This one?

What is your question?

Let me try. Euclid is assuming he can construct ADB as identical to ACB and then show the contradiction with how angles add up. I am not sure Proposition 5 is enough to prove this. Up to prop 7 I haven't seen a fully proven system yet. Even what it means for point A of ADB to be right there with point A of ACB seems ambiguous to me. I learned up to book IX in college 16 years ago and I've read a tiny bit about non-Euclidean geometry, and pondered Zeno's paradoxes endless times in my life. I wanted to start very basic and learn the essential nature of math from the ground up but Euclid hasn't been doing it for me. This is the first time I've read him since I was 19

Non standard analysis now says that it has rigorously examined infinitesimals and find them necessary, useful, and logical — Gregory

Not "necessary". Useful as an alternative to normal calculus. Logical - in a sense, I suppose.

Well I tried to read over Alezandre Bolovak's article The Inevitability of Infinitesimal but I could my grasp her main argument. Check it out on Google if your interested

I. prop. 7:
"Given two straight lines constructed from the ends of a straight line and meeting in a point, there cannot be constructed from the ends of the same straight line, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each equal to that from the same end."

----

Considering the two lines as radii, and drawing circles centered on the ends of the original lines, the two circles will always intersect at the same point. In picturing it all seems obvious - not so easy to put into words.

https://www.academia.edu/343657/Inevitability_of_infinitesimals

There's the article, I got the name slightly wrong. It goes into a lot of stuff. Very little was useful for me. I didn't get what was meant by "of course, with ultraproducts of rational numbers come infinitesimals." It makes intuitive sense to me, but that's all. I thought Newton did fine talking about infinitesimals as fluxons, half there and half not. Leibniz might have understood this the best. People latter on added stuff about "limits" but I don't know anything that it added to what Newton and Liebniz had already said. Newton already had a limit towards which an infinite series goes towards. Non-standard analysis, it seems to me, claims to show that Cantor was right in arguing that actual objects have an infinity of points, and the Britannica Encyclopedia says it uses the mathematical logic of Godel.

What does Banach-Tarki's paradox do to Euclid's system and proposition 7? I only know about the paradox from Vsauce on youtube, but at least Zeno's cubes are getting a closer look into

Zeno the century before had introduced the world to infinitesimals — Gregory

I'm thinking he did no such thing. Please show me mistaken. In particular, had he understood them, and other things as well, he would not have bothered with his paradoxes.

I might be able to do the paradox better than Zeno did. To get from A to B you have to go half the distance, since if there is any distance at all it has a half. But the half, if it is any distance at all, has a half. There is no end to the process. If you don't go another half you are already there. Hence the paradox: the finite appears to be infinite. What we call those "things" at the bottom are infinitesimals nowadays

https://www.youtube.com/watch?v=ffUnNaQTfZE

That video is good because he is smart and has great visuals on this. I don't see how someone can watch these 20 minutes and not recognize a paradox in space itself

What does Banach-Tarki's paradox do to Euclid's system and proposition 7? — Gregory

The B-T paradox, as has been said several times in this forum, depends upon the Axiom of Choice. Not one of Euclid's. Sometimes the Axiom of Choice is involved in what are called "pathological" examples in mathematics.

I haven't read that paper on nonstandard analysis, but perhaps the "necessary" part refers to the following (Wiki): "The real contributions of nonstandard analysis lie however in the concepts and theorems that utilize the new extended language of nonstandard set theory." And: "Nonstandard set theory is an attempt to generalise nonstandard analysis to cover the whole of classical mathematics."
Too much for my tired old eyes and drifting mind.

I wasn't surprised when I watched Vsauces video specifically on Banach-Tarski, because it is just the conclusion to be drawn from Zeno. Kants second antimony is literally Zenos paradox. Kant considered it unanswerable. Ye I know there has been a lot of work done in math since Kant. I've only taken a semester of pre-calculus and that was in senior high school. It's mind boggling to me that something so simple as space and motion can only be explained by high level mathematics. I don't understand the way calculus moves. I understand space and motion, but Zeno has permanently scarred my relationship with them

On no, I have more to say. I feel like math lovers have so many equations in their heads that they can't see obvious contradictions at times. Take Gabriel's horn. Obviously the surface area can't be infinite while what it contains is finite. That's just stupid. But this is taught as reality in calculus class. Does it not occur to them that pi (volume within the horn) is not finite, but infinite? Taking this to Euclid, if you have a segment one inch long, it is one inch long. But wait, bend it into a circle and its a product of pi says Euclid. Pi is endless. So Zeno was right.

There is a limit to how many equations someone can balance. I barely keep the number 1 in my head and ponder 1+1=2 from time to time. Maybe I have the right angle to see the paradox in all this

Obviously the surface area can't be infinite while what it contains is finite. That's just stupid. — Gregory

A simpler example in a lower dimension is $\int\limits_{1}^{\infty }{\frac{1}{{{x}^{2}}}}dx=1$

This is the finite area between an infinite curve in the plane and the x-axis.

Well that's a contradiction. I'm saying everything, physical and abstract, is both finite and infinite at the same time and this has huge consequences for geometry

The B-T paradox, as has been said several times in this forum, depends upon the Axiom of Choice. Not one of Euclid's. Sometimes the Axiom of Choice is involved in what are called "pathological" examples in mathematics. — jgill

Just perusing recent threads so no worries if this topic's no longer of interest. But you should be aware that the negation of the axiom of choice is an even more fruitful source of pathologies: a vector space with no basis; a surjection with no right inverse; a commutative ring with unity with no maximal ideal; a Dedekind-finite set that is nevertheless infinite; infinite sets that are not bijectively equivalent to any Aleph; an infinite set whose cardinality changes if you remove a single element; an infinite set not bijectively equivalent to any of its proper subsets; and many others. The axiom of choice serves to make infinite sets behave as they should and is therefore the more natural choice than its negation. In fact without the axiom of choice and with a single inaccessible cardinal (which amounts to nothing more than assuming ZF is consistent) the real numbers can be expressed as a countable union of countable sets, making it impossible to get modern probability theory off the ground.

Ever see the infinite hat problem? That's my favorite bizarre consequence of the axiom of choice.

The Banach-Tarski theorem uses the axiom of choice in a very natural way. You have an equivalence relation and you choose a representative from each nonempty equivalence class. It would be much more counterintutive if you couldn't do that than if you could.

The actual heart of the paradox is much simpler and depends on nothing more than the fact that the free group on two letters is paradoxical. The core idea is in fact syntactic and has little to do with 3-space and nothing to do with the axiom of choice. I wanted to write this up for this forum a while back but the person I was talking to lost interest.

You only need the axiom of choice to lift the paradoxical decomposition of the free group on two letters to the group of rotations of the unit sphere. Wikipedia has an excellent overview of the entire proof that is quite accessible. It's a long proof made up of very simple steps that can be followed one by one.

https://en.wikipedia.org/wiki/Free_group

https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

Also nonstandard analysis and in particular the construction of the hyperreals is about an hour's worth of work for a competent undergrad in math. Given the basic notions of set theory like set inclusion and partial orders and so forth, one defines filters, ultrafilters, and nonprincipal ultrafilters. You then mod out the reals by a nonprincipal ultrafilter in a manner analogous to how you construct the reals from equivalence classes of Cauchy sequences of rationals. The entire construction is surprisingly simply. The only hard part is the model theory that tells you exactly when you can transfer results back and forth from the reals to the hyperreals. Terence Tao has written some beautiful expository articles on nonstandard analysis and ultrafilters that are extremely enlightening, which I can link if anyone's interested.

The tl;dr here: (1) The axiom of choice is far more natural than its negation; (2) The Banach-Tarski theorem can be understood by a high school student who's willing to put in some work. There are no advanced concepts involved. The Wiki proof is very good. There are a lot of steps but each is relatively easy. (3) The construction of the hyperreals can be understood easily by a math undergrad in about an hour. Or a few days if you want to go back and nail down every detail and read the Tao articles and achieve some grokitude. There's much less to the subject than the folklore around it would indicate.

I think Banach-Tarski disproves common mathematics the way Godel set out to do (but failed). It breaks math

The axiom of choice serves to make infinite sets behave as they should and is therefore the more natural choice than its negation — fishfry

Nice defense of the AOC ! :cool: I think that those who spend time studying set theory and foundations have a much deeper appreciation for those subjects than do those of us who are briefly exposed to it and move on to different topics. It may have been a shift towards foundations the math department at the U of Chicago initiated back in the late 1950s that caused a rift with the physics department and resulted in physics students being required to take all their math courses in the school of physics. But I don't know how long that lasted; it may have been a temporary policy.

And without the AOC or something similar we wouldn't have non-measurable sets. :sad:

Nice defense of the AOC ! [/url]

LOL. When I read that sentence in my mentions I thought I must have said something good about Alexandria Ocasio-Cortez in one of the political threads!

— jgill

Nice defense of the AOC ! [/url]

:cool: I think that those who spend time studying set theory and foundations have a much deeper appreciation for those subjects than do those of us who are briefly exposed to it and move on to different topics. It may have been a shift towards foundations the math department at the U of Chicago initiated back in the late 1950s that caused a rift with the physics department and resulted in physics students being required to take all their math courses in the school of physics. But I don't know how long that lasted; it may have been a temporary policy. — jgill

As a math undergrad I called up the physics dept and asked them if there was way to study physics in an accelerated manner since I already knew a lot of the math. They told me absolutely no way, I had to start with freshman physics and take only physics classes. They were quite snippy about it! There was definitely some friction between the math and physics departments.

And without the AOC or something similar we wouldn't have non-measurable sets. :sad: — jgill

The question of why we choose one axiom over another gives insight into mathematical philosophy. As I've mentioned, AC regularizes infinite sets. It makes them well-behaved and gets rid of pathologies. So we adopt AC based on principles like convenience and naturality. That is to say, highly pragmatic and human grounds, not logical ones. Emphasizing again that math is in large part a human activity. Raising the question of which parts aren't, if any!

Why do I say that AC is natural? Because a choice set is a legislature. Let me explain. Say you have a country that's divided up into states. We say that two people are in the same "equivalence class" if they live in the same state. This is an equivalence relation that partitions the country into pairwise disjoint equivalence classes.

Can we choose a representative from each state? Sure, that's the US Senate. The US Senate is two applications of the axiom of choice. And it's perfectly natural. Each state is nonempty, so we can pick a representative from each state and send the representatives to Washington.

The House of Representatives is a choice set on the nation's Congressional districts. We say two people are equivalent if they live in the same CD, then we choose a representative from each district.

Now suppose there's a country somewhere hat has infinitely many states. For that matter suppose there are uncountably many nonempty states. Why can't we simply pick a legislature? It would be wildly unnatural for you to tell me that we can't pick a representative of each state. Of course we can, whether there are finitely many or infinitely many states.

A choice set is just a legislature. And you can always pick a legislature. Adopting AC is natural; adopting its negation would be unnatural.

I think Banach-Tarski disproves common mathematics the way Godel set out to do (but failed). It breaks math — Gregory

Oh gosh no. It's a very simple theorem. It just shows that the universe isn't literally identical to three-dimensional Euclidean space. And as I've said, the core paradox is in the free group on two letters. It's not a difficult argument.

https://en.wikipedia.org/wiki/Free_group

https://en.wikipedia.org/wiki/Paradoxical_set

The so-called "paradox" just shows that our mathematical intuition of physical space isn't actually true. Mathematically it's an easy theorem, accessible to anyone who puts in a little effort. The Wiki page has a very accessible outline of the proof.

It doesn't "disprove common mathematics." It disproves the common intuition that physical space is the same as 3D Euclidean space. It isn't. The paradox depends on the set-theoretical viewpoint that we can form arbitrary subsets of such a space. There's no physics theory to match or allow that.

What if they prove that you can take a greater object from a smaller? How is that much different than B\T? To me B\T says that is nothing discrete in geometry anymore

What if they prove that you can take a greater object from a smaller? — Gregory

Who's they and what would such a proof involve?

How is that much different than B\T? To me B\T says that is nothing discrete in geometry anymore — Gregory

I don't know what you mean by that. The best I can suggest is that you read through the Wiki pages that I've already linked on the Banach-Tarski paradox, the free group on two letters; and the idea of a paradoxical decomposition of a set.

The core of the proof is the paradoxical decomposition of the free group on two letters, which is explained in step 1 of the Wiki proof. As I've noted, this step does not require the axiom of choice and barely even uses infinite sets. It's just a surprising fact about finite-length strings made out of two arbitrary symbols and their symbolic inverses.

Another basic notion that might help is the idea of a nonmeasurable set. It turns out that there are sets of real numbers (and Euclidean space in general) such that it's not possible to assign them any sensible notion of size that analogizes the familiar length, area, and volume. The pieces involved in the B-T decomposition are such nonmeasurable sets. Our intuitions about volume fail on these types of sets, and that's why it's called a paradox.

I'm not sure how to respond to your concerns to repeat that the proof of B-T is a straightforward application of a number of very simple ideas. There's hardly any difficult math involved at all. But the key point really is that there are nonmeasurable sets that don't behave property with respect to intuition. To say anything more specific I'd have to go into the symbology but that's all on Wiki.

The B-T theorem doesn't break math or abolish discrete geometry, whatever you meant by that. It's just a counterintuitive fact about Euclidean space.

By the way have you seen the Vsauce video on B-T? He usually annoys me but his video on this topic is very good.

https://www.youtube.com/watch?v=s86-Z-CbaHA

ps -- It occurs to me that the word Euclidean could be part of the conceptual problem. What we think of as modern Euclidean space is very much different than Euclid's geometry. Euclid considered subsets of space like points, lines, and cones and so forth. But he never considered completely arbitrary sets of points that could not be assigned a sensible volume or size! Perhaps if we called $\mathbb R^3$ by the name set-theoretic space, we wouldn't expect it to behave as nicely as Euclid's space. We call $\mathbb R^3$ Euclidean space, but it's very different than what Euclid was thinking of.

I like this (from Wikipedia) illustrating AC:

"Bertrand Russell coined an analogy: for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection; this makes it possible to directly define a choice function. For an infinite collection of pairs of socks (assumed to have no distinguishing features), there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice."

Banach tarski has immeasurable things form new objects. Are they saying they can take one object and only one object out of an object of the same size? How big was the original object. I don't think there is anything purely finite, inside math and outside.