<story><title>The Point of the Banach-Tarski Theorem</title><url>https://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheorem.html?wa22hn</url></story><parent_chain></parent_chain><comment><author>soVeryTired</author><text>Mathoverflow has some good discussion of Banach-Tarski [0]. The top-rated posts argue that&lt;p&gt;- The &amp;#x27;problem&amp;#x27; with B-T may be related to our notion of &amp;quot;space&amp;quot; (i.e. point-set topology) rather than any issues with the axiom of choice&lt;p&gt;- Most practical uses of the axiom of choice could get by with the axiom of countable choice, under which B-T doesn&amp;#x27;t hold&lt;p&gt;- Mathematics need not directly model the &amp;#x27;real&amp;#x27; world or match our physical intuition&lt;p&gt;[0] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;mathoverflow.net&amp;#x2F;questions&amp;#x2F;260057&amp;#x2F;axiom-of-choice-banach-tarski-and-reality&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;mathoverflow.net&amp;#x2F;questions&amp;#x2F;260057&amp;#x2F;axiom-of-choice-ba...&lt;/a&gt;</text></comment>
<story><title>The Point of the Banach-Tarski Theorem</title><url>https://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheorem.html?wa22hn</url></story><parent_chain></parent_chain><comment><author>karmakaze</author><text>I&amp;#x27;m glad to have found this post. I discovered the Banach-Tarski theorem via Vsauce[0]. It was interesting but I couldn&amp;#x27;t get the significance of it. It either didn&amp;#x27;t seem like an unexpected result or too esoteric to appreciate.&lt;p&gt;There&amp;#x27;s phrasing in the post that could be misunderstood (later clarified) but can leave unclarity from assumed understanding of the earlier description.&lt;p&gt;&amp;gt; In R3, given a solid ball B of radius R, it is possible to partition B into finitely many pieces such that those pieces can be reassembled to form two solid balls B1 and B2 each of radius R.&lt;p&gt;&amp;quot;finitely many pieces&amp;quot; could be confusing because though we may be talking about 6 &amp;#x27;pieces&amp;#x27; those pieces have an uncountable infinity of radial line slices. It&amp;#x27;s also missing the rigid motion part which is key.&lt;p&gt;The part that didn&amp;#x27;t seem surprising (and is considered trivial) was in handling uncountable infinities of things. The &amp;#x27;number&amp;#x27; of points on a line [0, 1) is the same as the number on a line [0, 2) so I wouldn&amp;#x27;t be surprised to map points from [0, 1) to [0, 2) filling the latter without &amp;#x27;gaps&amp;#x27;. Similarly for areas. But what the theorem is saying is that this kind of mapping doesn&amp;#x27;t work in R1 nor R2 but does work in R3 (with rigid motions).&lt;p&gt;The part that makes B-T surprising is that the extra volume&amp;#x2F;ball can be constructed with rigid motions of those uncountably infinite sets. This is where it seems beyond me to appreciate: that one or two balls have the same uncountably number of radial line slices is considered trivial, and mapping using rigid motions is surprising.&lt;p&gt;For example, if you do the same rearrangement but instead of sets of radial lines, consider the set of points at the surface of those radial lines, we&amp;#x27;re basically working in R2. The reason why it can&amp;#x27;t be done is because if we instead of being on the surface of a sphere we&amp;#x27;re on a plane, then those same motions aren&amp;#x27;t distance preserving. Thinking geometrically doesn&amp;#x27;t seem weird: an extra dimension let&amp;#x27;s you do something you can&amp;#x27;t in lower ones.&lt;p&gt;I think the algebraic description that the post makes may be clearer to appreciate the difference, and I&amp;#x27;ll be giving it another read and more thought.&lt;p&gt;[0] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;www.youtube.com&amp;#x2F;watch?v=s86-Z-CbaHA&quot;&gt;https:&amp;#x2F;&amp;#x2F;www.youtube.com&amp;#x2F;watch?v=s86-Z-CbaHA&lt;/a&gt;</text></comment>