added. The infinite series that you described adds up to 1, which is a perfectly finite number, even though the series itself has an infinite number of terms. That is, 1/2 + 1/4 + 1/8 + ... +1/(2^n) + ... converges to 1, and this can be proven quite easily.

A seemingly similar series, 1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... also contains an infinite number of terms, but does not converge to a particular number. The more terms you add, the larger the sum gets, without bound. The first series, from Thomson's Paradox, also gets larger as you add more terms, but the sum is bounded above by 2. The more terms you add, the closer the partial sums (sums of a finite number of terms) get to 2.
thinkandmull said:
An answer I've heard is that standard analysis doesn't bother about a final term, but peering at the end of the line makes one curious how it would end.
If we have an infinite sum, how can there be a last term?
thinkandmull said:
HOWEVER, what I've gather here is that I am on the wrong track in understanding Cantors theory of uncountable points in an object by reducing everything to cardinality, instead of two other concepts of measure and density. I don't know if Banach-Tarski aids in this or makes it harder.
Cardinality plays a role in Cantor's proof of the uncountability of the real numbers. Since it doesn't matter which interval you take, we usually talk about the real numbers in the interval [0, 1]. This interval contains a countably infinite set, the rationals between 0 and 1, as well as an uncountably infinite set, the reals between 0 and 1. The two sets have different cardinalities. Both sets are dense in the interval [0, 1], but have different measures.
And, yes, you're on the wrong track in understanding Cantor's proof. I don't think Banach-Tarski has anything to do with Cantor's proof.

Further thoughts...
To understand Cantor's diagonal argument, I think you really need to know only three concepts:
1. The difference between countable sets (i.e., countably infinite) and uncountable sets (uncountably infinite).
2. How to determine whether a set is countable.
3. Understanding a proof by contradiction
To determine that a set is countable, you need to show that there is a one-to-one pairing of the members of the set with the positive integers.

As already mentioned, if your goal is to understand Cantor's diagonal argument, Banach-Tarski is a wild goose chase -- it's not related to Cantor's argument.

@thinkandmull if you want to understand purely mathematical theorems you should stop mixing it up with physics and physical reality. Math is the language of physics, not the other way around.

Its easiest for me to think the odd numbers are not less them all the whole numbers because of Hilbert's principle that you can add something less than an uncountable infinity to any infinity and have the same cardinality. I am definitely geometrizing the image of odd and whole numbers in two lines. If I were literally to do a supertask and past by between the lines, the "second after every other" odd numbers would have been past by by me just as often as the whole numbers because of the whole (that is, infinity). Am I right in thinking that the one to one correspondence comes from taking the whole prior to the part? Solving Zeno's paradox does this by taking the whole (finite) as prior to the parts (points).

This might be a more geometrical question (which itself might be confusing things), but if you had the odd and whole numbers lined up like I said (each being like a little box), if you pulled all odd numbers back so that the "three" on the odd line lined up with the number two on the whole number line, where in the horizon would there be no numbers at all on the odd side?

thinkandmull said:
Its easiest for me to think the odd numbers are not less them all the whole numbers because of Hilbert's principle
Correct, but it's not "Hilbert's principle" -- it's just the idea that you can do a one-to-one pairing of each odd integer with its counterpart in the integers. I showed you how in another thread you started.
thinkandmull said:
that you can add something less than an uncountable infinity to any infinity and have the same cardinality. I am definitely geometrizing the image of odd and whole numbers in two lines. If I were literally to do a supertask and past by between the lines, the "second after every other" odd numbers would have been past by by me just as often as the whole numbers because of the whole (that is, infinity). Am I right in thinking that the one to one correspondence comes from taking the whole prior to the part?
I don't know what this means.
thinkandmull said:
Solving Zeno's paradox does this by taking the whole (finite) as prior to the parts (points).
I don't understand what this means, either. The solution of Zeno's paradox is that as the arrow passes across smaller and smaller increments of the distance, the time required is also less.

thinkandmull said:
This might be a more geometrical question (which itself might be confusing things), but if you had the odd and whole numbers lined up like I said (each being like a little box), if you pulled all odd numbers back so that the "three" on the odd line lined up with the number two on the whole number line, where in the horizon would there be no numbers at all on the odd side?
Are you just shifting the position of the odd numbers? That's not how the pairing goes.
Code:
Pos. integers 1 2 3 4 5 ...  n   ...
Neg. integers 1 3 5 7 9 ... 2n-1 ...
You tell me a pos. integer, and I'll tell you what its odd counterpart is. Conversely, you tell me an odd integer, and I'll tell you its integer counterpart.

When we say 1/2 plus 1/4 plus 1/8 ect equals 1, are we doing more than making a set? When those fractions represent space, how can the spaceship approaching a limit even consider the limit to be a limit without a final term? It seems we see the limit from the outside dimension. I started this thread because the modern B/T paradox had blown my mind as to what a limit even is. If an object has uncountable points (Cantor), and we can take an infinity out of it (Hilbert), then Banach-Tarski can create a hotel for new guests. There is something fundamentally weird though about getting to this through division of the spatial. (Why must it be in that direction? Can an infinity of non-curves equal a curve like Archimedes writes about?) Now that I know infinity has three aspects (density, measure, and cardinality) though, I have to tell myself that maybe my personal concept of infinity is so infirm it doesn't represent mathematical truth. I'll try to research a lot more before forming new questions.

thinkandmull said:
When we say 1/2 plus 1/4 plus 1/8 ect equals 1, are we doing more than making a set?
This isn't a set -- it's an infinite sum that represents a number.
thinkandmull said:
When those fractions represent space, how can the spaceship approaching a limit even consider the limit to be a limit without a final term?
It seems you're still struggling to understand Zeno's paradox. Zeno's mathematics wasn't sophisitacted enough to comprehend the addition of an infinite number of things, and that even though the arrow would have to cover an infinite number of intervals, both the distance covered and the total time of flight were finite. Instead of intervals of length 1/2, 1/4, 1/8, etc., consider the arrow covering 9/10 of the distance, then 9/10 of the remaining tenth, the 9/10 of the remaining 1/100, etc. The running totals of the distances covered would be .9, .99, .999, and so on. Each running total would be getting closer and closer to the final 100% of the distance between the archer and the target. It's also important to realize that as the intervals get shorter, the time to cover that distance also grows shorter, assuming the arrow is traveling at a constant rate. Zeno's "paradox" isn't really a paradox if you understand the concept of infinite series, a topic that is presented in undergraduate calculus courses.
thinkandmull said:
It seems we see the limit from the outside dimension. I started this thread because the modern B/T paradox had blown my mind as to what a limit even is.

If an object has uncountable points (Cantor), and we can take an infinity out of it (Hilbert), then Banach-Tarski can create a hotel for new guests. There is something fundamentally weird though about getting to this through division of the spatial. (Why must it be in that direction? Can an infinity of non-curves equal a curve like Archimedes writes about?) Now that I know infinity has three aspects (density, measure, and cardinality)
These are aspects of sets, not of infinity. Consider the interval [0, 1], which is made up of an infinite number of rational numbers like 1/2, 2/3, and so on, and an infinite number of irrational numbers, like ##\sqrt 2 /2## and many others.
Although there are an infinite number of rationals and irrationals, there are far more irrational numbers in this interval -- the cardinalities of the two sets are different, due to the irrationals being uncountably infinite, versus only countably infinite for the rationals. Although both sets are dense in this interval (density has a precise definition), the measure of the irrationals is 1, but the measure of the rationals is 0.

thinkandmull said:
though, I have to tell myself that maybe my personal concept of infinity is so infirm it doesn't represent mathematical truth. I'll try to research a lot more before forming new questions.
Good idea...

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thinkandmull said:
Aristotle said something can potentially be divided into an infinity, but not actually.
Aristotle distinguished two forms of existence in Metaphysics: potential existence and actual existence. The housebuilder is potentially a builder even if he is not building. Because he has the ability to build. When the builder makes a house, then he naturally has an actual existence as a builder. I think this distinction in science is only seen as words. I do not believe that Georg Cantor, David Hilbert, Ernst Zermelo, John von Neumann, Abraham Fraenkel were well versed in ancient Greek philosophy.

General comment. What have all the recent posts to do with the Banach-Tarski paradox?

Periwinkle said:
I do not believe that Georg Cantor, David Hilbert, Ernst Zermelo, John von Neumann, Abraham Fraenkel were well versed in ancient Greek philosophy.
I would not bet! Maybe not John von Neumann as he was the youngest in this list, but a classic education was certainly the standard for the others!

An example:
David Hilbert said:
For us there is no ignorabimus, and in my opinion none whatever in natural science. In opposition to the foolish ignorabimus our slogan shall be: "We must know - we will know!"

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Periwinkle
mathman said:
General comment. What have all the recent posts to do with the Banach-Tarski paradox?
This holds true for the entire thread, starting with post #1.

I am an avid student of philosophy, not mathematics. I was more interested in these topics for there philosophical implications. It seems to me that Zeno showed the uncountable infinity of space long before Cantor, and that that infinity alone leads directly to Banach-Tarski. I don't know why one's a paradox but not the other. Pierre Bayle (his 1696 article on Zeno), Kant, Hegel and others thought these arguments show that reality is contradictory in the sense that Escher's "Ascending, Descending" picture is. Mathematicians don't always understand the shock waves they send through the field of philosophy. The Zenonian box: half of it stacked progressively on top of the other half, one fraction at a time, first blue then green, then blue again ect. When the box is whole again, will the top be blue or green? If traditional philosophy is true, there must be one color left on top. Period. But this is definitely a discussion for a different forum. Thanks

thinkandmull said:
I am an avid student of philosophy, not mathematics.
We don't discuss philosophy at PF, so I'm closing this thread.
thinkandmull said:
I was more interested in these topics for there philosophical implications. It seems to me that Zeno showed the uncountable infinity of space long before Cantor, and that that infinity alone leads directly to Banach-Tarski.
I don't think so. What Zeno's Paradox shows is a lack of understanding about how an infinite sum can add up to a finite number.
thinkandmull said:
I don't know why one's a paradox but not the other. Pierre Bayle (his 1696 article on Zeno), Kant, Hegel and others thought these arguments show that reality is contradictory in the sense that Escher's "Ascending, Descending" picture is.
If you really want to understand mathematics, you would be better off reading articles newer than 1696, written by mathematicians.
thinkandmull said:
Mathematicians don't always understand the shock waves they send through the field of philosophy.
I don't think this keeps them up at night.
thinkandmull said:
The Zenonian box: half of it stacked progressively on top of the other half, one fraction at a time, first blue then green, then blue again ect.
Et cetera is abbreviated as etc., not ect.
thinkandmull said:
When the box is whole again, will the top be blue or green?
If traditional philosophy is true, there must be one color left on top. Period.
Since there can be no last box in an infinite sequence of ever smaller boxes, one can conclude only that traditional philosophy is not true, using the principles of logical arguments. That is, ##p \Rightarrow q## is equivalent to ##\neg q \Rightarrow \neg p##.
thinkandmull said:
But this is definitely a discussion for a different forum. Thanks

## 1. What is the Banach-Tarski Paradox?

The Banach-Tarski Paradox is a mathematical paradox that states a solid ball can be divided into a finite number of pieces, and then reassembled into two identical copies of the original ball. This goes against our intuitive understanding of geometry and conservation of volume.

## 2. Who discovered the Banach-Tarski Paradox?

The Banach-Tarski Paradox was discovered by two mathematicians, Stefan Banach and Alfred Tarski, in 1924. They were both Polish mathematicians and logicians.

## 3. How does the Banach-Tarski Paradox work?

The Banach-Tarski Paradox is based on the concept of non-measurable sets, which are sets that cannot be assigned a numerical value for their size. By using these non-measurable sets, it is possible to divide a solid ball into pieces that can be rearranged to form two identical copies of the original ball.

## 4. What are the implications of the Banach-Tarski Paradox?

The Banach-Tarski Paradox challenges our understanding of geometry and the concept of volume. It also has implications for the foundations of mathematics, as it shows that seemingly basic concepts, such as the size of a set, can be counterintuitive and difficult to define.

## 5. Is the Banach-Tarski Paradox possible in the physical world?

No, the Banach-Tarski Paradox is a mathematical concept and cannot be replicated in the physical world. It relies on non-measurable sets, which do not exist in the physical world. This paradox is purely theoretical and has no practical applications.

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