1567:
38:
4385:
used to prove the Claim, one can see that the full circle is equidecomposable with the circle minus the point at the ball's center. (Basically, a countable set of points on the circle can be rotated to give itself plus one more point.) Note that this involves the rotation about a point other than the origin, so the Banach–Tarski paradox involves isometries of
Euclidean 3-space rather than just
114:. The intuition that such operations preserve volumes is not mathematically absurd and it is even included in the formal definition of volumes. However, this is not applicable here because in this case it is impossible to define the volumes of the considered subsets. Reassembling them reproduces a set that has a volume, which happens to be different from the volume at the start.
1282:) defined on all subsets of a Euclidean space of three (and greater) dimensions that is invariant with respect to Euclidean motions and takes the value one on a unit cube. In his later work, Tarski showed that, conversely, non-existence of paradoxical decompositions of this type implies the existence of a finitely-additive invariant measure.
4675:), hence one cannot simply transfer a paradoxical decomposition from the group to the square, as in the third step of the above proof of the Banach–Tarski paradox. Moreover, the fixed points of the group present difficulties (for example, the origin is fixed under all linear transformations). This is why von Neumann used the larger group
79:, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them, without changing their original shape. However, the pieces themselves are not "solids" in the traditional sense, but infinite scatterings of
1285:
The heart of the proof of the "doubling the ball" form of the paradox presented below is the remarkable fact that by a
Euclidean isometry (and renaming of elements), one can divide a certain set (essentially, the surface of a unit sphere) into four parts, then rotate one of them to become itself plus
1269:
On the other hand, in the Banach–Tarski paradox, the number of pieces is finite and the allowed equivalences are
Euclidean congruences, which preserve the volumes. Yet, somehow, they end up doubling the volume of the ball. While this is certainly surprising, some of the pieces used in the paradoxical
4384:
For step 4, it has already been shown that the ball minus a point admits a paradoxical decomposition; it remains to be shown that the ball minus a point is equidecomposable with the ball. Consider a circle within the ball, containing the point at the center of the ball. Using an argument like that
147:
rather than topological spaces. In this abstract setting, it is possible to have subspaces without point but still nonempty. The parts of the paradoxical decomposition do intersect a lot in the sense of locales, so much that some of these intersections should be given a positive mass. Allowing for
86:
An alternative form of the theorem states that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the
1497:
points out at the end of his monograph, the Banach–Tarski paradox has been more significant for its role in pure mathematics than for foundational questions: it motivated a fruitful new direction for research, the amenability of groups, which has nothing to do with the foundational questions.
479:
The Banach–Tarski paradox states that a ball in the ordinary
Euclidean space can be doubled using only the operations of partitioning into subsets, replacing a set with a congruent set, and reassembling. Its mathematical structure is greatly elucidated by emphasizing the role played by the
4594:
are necessarily of the same area, and therefore, a paradoxical decomposition of a square or disk of Banach–Tarski type that uses only
Euclidean congruences is impossible. A conceptual explanation of the distinction between the planar and higher-dimensional cases was given by
5121:
2019: Banach–Tarski paradox uses finitely many pieces in the duplication. In the case of countably many pieces, any two sets with non-empty interiors are equidecomposable using translations. But allowing only
Lebesgue measurable pieces one obtains: If A and B are subsets of
4619:
which is invariant under translations and rotations, and rules out paradoxical decompositions of non-negligible sets. Von
Neumann then posed the following question: can such a paradoxical decomposition be constructed if one allows a larger group of equivalences?
4443:
The proof sketched above requires 2 × 4 × 2 + 8 = 24 pieces - a factor of 2 to remove fixed points, a factor 4 from step 1, a factor 2 to recreate fixed points, and 8 for the center point of the second ball. But in step 1 when moving
935:
4666:
with two generators as a subgroup. This makes it plausible that the proof of Banach–Tarski paradox can be imitated in the plane. The main difficulty here lies in the fact that the unit square is not invariant under the action of the linear group
4722:
of sets in the plane, which is preserved by translations and rotations, is not preserved by non-isometric transformations even when they do preserve the area of polygons. The points of the plane (other than the origin) can be divided into two
4464:} of this last orbit to the center point of the second ball. This brings the total down to 16 + 1 pieces. With more algebra, one can also decompose fixed orbits into 4 sets as in step 1. This gives 5 pieces and is the best possible.
3172:
5134:
2024: Robert Samuel Simon and
Grzegorz Tomkowicz introduced a colouring rule of points in a Cantor space that links paradoxical decompositions with optimisation. This allows one to find an application of paradoxical decompositions in
4682:
including the translations, and he constructed a paradoxical decomposition of the unit square with respect to the enlarged group (in 1929). Applying the Banach–Tarski method, the paradox for the square can be strengthened as follows:
5126:
with non-empty interiors, then they have equal
Lebesgue measures if and only if they are countably equidecomposable using Lebesgue measurable pieces. Jan Mycielski and Grzegorz Tomkowicz extended this result to finite dimensional
4994:. This was shown by Jan Mycielski and Grzegorz Tomkowicz. Tomkowicz proved also that most of the classical paradoxes are an easy consequence of a graph theoretical result and the fact that the groups in question are rich enough.
4989:
2017: Von
Neumann's paradox concerns the Euclidean plane, but there are also other classical spaces where the paradoxes are possible. For example, one can ask if there is a Banach–Tarski paradox in the hyperbolic plane
4758:, or groups with an invariant mean, and include all finite and all solvable groups. Generally speaking, paradoxical decompositions arise when the group used for equivalences in the definition of equidecomposability is
829:
1983:
3021:
4638:. Since the area is preserved, any paradoxical decomposition of a square with respect to this group would be counterintuitive for the same reasons as the Banach–Tarski decomposition of a ball. In fact, the group
1053:
4707:"In accordance with this, already in the plane there is no non-negative additive measure (for which the unit square has a measure of 1), which is invariant with respect to all transformations belonging to
1278:) is not defined for them, and the partitioning cannot be accomplished in a practical way. In fact, the Banach–Tarski paradox demonstrates that it is impossible to find a finitely-additive measure (or a
2919:
2767:
467:
Tarski proved that amenable groups are precisely those for which no paradoxical decompositions exist. Since only free subgroups are needed in the Banach–Tarski paradox, this led to the long-standing
3226:
4572:
using rotations. These results then extend to the unit ball deprived of the origin. A 2010 article by Valeriy Churkin gives a new proof of the continuous version of the Banach–Tarski paradox.
4097:
4027:
3934:
1308:
with two generators. Banach and Tarski's proof relied on an analogous fact discovered by Hausdorff some years earlier: the surface of a unit sphere in space is a disjoint union of three sets
3864:
4747:
points in two new polygons. The new polygons have the same area as the old polygon, but the two transformed sets cannot have the same measure as before (since they contain only part of the
3770:
326:
274:
3664:
4718:
To explain further, the question of whether a finitely additive measure (that is preserved under certain transformations) exists or not depends on what transformations are allowed. The
2141:
2062:
4127:. However, there are only countably many such points, and like the case of the point at the center of the ball, it is possible to patch the proof to account for them all. (See below.)
3712:
3602:
3418:
This step cannot be performed in two dimensions since it involves rotations in three dimensions. If two rotations are taken about the same axis, the resulting group is the abelian
184:, and discussed a number of related questions concerning decompositions of subsets of Euclidean spaces in various dimensions. They proved the following more general statement, the
1529:
Here a proof is sketched which is similar but not identical to that given by Banach and Tarski. Essentially, the paradoxical decomposition of the ball is achieved in four steps:
3957:
The (majority of the) sphere has now been divided into four sets (each one dense on the sphere), and when two of these are rotated, the result is double of what was had before:
728:
682:
5076:
837:
5104:
6415:
4968:
4910:
4566:
3380:
2437:
2509:
2361:
2249:
971:
4754:
The class of groups isolated by von Neumann in the course of study of Banach–Tarski phenomenon turned out to be very important for many areas of Mathematics: these are
3425:
An alternative arithmetic proof of the existence of free groups in some special orthogonal groups using integral quaternions leads to paradoxical decompositions of the
1697:. Two such strings can be concatenated and converted into a string of this type by repeatedly replacing the "forbidden" substrings with the empty string. For instance:
2207:
140:
It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved continuously into place without running into one another.
3346:
3318:
3298:
3079:
3041:
2834:
2795:
2589:
2559:
2467:
2391:
2299:
1246:, then all sets with non-empty interior become congruent. Likewise, one ball can be made into a larger or smaller ball by stretching, or in other words, by applying
148:
this hidden mass to be taken into account, the theory of locales permits all subsets (and even all sublocales) of the Euclidean space to be satisfactorily measured.
2685:
2658:
1833:
1798:
3252:
3084:
1431:
should not be rejected solely because it produces a paradoxical decomposition, for such an argument also undermines proofs of geometrically intuitive statements.
3272:
2609:
2529:
2319:
2269:
415:
subsets are allowed. The difference between dimensions 1 and 2 on the one hand, and 3 and higher on the other hand, is due to the richer structure of the group
4115:
then yields a paradoxical decomposition of the solid unit ball minus the point at the ball's center. (This center point needs a bit more care; see below.)
4695:"Infolgedessen gibt es bereits in der Ebene kein nichtnegatives additives Maß (wo das Einheitsquadrat das Maß 1 hat), das gegenüber allen Abbildungen von
411:
The strong form of the Banach–Tarski paradox is false in dimensions one and two, but Banach and Tarski showed that an analogous statement remains true if
6364:
by David Morgan-Mar provides a non-technical explanation of the paradox. It includes a step-by-step demonstration of how to create two spheres from one.
4771:
2000: Von Neumann's paper left open the possibility of a paradoxical decomposition of the interior of the unit square with respect to the linear group
4123:
This sketch glosses over some details. One has to be careful about the set of points on the sphere which happen to lie on the axis of some rotation in
4835:
2011: Laczkovich's paper left open the possibility that there exists a free group F of piecewise linear transformations acting on the punctured disk
1380:, Hausdorff's paradox (1914), and an earlier (1923) paper of Banach as the precursors to their work. Vitali's and Hausdorff's constructions depend on
736:
1869:
5009:. A similar paradox was obtained in 2018 by Grzegorz Tomkowicz, who constructed a free properly discontinuous subgroup G of the affine group
4687:
Any two bounded subsets of the Euclidean plane with non-empty interiors are equidecomposable with respect to the area-preserving affine maps.
6266:
5206:
2924:
31:
4627:, any two squares in the plane become equivalent even without further subdivision. This motivates restricting one's attention to the group
1551:
Use the paradoxical decomposition of that group and the axiom of choice to produce a paradoxical decomposition of the hollow unit sphere.
5576:
5175:
4139:. To streamline the proof, the discussion of points that are fixed by some rotation was omitted; since the paradoxical decomposition of
979:
1224:
The Banach–Tarski paradox can be put in context by pointing out that for two sets in the strong form of the paradox, there is always a
6369:
1181:
While apparently more general, this statement is derived in a simple way from the doubling of a ball by using a generalization of the
6325:
6299:
6276:
5241:
Wilson, Trevor M. (September 2005). "A continuous movement version of the Banach–Tarski paradox: A solution to De Groot's problem".
5131:
and second countable locally compact topological groups that are totally disconnected or have countably many connected components.
456:
studied the properties of the group of equivalences that make a paradoxical decomposition possible, and introduced the notion of
396:
can be divided into a certain number of pieces and then be rotated and translated in such a way that the result is the whole set
2847:
1435:
1182:
6410:
6173:
6380:
6430:
6355:
3177:
2725:
6216:
4821:, and that the minimal number of pieces would equal four provided that there exists a locally commutative free subgroup of
6420:
4787:
be the family of all bounded subsets of the plane with non-empty interior and at a positive distance from the origin, and
1392:"), which is also crucial to the Banach–Tarski paper, both for proving their paradox and for the proof of another result:
98:
because it contradicts basic geometric intuition. "Doubling the ball" by dividing it into parts and moving them around by
1596:). Traversing a horizontal edge of the graph in the rightward direction represents left multiplication of an element of
6395:
4146:
relies on shifting certain subsets, the fact that some points are fixed might cause some trouble. Since any rotation of
6002:
Mycielski, Jan; Tomkowicz, Grzegorz (2019). "On the equivalence of sets of equal measures by countable decomposition".
41:"Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original?"
6425:
4033:
3963:
3875:
4510:
can be partitioned into countably infinitely many pieces, each of which is equidecomposable (with two pieces) to the
3781:
1266:
can be made into two copies of itself, one might expect that using countably many pieces could somehow do the trick.
5296:
Théorie de la mesure dans les lieux réguliers. ou : Les intersections cachées dans le paradoxe de Banach-Tarski
4839:\ {(0,0)} without fixed points. Grzegorz Tomkowicz constructed such a group, showing that the system of congruences
4492:
pieces so that each of them is equidecomposable to a ball of the same size as the original. Using the fact that the
3718:
3449:
1442:
503:
279:
227:
143:
As proved independently by Leroy and Simpson, the Banach–Tarski paradox does not violate volumes if one works with
1607:; traversing a vertical edge of the graph in the upward direction represents left multiplication of an element of
133:, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an
4278:
4151:
3608:
3505:
1655:
1538:
1454:
4791:
the family of all planar sets with the property that a union of finitely many translates under some elements of
2073:
1994:
1513:. The Hahn–Banach theorem does not rely on the full axiom of choice but can be proved using a weaker version of
3670:
144:
1510:
390:
be the union of two translated copies of the original ball. Then the proposition means that the original ball
5347:
5215:
4997:
2018: In 1984, Jan Mycielski and Stan Wagon constructed a paradoxical decomposition of the hyperbolic plane
3550:
6405:
6239:
6121:
5599:
5378:
5170:
106:, without any stretching, bending, or adding new points, seems to be impossible, since all these operations
65:
6037:
Simon, Robert; Tomkowicz, Grzegorz (2024). "A measure theoretic paradox from a continuous colouring rule".
5250:
5002:
930:{\displaystyle \quad A_{i}\cap A_{j}=B_{i}\cap B_{j}=\emptyset \quad {\text{for all }}1\leq i<j\leq k,}
468:
103:
2801:
axis (there are many other suitable pairs of irrational multiples of π that could be used here as well).
1228:
function that can map the points in one shape into the other in a one-to-one fashion. In the language of
6400:
6309:
4624:
1247:
687:
641:
562:
461:
371:
122:
5040:
1159:: doubling the ball can be accomplished with five pieces, and fewer than five pieces will not suffice.
5081:
6341:
5814:
Tomkowicz, Grzegorz (2017). "On decompositions of the hyperbolic plane satisfying many congruences".
4974:. Grzegorz Tomkowicz showed that Adams and Mycielski construction can be generalized to obtain a set
4939:
4881:
4780:
4635:
4537:
3445:
3426:
1174:
629:
5255:
6177:
5407:
5181:
4581:
3359:
2396:
1156:
1152:
481:
209:
1434:
However, in 1949, A. P. Morse showed that the statement about Euclidean polygons can be proved in
1423:
They point out that while the second result fully agrees with geometric intuition, its proof uses
6351:
6201:
6161:
6046:
6019:
5984:
5831:
5566:
5299:
5276:
5268:
4503:
1427:
in an even more substantial way than the proof of the paradox. Thus Banach and Tarski imply that
1271:
130:
118:
61:
4829:
2472:
2324:
2212:
1566:
943:
17:
6321:
6295:
6272:
5549:
5435:
5160:
4739:
points of a given polygon are transformed by a certain area-preserving transformation and the
3485:
2176:
1545:
1518:
1361:
69:
5967:
Tomkowicz, Grzegorz (2018). "A properly discontinuous free group of affine transformations".
5948:
Mycielski, Jan; Wagon, Stan (1984). "Large free groups of isometries and their geometrical".
3331:
3303:
3277:
3167:{\textstyle \left({\frac {k}{3^{N}}},{\frac {l{\sqrt {2}}}{3^{N}}},{\frac {m}{3^{N}}}\right)}
3046:
3026:
2819:
2780:
1505:
and Friedrich Wehrung, Janusz Pawlikowski proved that the Banach–Tarski paradox follows from
6347:
6248:
6227:
6193:
6153:
6130:
6083:
6056:
6011:
5976:
5928:
5895:
5862:
5823:
5794:
5763:
5734:
5671:
5608:
5558:
5509:
5473:
5423:
5387:
5342:
5332:
5260:
4921:
4814:
2003: It had been known for a long time that the full plane was paradoxical with respect to
4596:
4591:
2564:
2534:
2442:
2366:
2274:
1756:
1401:
1275:
485:
453:
427:
80:
5138:
2024: Grzegorz Tomkowicz proved that in the case of non-supramenable connected Lie groups
2663:
2636:
1811:
1776:
4587:
4523:
3473:
1634:) are blue dots or red dots with blue border. Red dots with blue border are elements of
1502:
1385:
1373:
1167:
493:
181:
173:
134:
126:
4424:
can be done using number of pieces equal to the product of the numbers needed for taking
3231:
5494:
5458:
3476:
can be used to pick exactly one point from every orbit; collect these points into a set
6288:
4755:
4719:
4608:
4318:
4227:
3457:
3257:
2594:
2514:
2304:
2254:
1279:
457:
438:
412:
6231:
6144:
Churkin, V. A. (2010). "A continuous version of the Hausdorff–Banach–Tarski paradox".
5591:
5547:
Churkin, V. A. (2010). "A continuous version of the Hausdorff–Banach–Tarski paradox".
37:
6389:
6165:
6113:
6109:
6105:
6023:
5988:
5835:
5570:
5370:
5366:
5362:
4925:
4290:
1755:. One can check that the set of those strings with this operation forms a group with
1263:
177:
161:
157:
73:
6114:"Sur la décomposition des ensembles de points en parties respectivement congruentes"
5371:"Sur la décomposition des ensembles de points en parties respectivement congruentes"
5280:
3419:
1570:
1229:
4870:
2017: It has been known for a long time that there exists in the hyperbolic plane
3464:
which moves the first point into the second. (Note that the orbit of a point is a
1554:
Extend this decomposition of the sphere to a decomposition of the solid unit ball.
1414:
Le rôle que joue cet axiome dans nos raisonnements nous semble mériter l'attention
1143:
Using this terminology, the Banach–Tarski paradox can be reformulated as follows:
1147:
A three-dimensional Euclidean ball is equidecomposable with two copies of itself.
208:
of a Euclidean space in at least three dimensions, both of which have a nonempty
5459:"The Hahn–Banach theorem implies the existence of a non-Lebesgue measurable set"
5295:
5202:
5165:
3438:
2700:
2163:
1237:
193:
6087:
6060:
5337:
5320:
4799:) contains a punctured neighborhood of the origin. Then all sets in the family
6313:
6262:
6213:
6157:
5980:
5767:
5562:
4657:
4493:
2696:
1534:
1494:
1377:
1305:
1233:
460:. He also found a form of the paradox in the plane which uses area-preserving
449:
169:
54:
6361:
5867:
5850:
5264:
1419:(The role this axiom plays in our reasoning seems to us to deserve attention)
5799:
5782:
5428:
5128:
4724:
4527:
4166:
4119:
3465:
1225:
1171:
6253:
6135:
5613:
5514:
5478:
5392:
4534:
can be partitioned into as many pieces as there are real numbers (that is,
4214:
Let λ be some line through the origin that does not intersect any point in
561:. Two geometric figures that can be transformed into each other are called
6074:
Tomkowicz, Grzegorz (2024). "On bounded paradoxical sets and Lie groups".
3950:
as an extra piece after doubling, owing to the presence of the singleton {
1262:-equidecomposable sets may be found whose "size"s vary. Moreover, since a
824:{\displaystyle A=\bigcup _{i=1}^{k}A_{i},\quad B=\bigcup _{i=1}^{k}B_{i},}
5933:
5916:
4111:
with a half-open segment to the origin; the paradoxical decomposition of
1978:{\displaystyle F_{2}=\{e\}\cup S(a)\cup S(a^{-1})\cup S(b)\cup S(b^{-1})}
535:
99:
57:
5272:
5017:). The existence of such a group implies the existence of a subset E of
4916:. The requirement was satisfied by orientation-preserving isometries of
3300:. The same argument repeated (by symmetry of the problem) is valid when
1665:
consists of all finite strings that can be formed from the four symbols
6373:
6205:
6015:
5827:
3496:
can be reached in exactly one way by applying the proper rotation from
1397:
1381:
95:
50:
5900:
5883:
5739:
5722:
5676:
5659:
3016:{\displaystyle k_{1}>0,\ k_{2},k_{3},\ldots ,k_{n}\neq 0,\ n\geq 1}
4646:
4590:, two figures that are equidecomposable with respect to the group of
2173:
This is at the core of the proof. For example, there may be a string
572:
111:
76:
6197:
4514:
using rotations. By using analytic properties of the rotation group
6051:
4568:
pieces), so that each piece is equidecomposable with two pieces to
1240:. Thus, if one enlarges the group to allow arbitrary bijections of
5304:
4600:
4515:
4386:
1565:
1048:{\displaystyle g_{i}(A_{i})=B_{i}{\text{ for all }}1\leq i\leq k,}
36:
4832:
constructed such a subgroup, confirming that four pieces suffice.
5178: – Problem of cutting and reassembling a disk into a square
5144:
acting continuously and transitively on a metric space, bounded
5001:
that uses Borel sets. The paradox depends on the existence of a
4611:, which implies the existence of a finitely-additive measure on
1368:
Connection with earlier work and the role of the axiom of choice
5411:
4135:
In Step 3, the sphere was partitioned into orbits of our group
622:
can be partitioned into the same finite number of respectively
2699:
of rotations of 3D space, i.e. that behaves just like (or "is
5754:
Adams, John Frank (1954). "On decompositions of the sphere".
4783:
proved that such a decomposition exists. More precisely, let
5537:
INVARIANT MEASURES, EXPANDERS AND PROPERTY T MAXIME BERGERON
4811:. It follows that both families consist of paradoxical sets.
6377:
gives an overview on the fundamental basics of the paradox.
6184:
Stromberg, Karl (March 1979). "The Banach–Tarski paradox".
5495:"The Hahn–Banach theorem implies the Banach–Tarski paradox"
1469:
sufficient for proving the Banach–Tarski paradox, that is,
83:. The reconstruction can work with as few as five pieces.
4472:
Using the Banach–Tarski paradox, it is possible to obtain
2914:{\displaystyle \omega =\ldots b^{k_{3}}a^{k_{2}}b^{k_{1}}}
1286:
two of the other parts. This follows rather easily from a
5884:"The Banach-Tarski paradox for the hyperbolic plane (II)"
4751:
points), and therefore there is no measure that "works".
550:
into itself that preserve the distances, usually denoted
2762:{\textstyle \theta =\arccos \left({\frac {1}{3}}\right)}
1441:
and thus does not require the axiom of choice. In 1964,
4743:
points by another, both sets can become subsets of the
3221:{\displaystyle k,l,m\in \mathbb {Z} ,N\in \mathbb {N} }
2633:, then "reassembled" as two pieces to make one copy of
565:, and this terminology will be extended to the general
180:
and on the paradoxical decompositions of the sphere by
5660:"A locally commutative free group acting on the plane"
5438:, settled a question put forth by von Neumann in 1929:
3087:
2728:
2687:. That is exactly what is intended to do to the ball.
5084:
5043:
4942:
4884:
4540:
4273:. Let ρ be the rotation about λ by θ. Then ρ acts on
4036:
3966:
3878:
3869:
and likewise for the other sets, and where we define
3784:
3722:
3721:
3674:
3673:
3611:
3553:
3362:
3334:
3306:
3280:
3260:
3234:
3180:
3049:
3029:
2927:
2850:
2822:
2783:
2666:
2639:
2597:
2567:
2537:
2517:
2475:
2445:
2399:
2369:
2327:
2307:
2277:
2257:
2215:
2179:
2076:
1997:
1872:
1814:
1779:
982:
946:
840:
739:
690:
644:
282:
230:
5851:"The Banach-Tarski paradox for the hyperbolic plane"
1445:
proved that the axiom of choice is independent from
1400:, one of which strictly contains the other, are not
72:
a decomposition of the ball into a finite number of
6192:(3). Mathematical Association of America: 151–161.
5691:Laczkovich, Miklós (1999). "Paradoxical sets under
5628:Laczkovich, Miklós (1999). "Paradoxical sets under
3422:and does not have the property required in step 1.
2625:}), then two of them "shifted" by multiplying with
2621:
has been cut into four pieces (plus the singleton {
6287:
5917:"Banach-Tarski paradox in some complete manifolds"
5723:"A free group of piecewise linear transformations"
5098:
5070:
4962:
4904:
4560:
4506:rank, a similar proof yields that the unit sphere
4091:
4021:
3928:
3858:
3764:
3706:
3658:
3596:
3374:
3340:
3312:
3292:
3266:
3246:
3220:
3166:
3073:
3035:
3015:
2913:
2828:
2789:
2761:
2679:
2652:
2603:
2583:
2553:
2523:
2503:
2461:
2431:
2385:
2355:
2313:
2293:
2263:
2243:
2201:
2135:
2056:
1977:
1827:
1800:can be "paradoxically decomposed" as follows: Let
1792:
1047:
965:
929:
823:
722:
676:
320:
268:
6381:Banach-Tarski and the Paradox of Infinite Cloning
5348:20.500.11820/47f5df74-8a53-452a-88c0-d5489ee5d659
4807:)-equidecomposable, and likewise for the sets in
4936:that is a half, a third, a fourth and ... and a
4645:contains as a subgroup the special linear group
4269:is countable. So there exists an angle θ not in
4092:{\displaystyle bA_{4}=A_{1}\cup A_{2}\cup A_{4}}
4022:{\displaystyle aA_{2}=A_{2}\cup A_{3}\cup A_{4}}
3929:{\displaystyle B=a^{-1}M\cup a^{-2}M\cup \dots }
3396:The two rotations behave just like the elements
1558:These steps are discussed in more detail below.
1453:. A weaker version of an axiom of choice is the
4150:(other than the null rotation) has exactly two
3859:{\displaystyle S(a)M=\{s(x)|s\in S(a),x\in M\}}
2840:that starts with a positive rotation about the
1342:are pairwise congruent, and on the other hand,
3765:{\displaystyle \displaystyle A_{4}=S(b^{-1})M}
3411:: there is now a paradoxical decomposition of
1473:The Banach–Tarski paradox is not a theorem of
125:in a critical way. It can be proven using the
321:{\displaystyle B=B_{1}\cup \cdots \cup B_{k}}
269:{\displaystyle A=A_{1}\cup \cdots \cup A_{k}}
8:
5882:Mycielski, Jan; Tomkowicz, Grzegorz (2013).
4603:of rotations in three dimensions, the group
4381:, where ~ denotes "is equidecomposable to".
4226:be the set of angles, α, such that for some
3946:were not used directly, as they would leave
3853:
3803:
1892:
1886:
1412:
60:, which states the following: Given a solid
6290:The Pea and the Sun: A Mathematical Paradox
6004:Bulletin of the London Mathematical Society
5816:Bulletin of the London Mathematical Society
5704:Ann. Univ. Sci. Budapest. Eötvös Sect. Math
5641:Ann. Univ. Sci. Budapest. Eötvös Sect. Math
5575:Full text in Russian is available from the
4460:), do this to all orbits except one. Move {
3659:{\displaystyle A_{2}=S(a^{-1})M\setminus B}
1274:, so the notion of volume (more precisely,
524:-dimensional Euclidean space (for integral
6416:Theorems in the foundations of mathematics
5434:This article, based on an analysis of the
4576:Von Neumann paradox in the Euclidean plane
3488:and so each orbit can be identified with
3320:starts with a negative rotation about the
2710:, two orthogonal axes are taken (e.g. the
2660:and the other two to make another copy of
2136:{\displaystyle F_{2}=bS(b^{-1})\cup S(b),}
2057:{\displaystyle F_{2}=aS(a^{-1})\cup S(a),}
1835:consisting of all strings that start with
1162:The strong version of the paradox claims:
638:. Formally, if there exist non-empty sets
224:into a finite number of disjoint subsets,
6320:. Cambridge: Cambridge University Press.
6294:. Wellesley, Massachusetts: A.K. Peters.
6271:. Cambridge: Cambridge University Press.
6252:
6134:
6050:
5932:
5899:
5866:
5798:
5738:
5675:
5612:
5513:
5477:
5427:
5391:
5346:
5336:
5303:
5254:
5092:
5088:
5083:
5042:
4952:
4947:
4941:
4894:
4889:
4883:
4607:(2) of Euclidean motions of the plane is
4550:
4545:
4539:
4083:
4070:
4057:
4044:
4035:
4013:
4000:
3987:
3974:
3965:
3908:
3889:
3877:
3818:
3783:
3746:
3727:
3720:
3707:{\displaystyle \displaystyle A_{3}=S(b)M}
3679:
3672:
3635:
3616:
3610:
3558:
3552:
3361:
3333:
3305:
3279:
3259:
3233:
3214:
3213:
3200:
3199:
3179:
3151:
3142:
3131:
3119:
3113:
3102:
3093:
3086:
3048:
3028:
2986:
2967:
2954:
2932:
2926:
2903:
2898:
2886:
2881:
2869:
2864:
2849:
2821:
2782:
2745:
2727:
2671:
2665:
2644:
2638:
2596:
2572:
2566:
2542:
2536:
2516:
2511:contains all the strings that start with
2489:
2474:
2450:
2444:
2420:
2407:
2398:
2374:
2368:
2363:contains all the strings that start with
2341:
2326:
2306:
2282:
2276:
2256:
2229:
2214:
2187:
2178:
2103:
2081:
2075:
2024:
2002:
1996:
1963:
1923:
1877:
1871:
1819:
1813:
1784:
1778:
1372:Banach and Tarski explicitly acknowledge
1166:Any two bounded subsets of 3-dimensional
1022:
1016:
1000:
987:
981:
951:
945:
898:
885:
872:
859:
846:
839:
812:
802:
791:
771:
761:
750:
738:
714:
695:
689:
668:
649:
643:
312:
293:
281:
260:
241:
229:
6172:Edward Kasner & James Newman (1940)
4878:that is a third, a fourth and ... and a
4530:, one can further prove that the sphere
4468:Obtaining infinitely many balls from one
1533:Find a paradoxical decomposition of the
1449:– that is, choice cannot be proved from
186:strong form of the Banach–Tarski paradox
121:of this result depends on the choice of
110:, intuitively speaking, to preserve the
5194:
5005:subgroup of the group of isometries of
3650:
3597:{\displaystyle A_{1}=S(a)M\cup M\cup B}
1544:Find a group of rotations in 3-d space
129:, which allows for the construction of
4636:area-preserving affine transformations
3512:yields a paradoxical decomposition of
3456:: two points belong to the same orbit
2844:axis, that is, an element of the form
1626:) are green dots; elements of the set
1580:, showing decomposition into the sets
1501:In 1991, using then-recent results by
512:. In the most important special case,
117:Unlike most theorems in geometry, the
6318:The Banach–Tarski Paradox 2nd Edition
4920:. Analogous results were obtained by
4177:. Denote this set of fixed points as
4169:, there are countably many points of
1250:transformations. Hence, if the group
1203:is equidecomposable with a subset of
1191:is equidecomposable with a subset of
7:
6232:"Zur allgemeinen Theorie des Masses"
5592:"Zur allgemeinen Theorie des Masses"
5321:"Measure, randomness and sublocales"
5106:denotes the symmetric difference of
4502:of rank 2 admits a free subgroup of
4189:admits a paradoxical decomposition.
3023:. It can be shown by induction that
2804:The group of rotations generated by
1548:to the free group in two generators.
30:For the book about the paradox, see
4173:that are fixed by some rotation in
1185:due to Banach that implies that if
628:-congruent pieces. This defines an
464:in place of the usual congruences.
5089:
5062:
4949:
4891:
4779:) (Wagon, Question 7.4). In 2000,
4547:
4480:-space from one, for any integers
4476:copies of a ball in the Euclidean
3348:is given by a non-trivial word in
894:
723:{\displaystyle B_{1},\dots ,B_{k}}
677:{\displaystyle A_{1},\dots ,A_{k}}
25:
6186:The American Mathematical Monthly
5457:Foreman, M.; Wehrung, F. (1991).
5412:"On the Decomposition of Spheres"
5319:Simpson, Alex (1 November 2012).
5208:An introduction to measure theory
5071:{\displaystyle g(E)=E\triangle F}
4656:, which in its turn contains the
4488:≥ 1, i.e. a ball can be cut into
3939:(The five "paradoxical" parts of
3492:. In other words, every point in
1489:Large amounts of mathematics use
603:equidecomposable with respect to
5325:Annals of Pure and Applied Logic
5176:Tarski's circle-squaring problem
5099:{\displaystyle E\,\triangle \,F}
4928:who showed that the unit sphere
4623:It is clear that if one permits
4392:Use is made of the fact that if
4192:What remains to be shown is the
4107:Finally, connect every point on
2251:which, because of the rule that
2154:) means take all the strings in
332:), such that for each (integer)
32:The Banach–Tarski Paradox (book)
6174:Mathematics and the Imagination
4982:with the same properties as in
4963:{\displaystyle 2^{\aleph _{0}}}
4905:{\displaystyle 2^{\aleph _{0}}}
4561:{\displaystyle 2^{\aleph _{0}}}
3386:is a free group, isomorphic to
1741:, which contains the substring
1724:, which contains the substring
1348:is congruent with the union of
897:
841:
780:
488:and introducing the notions of
471:, which was disproved in 1980.
402:, which contains two copies of
18:Hausdorff–Banach–Tarski paradox
6356:Wolfram Demonstrations Project
5783:"On the paradox of the sphere"
5053:
5047:
5021:such that for any finite F of
4448:} and all strings of the form
3838:
3832:
3819:
3815:
3809:
3794:
3788:
3755:
3739:
3697:
3691:
3644:
3628:
3576:
3570:
3324:axis, or a rotation about the
3068:
3050:
2591:, as well as the empty string
2498:
2482:
2350:
2334:
2238:
2222:
2127:
2121:
2112:
2096:
2048:
2042:
2033:
2017:
1972:
1956:
1947:
1941:
1932:
1916:
1907:
1901:
1295:-paradoxical decomposition of
1006:
993:
544:, i.e. the transformations of
164:gave a construction of such a
156:In a paper published in 1924,
1:
5150:paradoxical sets are generic.
3375:{\displaystyle \omega \neq e}
2722:is taken to be a rotation of
2432:{\displaystyle aa^{-1}a^{-1}}
1485:, assuming their consistency.
1461:, and it has been shown that
152:Banach and Tarski publication
5915:Tomkowicz, Grzegorz (2017).
5721:Tomkowicz, Grzegorz (2011).
5493:Pawlikowski, Janusz (1991).
4859:can be realized by means of
4261:α) is a rotation about λ of
3274:modulo 3, one can show that
1685:appears directly next to an
1376:'s 1905 construction of the
1328:such that, on the one hand,
1236:, these two sets have equal
1155:result in this case, due to
6370:"The Banach–Tarski Paradox"
6286:Wapner, Leonard M. (2005).
6217:"The Banach–Tarski Paradox"
5214:. p. 3. Archived from
3500:to the proper element from
1763:. This group may be called
1693:appears directly next to a
1360:. This is often called the
1183:Bernstein–Schroeder theorem
6447:
6088:10.1007/s10711-024-00923-1
6061:10.1007/s00208-023-02644-4
5338:10.1016/j.apal.2011.12.014
4579:
2504:{\displaystyle aS(a^{-1})}
2356:{\displaystyle aS(a^{-1})}
2244:{\displaystyle aS(a^{-1})}
966:{\displaystyle g_{i}\in G}
212:, there are partitions of
29:
6362:Irregular Webcomic! #2339
6348:The Banach-Tarski Paradox
6268:The Banach–Tarski Paradox
6158:10.1007/s10469-010-9080-y
5981:10.1007/s10711-018-0320-y
5563:10.1007/s10469-010-9080-y
5243:Journal of Symbolic Logic
5184: – Geometric theorem
4218:. This is possible since
4204:is equidecomposable with
4158:, which is isomorphic to
4131:Some details, fleshed out
3506:paradoxical decomposition
3328:axis. This shows that if
2393:(for example, the string
1731:, and so gets reduced to
1455:axiom of dependent choice
1088:has two disjoint subsets
1058:then it can be said that
940:and there exist elements
384:be the original ball and
166:paradoxical decomposition
5868:10.4064/fm-132-2-143-149
5025:there exists an element
2301:, reduces to the string
2271:must not appear next to
2202:{\displaystyle aa^{-1}b}
1748:, which gets reduced to
1654:The free group with two
1642:), which is a subset of
1130:-equidecomposable, then
1076:-equidecomposable using
94:The theorem is called a
6240:Fundamenta Mathematicae
6122:Fundamenta Mathematicae
5849:Mycielski, Jan (1989).
5800:10.4064/fm-42-2-348-355
5781:Mycielski, Jan (1955).
5768:10.1112/jlms/s1-29.1.96
5727:Colloquium Mathematicum
5664:Fundamenta Mathematicae
5600:Fundamenta Mathematicae
5590:Neumann, J. v. (1929).
5502:Fundamenta Mathematicae
5466:Fundamenta Mathematicae
5429:10.4064/fm-34-1-246-260
5379:Fundamenta Mathematicae
5294:Olivier, Leroy (1995).
5171:Paradoxes of set theory
4416:. The decomposition of
3504:. Because of this, the
3460:there is a rotation in
3382:. Therefore, the group
3341:{\displaystyle \omega }
3313:{\displaystyle \omega }
3293:{\displaystyle l\neq 0}
3074:{\displaystyle (1,0,0)}
3036:{\displaystyle \omega }
2829:{\displaystyle \omega }
2790:{\displaystyle \theta }
1618:. Elements of the set
89:pea and the Sun paradox
66:three-dimensional space
6411:Mathematical paradoxes
6254:10.4064/fm-13-1-73-116
6136:10.4064/fm-6-1-244-277
5614:10.4064/fm-13-1-73-116
5515:10.4064/fm-138-1-21-22
5479:10.4064/fm-138-1-13-19
5393:10.4064/fm-6-1-244-277
5265:10.2178/jsl/1122038921
5100:
5072:
5003:properly discontinuous
4964:
4906:
4691:As von Neumann notes:
4562:
4329:= 0, 1, 2, ... . Then
4093:
4023:
3930:
3860:
3766:
3708:
3660:
3598:
3376:
3342:
3314:
3294:
3268:
3248:
3222:
3168:
3075:
3037:
3017:
2915:
2830:
2791:
2763:
2681:
2654:
2605:
2585:
2584:{\displaystyle a^{-1}}
2555:
2554:{\displaystyle b^{-1}}
2525:
2505:
2463:
2462:{\displaystyle a^{-1}}
2433:
2387:
2386:{\displaystyle a^{-1}}
2357:
2315:
2295:
2294:{\displaystyle a^{-1}}
2265:
2245:
2203:
2166:them on the left with
2137:
2058:
1979:
1863:) similarly. Clearly,
1829:
1794:
1651:
1413:
1221:are equidecomposable.
1049:
967:
931:
825:
807:
766:
724:
678:
469:von Neumann conjecture
462:affine transformations
322:
270:
42:
6431:Paradoxes of infinity
6342:Banach–Tarski paradox
5921:Proc. Amer. Math. Soc
5447:Wagon, Corollary 13.3
5231:Wagon, Corollary 13.3
5101:
5073:
4965:
4907:
4563:
4309:) is disjoint from ρ(
4181:. Step 3 proves that
4094:
4024:
3931:
3861:
3767:
3709:
3661:
3599:
3377:
3343:
3315:
3295:
3269:
3249:
3223:
3169:
3076:
3038:
3018:
2916:
2831:
2792:
2764:
2682:
2680:{\displaystyle F_{2}}
2655:
2653:{\displaystyle F_{2}}
2606:
2586:
2556:
2526:
2506:
2464:
2434:
2388:
2358:
2316:
2296:
2266:
2246:
2204:
2138:
2059:
1980:
1830:
1828:{\displaystyle F_{2}}
1795:
1793:{\displaystyle F_{2}}
1569:
1525:A sketch of the proof
1177:are equidecomposable.
1050:
968:
932:
826:
787:
746:
725:
679:
632:among all subsets of
490:equidecomposable sets
452:with two generators.
430:in 3 dimensions. For
323:
271:
123:axioms for set theory
47:Banach–Tarski paradox
40:
6421:Geometric dissection
6178:Simon & Schuster
5658:Satô, Kenzi (2003).
5408:Robinson, Raphael M.
5082:
5041:
4940:
4882:
4727:which may be called
4538:
4034:
3964:
3876:
3782:
3719:
3671:
3609:
3551:
3484:on a given orbit is
3444:is partitioned into
3360:
3332:
3304:
3278:
3258:
3232:
3178:
3085:
3047:
3027:
2925:
2848:
2820:
2781:
2777:to be a rotation of
2726:
2703:to") the free group
2664:
2637:
2595:
2565:
2535:
2515:
2473:
2443:
2397:
2367:
2325:
2305:
2275:
2255:
2213:
2177:
2074:
1995:
1870:
1812:
1777:
1378:set bearing his name
1322:and a countable set
1151:In fact, there is a
980:
944:
838:
737:
688:
642:
630:equivalence relation
280:
228:
6396:Eponymous paradoxes
6310:Tomkowicz, Grzegorz
5756:J. London Math. Soc
5182:Von Neumann paradox
4599:: unlike the group
4582:Von Neumann paradox
3486:free and transitive
3247:{\displaystyle k,l}
2695:In order to find a
2146:where the notation
1808:) be the subset of
1511:Hahn–Banach theorem
1272:non-measurable sets
1157:Raphael M. Robinson
1024: for all
137:number of choices.
131:non-measurable sets
6426:1924 introductions
6352:Macalester College
6016:10.1112/blms.12289
5934:10.1090/proc/13657
5828:10.1112/blms.12024
5096:
5068:
4960:
4902:
4558:
4504:countably infinite
4297:, and for natural
4222:is countable. Let
4089:
4019:
3926:
3856:
3762:
3761:
3704:
3703:
3656:
3594:
3372:
3338:
3310:
3290:
3264:
3244:
3218:
3164:
3071:
3033:
3013:
2911:
2826:
2787:
2759:
2677:
2650:
2601:
2581:
2551:
2521:
2501:
2459:
2429:
2383:
2353:
2311:
2291:
2261:
2241:
2199:
2133:
2054:
1975:
1825:
1790:
1704:concatenated with
1652:
1270:decomposition are
1045:
963:
927:
821:
720:
674:
328:(for some integer
318:
266:
119:mathematical proof
43:
6228:von Neumann, John
6146:Algebra and Logic
5927:(12): 5359–5362.
5901:10.4064/fm222-3-5
5740:10.4064/cm125-2-1
5677:10.4064/fm180-1-3
5550:Algebra and Logic
5436:Hausdorff paradox
5331:(11): 1642–1659.
5161:Hausdorff paradox
4781:Miklós Laczkovich
4592:Euclidean motions
3516:into four pieces
3267:{\displaystyle m}
3157:
3137:
3124:
3108:
3003:
2949:
2836:be an element of
2753:
2604:{\displaystyle e}
2524:{\displaystyle b}
2439:which reduces to
2314:{\displaystyle b}
2264:{\displaystyle a}
1759:the empty string
1519:ultrafilter lemma
1362:Hausdorff paradox
1256:is large enough,
1082:pieces. If a set
1025:
901:
599:-equidecomposable
486:Euclidean motions
428:Euclidean motions
27:Geometric theorem
16:(Redirected from
6438:
6376:
6331:
6305:
6293:
6282:
6258:
6256:
6236:
6223:
6221:
6209:
6169:
6140:
6138:
6118:
6092:
6091:
6071:
6065:
6064:
6054:
6034:
6028:
6027:
5999:
5993:
5992:
5964:
5958:
5957:
5945:
5939:
5938:
5936:
5912:
5906:
5905:
5903:
5879:
5873:
5872:
5870:
5846:
5840:
5839:
5811:
5805:
5804:
5802:
5778:
5772:
5771:
5751:
5745:
5744:
5742:
5718:
5712:
5711:
5688:
5682:
5681:
5679:
5655:
5649:
5648:
5625:
5619:
5618:
5616:
5596:
5586:
5580:
5574:
5544:
5538:
5535:
5529:
5526:
5520:
5519:
5517:
5499:
5490:
5484:
5483:
5481:
5463:
5454:
5448:
5445:
5439:
5433:
5431:
5404:
5398:
5397:
5395:
5375:
5359:
5353:
5352:
5350:
5340:
5316:
5310:
5309:
5307:
5291:
5285:
5284:
5258:
5238:
5232:
5229:
5223:
5222:
5220:
5213:
5199:
5149:
5143:
5117:
5111:
5105:
5103:
5102:
5097:
5077:
5075:
5074:
5069:
5036:
5030:
4969:
4967:
4966:
4961:
4959:
4958:
4957:
4956:
4922:John Frank Adams
4911:
4909:
4908:
4903:
4901:
4900:
4899:
4898:
4702:invariant wäre."
4597:John von Neumann
4567:
4565:
4564:
4559:
4557:
4556:
4555:
4554:
4098:
4096:
4095:
4090:
4088:
4087:
4075:
4074:
4062:
4061:
4049:
4048:
4028:
4026:
4025:
4020:
4018:
4017:
4005:
4004:
3992:
3991:
3979:
3978:
3935:
3933:
3932:
3927:
3916:
3915:
3897:
3896:
3865:
3863:
3862:
3857:
3822:
3775:where we define
3771:
3769:
3768:
3763:
3754:
3753:
3732:
3731:
3713:
3711:
3710:
3705:
3684:
3683:
3665:
3663:
3662:
3657:
3643:
3642:
3621:
3620:
3603:
3601:
3600:
3595:
3563:
3562:
3480:. The action of
3381:
3379:
3378:
3373:
3347:
3345:
3344:
3339:
3319:
3317:
3316:
3311:
3299:
3297:
3296:
3291:
3273:
3271:
3270:
3265:
3253:
3251:
3250:
3245:
3227:
3225:
3224:
3219:
3217:
3203:
3173:
3171:
3170:
3165:
3163:
3159:
3158:
3156:
3155:
3143:
3138:
3136:
3135:
3126:
3125:
3120:
3114:
3109:
3107:
3106:
3094:
3080:
3078:
3077:
3072:
3042:
3040:
3039:
3034:
3022:
3020:
3019:
3014:
3001:
2991:
2990:
2972:
2971:
2959:
2958:
2947:
2937:
2936:
2920:
2918:
2917:
2912:
2910:
2909:
2908:
2907:
2893:
2892:
2891:
2890:
2876:
2875:
2874:
2873:
2835:
2833:
2832:
2827:
2796:
2794:
2793:
2788:
2768:
2766:
2765:
2760:
2758:
2754:
2746:
2686:
2684:
2683:
2678:
2676:
2675:
2659:
2657:
2656:
2651:
2649:
2648:
2610:
2608:
2607:
2602:
2590:
2588:
2587:
2582:
2580:
2579:
2560:
2558:
2557:
2552:
2550:
2549:
2530:
2528:
2527:
2522:
2510:
2508:
2507:
2502:
2497:
2496:
2469:). In this way,
2468:
2466:
2465:
2460:
2458:
2457:
2438:
2436:
2435:
2430:
2428:
2427:
2415:
2414:
2392:
2390:
2389:
2384:
2382:
2381:
2362:
2360:
2359:
2354:
2349:
2348:
2320:
2318:
2317:
2312:
2300:
2298:
2297:
2292:
2290:
2289:
2270:
2268:
2267:
2262:
2250:
2248:
2247:
2242:
2237:
2236:
2208:
2206:
2205:
2200:
2195:
2194:
2142:
2140:
2139:
2134:
2111:
2110:
2086:
2085:
2063:
2061:
2060:
2055:
2032:
2031:
2007:
2006:
1984:
1982:
1981:
1976:
1971:
1970:
1931:
1930:
1882:
1881:
1834:
1832:
1831:
1826:
1824:
1823:
1799:
1797:
1796:
1791:
1789:
1788:
1757:identity element
1416:
1402:equidecomposable
1359:
1353:
1347:
1341:
1327:
1321:
1303:
1294:
1276:Lebesgue measure
1261:
1255:
1245:
1220:
1214:
1208:
1202:
1196:
1190:
1135:
1129:
1123:
1117:
1111:
1105:
1099:
1093:
1087:
1081:
1075:
1069:
1063:
1054:
1052:
1051:
1046:
1026:
1023:
1021:
1020:
1005:
1004:
992:
991:
972:
970:
969:
964:
956:
955:
936:
934:
933:
928:
902:
899:
890:
889:
877:
876:
864:
863:
851:
850:
830:
828:
827:
822:
817:
816:
806:
801:
776:
775:
765:
760:
729:
727:
726:
721:
719:
718:
700:
699:
683:
681:
680:
675:
673:
672:
654:
653:
637:
627:
621:
615:
608:
598:
591:
585:
579:
570:
560:
549:
543:
534:consists of all
533:
523:
517:
511:
501:
475:Formal treatment
454:John von Neumann
447:
436:
425:
407:
401:
395:
389:
383:
369:
358:
347:
341:
337:
327:
325:
324:
319:
317:
316:
298:
297:
275:
273:
272:
267:
265:
264:
246:
245:
223:
217:
207:
201:
21:
6446:
6445:
6441:
6440:
6439:
6437:
6436:
6435:
6386:
6385:
6367:
6350:by Stan Wagon (
6338:
6328:
6308:
6302:
6285:
6279:
6261:
6234:
6226:
6219:
6212:
6198:10.2307/2321514
6183:
6143:
6116:
6104:
6101:
6096:
6095:
6073:
6072:
6068:
6036:
6035:
6031:
6001:
6000:
5996:
5966:
5965:
5961:
5947:
5946:
5942:
5914:
5913:
5909:
5881:
5880:
5876:
5848:
5847:
5843:
5813:
5812:
5808:
5780:
5779:
5775:
5753:
5752:
5748:
5720:
5719:
5715:
5697:
5690:
5689:
5685:
5657:
5656:
5652:
5634:
5627:
5626:
5622:
5594:
5589:
5587:
5583:
5577:Mathnet.ru page
5546:
5545:
5541:
5536:
5532:
5527:
5523:
5497:
5492:
5491:
5487:
5461:
5456:
5455:
5451:
5446:
5442:
5406:
5405:
5401:
5373:
5361:
5360:
5356:
5318:
5317:
5313:
5293:
5292:
5288:
5256:10.1.1.502.6600
5240:
5239:
5235:
5230:
5226:
5218:
5211:
5201:
5200:
5196:
5191:
5157:
5145:
5139:
5113:
5107:
5080:
5079:
5039:
5038:
5032:
5026:
4948:
4943:
4938:
4937:
4932:contains a set
4890:
4885:
4880:
4879:
4827:
4820:
4768:
4766:Recent progress
4756:amenable groups
4713:
4701:
4681:
4665:
4644:
4633:
4588:Euclidean plane
4584:
4578:
4546:
4541:
4536:
4535:
4501:
4470:
4432:and for taking
4249:α)P is also in
4164:
4145:
4133:
4105:
4079:
4066:
4053:
4040:
4032:
4031:
4009:
3996:
3983:
3970:
3962:
3961:
3944:
3904:
3885:
3874:
3873:
3780:
3779:
3742:
3723:
3717:
3716:
3675:
3669:
3668:
3631:
3612:
3607:
3606:
3554:
3549:
3548:
3543:
3536:
3529:
3522:
3474:axiom of choice
3435:
3410:
3392:
3358:
3357:
3330:
3329:
3302:
3301:
3276:
3275:
3256:
3255:
3230:
3229:
3176:
3175:
3147:
3127:
3115:
3098:
3092:
3088:
3083:
3082:
3045:
3044:
3043:maps the point
3025:
3024:
2982:
2963:
2950:
2928:
2923:
2922:
2899:
2894:
2882:
2877:
2865:
2860:
2846:
2845:
2818:
2817:
2812:will be called
2779:
2778:
2741:
2724:
2723:
2709:
2693:
2667:
2662:
2661:
2640:
2635:
2634:
2620:
2593:
2592:
2568:
2563:
2562:
2538:
2533:
2532:
2513:
2512:
2485:
2471:
2470:
2446:
2441:
2440:
2416:
2403:
2395:
2394:
2370:
2365:
2364:
2337:
2323:
2322:
2303:
2302:
2278:
2273:
2272:
2253:
2252:
2225:
2211:
2210:
2183:
2175:
2174:
2099:
2077:
2072:
2071:
2020:
1998:
1993:
1992:
1959:
1919:
1873:
1868:
1867:
1815:
1810:
1809:
1780:
1775:
1774:
1769:
1613:
1602:
1579:
1564:
1527:
1503:Matthew Foreman
1386:axiom of choice
1374:Giuseppe Vitali
1370:
1355:
1349:
1343:
1329:
1323:
1309:
1302:
1296:
1293:
1287:
1257:
1251:
1241:
1216:
1210:
1204:
1198:
1192:
1186:
1168:Euclidean space
1131:
1125:
1119:
1113:
1107:
1101:
1095:
1089:
1083:
1077:
1071:
1065:
1059:
1012:
996:
983:
978:
977:
947:
942:
941:
881:
868:
855:
842:
836:
835:
808:
767:
735:
734:
710:
691:
686:
685:
664:
645:
640:
639:
633:
623:
617:
611:
604:
594:
587:
581:
575:
566:
551:
545:
539:
529:
519:
513:
507:
497:
496:. Suppose that
494:paradoxical set
477:
458:amenable groups
442:
431:
416:
403:
397:
391:
385:
379:
368:
360:
357:
349:
343:
339:
333:
308:
289:
278:
277:
256:
237:
226:
225:
219:
213:
203:
197:
182:Felix Hausdorff
176:concerning the
174:Giuseppe Vitali
154:
127:axiom of choice
35:
28:
23:
22:
15:
12:
11:
5:
6444:
6442:
6434:
6433:
6428:
6423:
6418:
6413:
6408:
6406:Measure theory
6403:
6398:
6388:
6387:
6384:
6383:
6378:
6365:
6359:
6345:
6337:
6336:External links
6334:
6333:
6332:
6326:
6306:
6300:
6283:
6277:
6259:
6224:
6214:Su, Francis E.
6210:
6181:
6170:
6141:
6110:Tarski, Alfred
6106:Banach, Stefan
6100:
6097:
6094:
6093:
6076:Geom. Dedicata
6066:
6029:
5994:
5969:Geom. Dedicata
5959:
5940:
5907:
5894:(3): 289–290.
5874:
5861:(2): 143–149.
5841:
5806:
5793:(2): 348–355.
5773:
5746:
5733:(2): 141–146.
5713:
5695:
5683:
5650:
5632:
5620:
5581:
5539:
5530:
5521:
5485:
5449:
5440:
5399:
5367:Tarski, Alfred
5363:Banach, Stefan
5354:
5311:
5286:
5249:(3): 946–952.
5233:
5224:
5221:on 6 May 2021.
5193:
5192:
5190:
5187:
5186:
5185:
5179:
5173:
5168:
5163:
5156:
5153:
5152:
5151:
5136:
5132:
5119:
5095:
5091:
5087:
5067:
5064:
5061:
5058:
5055:
5052:
5049:
5046:
4995:
4987:
4955:
4951:
4946:
4897:
4893:
4888:
4868:
4833:
4825:
4818:
4812:
4767:
4764:
4720:Banach measure
4716:
4715:
4711:
4704:
4703:
4699:
4689:
4688:
4679:
4663:
4642:
4631:
4580:Main article:
4577:
4574:
4553:
4549:
4544:
4499:
4469:
4466:
4319:disjoint union
4228:natural number
4162:
4143:
4132:
4129:
4104:
4101:
4100:
4099:
4086:
4082:
4078:
4073:
4069:
4065:
4060:
4056:
4052:
4047:
4043:
4039:
4029:
4016:
4012:
4008:
4003:
3999:
3995:
3990:
3986:
3982:
3977:
3973:
3969:
3942:
3937:
3936:
3925:
3922:
3919:
3914:
3911:
3907:
3903:
3900:
3895:
3892:
3888:
3884:
3881:
3867:
3866:
3855:
3852:
3849:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3821:
3817:
3814:
3811:
3808:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
3773:
3772:
3760:
3757:
3752:
3749:
3745:
3741:
3738:
3735:
3730:
3726:
3714:
3702:
3699:
3696:
3693:
3690:
3687:
3682:
3678:
3666:
3655:
3652:
3649:
3646:
3641:
3638:
3634:
3630:
3627:
3624:
3619:
3615:
3604:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3561:
3557:
3541:
3534:
3527:
3520:
3458:if and only if
3434:
3431:
3427:rotation group
3408:
3390:
3371:
3368:
3365:
3337:
3309:
3289:
3286:
3283:
3263:
3243:
3240:
3237:
3216:
3212:
3209:
3206:
3202:
3198:
3195:
3192:
3189:
3186:
3183:
3162:
3154:
3150:
3146:
3141:
3134:
3130:
3123:
3118:
3112:
3105:
3101:
3097:
3091:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3032:
3012:
3009:
3006:
3000:
2997:
2994:
2989:
2985:
2981:
2978:
2975:
2970:
2966:
2962:
2957:
2953:
2946:
2943:
2940:
2935:
2931:
2906:
2902:
2897:
2889:
2885:
2880:
2872:
2868:
2863:
2859:
2856:
2853:
2825:
2786:
2757:
2752:
2749:
2744:
2740:
2737:
2734:
2731:
2707:
2692:
2689:
2674:
2670:
2647:
2643:
2618:
2600:
2578:
2575:
2571:
2548:
2545:
2541:
2520:
2500:
2495:
2492:
2488:
2484:
2481:
2478:
2456:
2453:
2449:
2426:
2423:
2419:
2413:
2410:
2406:
2402:
2380:
2377:
2373:
2352:
2347:
2344:
2340:
2336:
2333:
2330:
2321:. Similarly,
2310:
2288:
2285:
2281:
2260:
2240:
2235:
2232:
2228:
2224:
2221:
2218:
2198:
2193:
2190:
2186:
2182:
2144:
2143:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2109:
2106:
2102:
2098:
2095:
2092:
2089:
2084:
2080:
2065:
2064:
2053:
2050:
2047:
2044:
2041:
2038:
2035:
2030:
2027:
2023:
2019:
2016:
2013:
2010:
2005:
2001:
1986:
1985:
1974:
1969:
1966:
1962:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1929:
1926:
1922:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1880:
1876:
1822:
1818:
1787:
1783:
1767:
1611:
1600:
1577:
1563:
1560:
1556:
1555:
1552:
1549:
1542:
1526:
1523:
1487:
1486:
1468:
1421:
1420:
1417:
1406:
1405:
1396:Two Euclidean
1369:
1366:
1300:
1291:
1280:Banach measure
1179:
1178:
1149:
1148:
1056:
1055:
1044:
1041:
1038:
1035:
1032:
1029:
1019:
1015:
1011:
1008:
1003:
999:
995:
990:
986:
962:
959:
954:
950:
938:
937:
926:
923:
920:
917:
914:
911:
908:
905:
896:
893:
888:
884:
880:
875:
871:
867:
862:
858:
854:
849:
845:
832:
831:
820:
815:
811:
805:
800:
797:
794:
790:
786:
783:
779:
774:
770:
764:
759:
756:
753:
749:
745:
742:
717:
713:
709:
706:
703:
698:
694:
671:
667:
663:
660:
657:
652:
648:
476:
473:
448:it contains a
413:countably many
376:
375:
364:
353:
315:
311:
307:
304:
301:
296:
292:
288:
285:
263:
259:
255:
252:
249:
244:
240:
236:
233:
192:Given any two
153:
150:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6443:
6432:
6429:
6427:
6424:
6422:
6419:
6417:
6414:
6412:
6409:
6407:
6404:
6402:
6399:
6397:
6394:
6393:
6391:
6382:
6379:
6375:
6371:
6366:
6363:
6360:
6357:
6353:
6349:
6346:
6343:
6340:
6339:
6335:
6329:
6327:9781107042599
6323:
6319:
6315:
6311:
6307:
6303:
6301:1-56881-213-2
6297:
6292:
6291:
6284:
6280:
6278:0-521-45704-1
6274:
6270:
6269:
6264:
6260:
6255:
6250:
6246:
6242:
6241:
6233:
6229:
6225:
6218:
6215:
6211:
6207:
6203:
6199:
6195:
6191:
6187:
6182:
6179:
6175:
6171:
6167:
6163:
6159:
6155:
6151:
6147:
6142:
6137:
6132:
6128:
6124:
6123:
6115:
6111:
6107:
6103:
6102:
6098:
6089:
6085:
6081:
6077:
6070:
6067:
6062:
6058:
6053:
6048:
6045:: 1441–1462.
6044:
6040:
6033:
6030:
6025:
6021:
6017:
6013:
6009:
6005:
5998:
5995:
5990:
5986:
5982:
5978:
5974:
5970:
5963:
5960:
5955:
5951:
5944:
5941:
5935:
5930:
5926:
5922:
5918:
5911:
5908:
5902:
5897:
5893:
5889:
5885:
5878:
5875:
5869:
5864:
5860:
5856:
5852:
5845:
5842:
5837:
5833:
5829:
5825:
5821:
5817:
5810:
5807:
5801:
5796:
5792:
5788:
5784:
5777:
5774:
5769:
5765:
5761:
5757:
5750:
5747:
5741:
5736:
5732:
5728:
5724:
5717:
5714:
5709:
5705:
5701:
5694:
5687:
5684:
5678:
5673:
5669:
5665:
5661:
5654:
5651:
5646:
5642:
5638:
5631:
5624:
5621:
5615:
5610:
5606:
5602:
5601:
5593:
5585:
5582:
5578:
5572:
5568:
5564:
5560:
5556:
5552:
5551:
5543:
5540:
5534:
5531:
5528:Wagon, p. 16.
5525:
5522:
5516:
5511:
5507:
5503:
5496:
5489:
5486:
5480:
5475:
5471:
5467:
5460:
5453:
5450:
5444:
5441:
5437:
5430:
5425:
5421:
5417:
5413:
5409:
5403:
5400:
5394:
5389:
5385:
5382:(in French).
5381:
5380:
5372:
5368:
5364:
5358:
5355:
5349:
5344:
5339:
5334:
5330:
5326:
5322:
5315:
5312:
5306:
5301:
5297:
5290:
5287:
5282:
5278:
5274:
5270:
5266:
5262:
5257:
5252:
5248:
5244:
5237:
5234:
5228:
5225:
5217:
5210:
5209:
5204:
5198:
5195:
5188:
5183:
5180:
5177:
5174:
5172:
5169:
5167:
5164:
5162:
5159:
5158:
5154:
5148:
5142:
5137:
5133:
5130:
5125:
5120:
5116:
5110:
5093:
5085:
5065:
5059:
5056:
5050:
5044:
5035:
5029:
5024:
5020:
5016:
5012:
5008:
5004:
5000:
4996:
4993:
4988:
4985:
4981:
4977:
4973:
4953:
4944:
4935:
4931:
4927:
4926:Jan Mycielski
4923:
4919:
4915:
4895:
4886:
4877:
4873:
4869:
4866:
4862:
4858:
4854:
4850:
4846:
4842:
4838:
4834:
4831:
4824:
4817:
4813:
4810:
4806:
4802:
4798:
4794:
4790:
4786:
4782:
4778:
4774:
4770:
4769:
4765:
4763:
4761:
4757:
4752:
4750:
4746:
4742:
4738:
4734:
4730:
4726:
4721:
4710:
4706:
4705:
4698:
4694:
4693:
4692:
4686:
4685:
4684:
4678:
4674:
4670:
4662:
4659:
4655:
4653:
4649:
4641:
4637:
4630:
4626:
4621:
4618:
4614:
4610:
4606:
4602:
4598:
4593:
4589:
4583:
4575:
4573:
4571:
4551:
4542:
4533:
4529:
4525:
4522:, which is a
4521:
4519:
4513:
4509:
4505:
4498:
4495:
4491:
4487:
4483:
4479:
4475:
4467:
4465:
4463:
4459:
4455:
4451:
4447:
4441:
4439:
4435:
4431:
4427:
4423:
4419:
4415:
4411:
4407:
4403:
4399:
4395:
4390:
4388:
4382:
4380:
4376:
4372:
4368:
4364:
4360:
4356:
4352:
4348:
4344:
4340:
4336:
4332:
4328:
4324:
4320:
4316:
4312:
4308:
4304:
4300:
4296:
4292:
4288:
4284:
4280:
4276:
4272:
4268:
4264:
4260:
4256:
4252:
4248:
4244:
4240:
4236:
4232:
4229:
4225:
4221:
4217:
4213:
4209:
4207:
4203:
4199:
4195:
4190:
4188:
4184:
4180:
4176:
4172:
4168:
4161:
4157:
4153:
4149:
4142:
4138:
4130:
4128:
4126:
4122:
4121:
4116:
4114:
4110:
4102:
4084:
4080:
4076:
4071:
4067:
4063:
4058:
4054:
4050:
4045:
4041:
4037:
4030:
4014:
4010:
4006:
4001:
3997:
3993:
3988:
3984:
3980:
3975:
3971:
3967:
3960:
3959:
3958:
3955:
3953:
3949:
3945:
3923:
3920:
3917:
3912:
3909:
3905:
3901:
3898:
3893:
3890:
3886:
3882:
3879:
3872:
3871:
3870:
3850:
3847:
3844:
3841:
3835:
3829:
3826:
3823:
3812:
3806:
3800:
3797:
3791:
3785:
3778:
3777:
3776:
3758:
3750:
3747:
3743:
3736:
3733:
3728:
3724:
3715:
3700:
3694:
3688:
3685:
3680:
3676:
3667:
3653:
3647:
3639:
3636:
3632:
3625:
3622:
3617:
3613:
3605:
3591:
3588:
3585:
3582:
3579:
3573:
3567:
3564:
3559:
3555:
3547:
3546:
3545:
3540:
3533:
3526:
3519:
3515:
3511:
3507:
3503:
3499:
3495:
3491:
3487:
3483:
3479:
3475:
3471:
3467:
3463:
3459:
3455:
3452:of our group
3451:
3447:
3443:
3440:
3432:
3430:
3428:
3423:
3421:
3416:
3414:
3407:
3404:in the group
3403:
3399:
3394:
3389:
3385:
3369:
3366:
3363:
3355:
3351:
3335:
3327:
3323:
3307:
3287:
3284:
3281:
3261:
3241:
3238:
3235:
3210:
3207:
3204:
3196:
3193:
3190:
3187:
3184:
3181:
3160:
3152:
3148:
3144:
3139:
3132:
3128:
3121:
3116:
3110:
3103:
3099:
3095:
3089:
3065:
3062:
3059:
3056:
3053:
3030:
3010:
3007:
3004:
2998:
2995:
2992:
2987:
2983:
2979:
2976:
2973:
2968:
2964:
2960:
2955:
2951:
2944:
2941:
2938:
2933:
2929:
2904:
2900:
2895:
2887:
2883:
2878:
2870:
2866:
2861:
2857:
2854:
2851:
2843:
2839:
2823:
2815:
2811:
2807:
2802:
2800:
2784:
2776:
2772:
2755:
2750:
2747:
2742:
2738:
2735:
2732:
2729:
2721:
2718:axes). Then,
2717:
2713:
2706:
2702:
2698:
2690:
2688:
2672:
2668:
2645:
2641:
2632:
2628:
2624:
2617:
2612:
2598:
2576:
2573:
2569:
2546:
2543:
2539:
2518:
2493:
2490:
2486:
2479:
2476:
2454:
2451:
2447:
2424:
2421:
2417:
2411:
2408:
2404:
2400:
2378:
2375:
2371:
2345:
2342:
2338:
2331:
2328:
2308:
2286:
2283:
2279:
2258:
2233:
2230:
2226:
2219:
2216:
2196:
2191:
2188:
2184:
2180:
2171:
2169:
2165:
2161:
2157:
2153:
2149:
2130:
2124:
2118:
2115:
2107:
2104:
2100:
2093:
2090:
2087:
2082:
2078:
2070:
2069:
2068:
2051:
2045:
2039:
2036:
2028:
2025:
2021:
2014:
2011:
2008:
2003:
1999:
1991:
1990:
1989:
1967:
1964:
1960:
1953:
1950:
1944:
1938:
1935:
1927:
1924:
1920:
1913:
1910:
1904:
1898:
1895:
1889:
1883:
1878:
1874:
1866:
1865:
1864:
1862:
1858:
1854:
1850:
1846:
1842:
1839:, and define
1838:
1820:
1816:
1807:
1803:
1785:
1781:
1771:
1766:
1762:
1758:
1754:
1751:
1747:
1744:
1740:
1737:
1734:
1730:
1727:
1723:
1720:
1717:
1714:
1710:
1707:
1703:
1700:
1696:
1692:
1688:
1684:
1681:such that no
1680:
1676:
1672:
1668:
1664:
1660:
1657:
1649:
1645:
1641:
1637:
1633:
1629:
1625:
1621:
1617:
1610:
1606:
1599:
1595:
1591:
1587:
1583:
1576:
1572:
1568:
1561:
1559:
1553:
1550:
1547:
1543:
1540:
1536:
1532:
1531:
1530:
1524:
1522:
1520:
1516:
1512:
1508:
1504:
1499:
1496:
1492:
1484:
1480:
1476:
1472:
1471:
1470:
1466:
1464:
1460:
1456:
1452:
1448:
1444:
1440:
1438:
1432:
1430:
1426:
1418:
1415:
1411:
1410:
1409:
1408:They remark:
1403:
1399:
1395:
1394:
1393:
1391:
1387:
1383:
1379:
1375:
1367:
1365:
1363:
1358:
1352:
1346:
1340:
1336:
1332:
1326:
1320:
1316:
1312:
1307:
1299:
1290:
1283:
1281:
1277:
1273:
1267:
1265:
1264:countable set
1260:
1254:
1249:
1244:
1239:
1235:
1231:
1227:
1222:
1219:
1213:
1207:
1201:
1195:
1189:
1184:
1176:
1173:
1169:
1165:
1164:
1163:
1160:
1158:
1154:
1146:
1145:
1144:
1141:
1139:
1134:
1128:
1122:
1116:
1112:, as well as
1110:
1104:
1098:
1092:
1086:
1080:
1074:
1068:
1062:
1042:
1039:
1036:
1033:
1030:
1027:
1017:
1013:
1009:
1001:
997:
988:
984:
976:
975:
974:
960:
957:
952:
948:
924:
921:
918:
915:
912:
909:
906:
903:
900:for all
891:
886:
882:
878:
873:
869:
865:
860:
856:
852:
847:
843:
834:
833:
818:
813:
809:
803:
798:
795:
792:
788:
784:
781:
777:
772:
768:
762:
757:
754:
751:
747:
743:
740:
733:
732:
731:
715:
711:
707:
704:
701:
696:
692:
669:
665:
661:
658:
655:
650:
646:
636:
631:
626:
620:
614:
609:
607:
600:
597:
590:
584:
578:
574:
571:-action. Two
569:
564:
558:
554:
548:
542:
537:
532:
527:
522:
516:
510:
505:
500:
495:
491:
487:
483:
474:
472:
470:
465:
463:
459:
455:
451:
445:
440:
437:the group is
434:
429:
423:
419:
414:
409:
406:
400:
394:
388:
382:
373:
367:
363:
356:
352:
346:
336:
331:
313:
309:
305:
302:
299:
294:
290:
286:
283:
261:
257:
253:
250:
247:
242:
238:
234:
231:
222:
216:
211:
206:
200:
195:
191:
190:
189:
187:
183:
179:
178:unit interval
175:
171:
167:
163:
162:Alfred Tarski
159:
158:Stefan Banach
151:
149:
146:
141:
138:
136:
132:
128:
124:
120:
115:
113:
109:
105:
101:
97:
92:
90:
84:
82:
78:
75:
71:
67:
63:
59:
56:
55:set-theoretic
52:
48:
39:
33:
19:
6401:Group theory
6372:– via
6344:at ProofWiki
6317:
6289:
6267:
6244:
6238:
6189:
6185:
6176:, pp 205–7,
6152:(1): 91–98.
6149:
6145:
6126:
6120:
6079:
6075:
6069:
6042:
6038:
6032:
6007:
6003:
5997:
5972:
5968:
5962:
5953:
5949:
5943:
5924:
5920:
5910:
5891:
5887:
5877:
5858:
5854:
5844:
5819:
5815:
5809:
5790:
5786:
5776:
5759:
5755:
5749:
5730:
5726:
5716:
5707:
5703:
5699:
5692:
5686:
5670:(1): 25–34.
5667:
5663:
5653:
5644:
5640:
5636:
5629:
5623:
5604:
5598:
5584:
5557:(1): 81–89.
5554:
5548:
5542:
5533:
5524:
5505:
5501:
5488:
5469:
5465:
5452:
5443:
5419:
5415:
5402:
5383:
5377:
5357:
5328:
5324:
5314:
5289:
5246:
5242:
5236:
5227:
5216:the original
5207:
5203:Tao, Terence
5197:
5146:
5140:
5123:
5114:
5108:
5033:
5027:
5022:
5018:
5014:
5010:
5006:
4998:
4991:
4983:
4979:
4975:
4971:
4970:-th part of
4933:
4929:
4917:
4913:
4912:-th part of
4875:
4871:
4864:
4860:
4856:
4852:
4848:
4844:
4840:
4836:
4822:
4815:
4808:
4804:
4800:
4796:
4792:
4788:
4784:
4776:
4772:
4759:
4753:
4748:
4744:
4740:
4736:
4732:
4728:
4717:
4708:
4696:
4690:
4676:
4672:
4668:
4660:
4651:
4647:
4639:
4628:
4625:similarities
4622:
4616:
4612:
4604:
4585:
4569:
4531:
4517:
4511:
4507:
4496:
4489:
4485:
4481:
4477:
4473:
4471:
4461:
4457:
4453:
4449:
4445:
4442:
4437:
4433:
4429:
4425:
4421:
4417:
4413:
4409:
4405:
4401:
4397:
4393:
4391:
4383:
4378:
4374:
4370:
4366:
4362:
4358:
4354:
4350:
4346:
4342:
4338:
4334:
4330:
4326:
4322:
4314:
4310:
4306:
4302:
4298:
4294:
4286:
4282:
4279:fixed points
4274:
4270:
4266:
4262:
4258:
4254:
4250:
4246:
4242:
4238:
4234:
4230:
4223:
4219:
4215:
4211:
4210:
4205:
4201:
4197:
4193:
4191:
4186:
4182:
4178:
4174:
4170:
4159:
4155:
4154:, and since
4152:fixed points
4147:
4140:
4136:
4134:
4124:
4118:
4117:
4112:
4108:
4106:
3956:
3951:
3947:
3940:
3938:
3868:
3774:
3544:as follows:
3538:
3531:
3524:
3517:
3513:
3509:
3501:
3497:
3493:
3489:
3481:
3477:
3469:
3461:
3453:
3441:
3436:
3424:
3420:circle group
3417:
3412:
3405:
3401:
3397:
3395:
3387:
3383:
3353:
3349:
3325:
3321:
3228:. Analyzing
2841:
2837:
2813:
2809:
2805:
2803:
2798:
2774:
2770:
2719:
2715:
2711:
2704:
2694:
2630:
2626:
2622:
2615:
2613:
2172:
2167:
2159:
2155:
2151:
2147:
2145:
2066:
1987:
1860:
1856:
1852:
1848:
1844:
1840:
1836:
1805:
1801:
1772:
1764:
1760:
1752:
1749:
1745:
1742:
1738:
1735:
1732:
1728:
1725:
1721:
1718:
1715:
1712:
1708:
1705:
1701:
1698:
1694:
1690:
1686:
1682:
1678:
1674:
1670:
1666:
1662:
1658:
1653:
1647:
1643:
1639:
1635:
1631:
1627:
1623:
1619:
1615:
1608:
1604:
1597:
1593:
1589:
1585:
1581:
1574:
1571:Cayley graph
1557:
1528:
1514:
1506:
1500:
1490:
1488:
1482:
1478:
1474:
1462:
1458:
1450:
1446:
1436:
1433:
1428:
1424:
1422:
1407:
1389:
1371:
1356:
1350:
1344:
1338:
1334:
1330:
1324:
1318:
1314:
1310:
1297:
1288:
1284:
1268:
1258:
1252:
1242:
1230:Georg Cantor
1223:
1217:
1211:
1205:
1199:
1193:
1187:
1180:
1161:
1150:
1142:
1137:
1132:
1126:
1120:
1114:
1108:
1102:
1096:
1090:
1084:
1078:
1072:
1066:
1060:
1057:
939:
634:
624:
618:
612:
605:
602:
595:
593:
588:
582:
576:
567:
556:
552:
546:
540:
530:
525:
520:
514:
508:
498:
489:
478:
466:
443:
432:
421:
417:
410:
404:
398:
392:
386:
380:
377:
365:
361:
354:
350:
344:
334:
329:
220:
214:
204:
198:
185:
170:earlier work
165:
155:
142:
139:
116:
107:
104:translations
93:
88:
85:
70:there exists
46:
44:
6314:Wagon, Stan
6263:Wagon, Stan
6129:: 244–277.
6010:: 961–966.
5822:: 133–140.
5422:: 246–260.
5386:: 244–277.
5166:Nikodym set
4233:, and some
3439:unit sphere
3174:, for some
2209:in the set
2164:concatenate
1517:called the
1238:cardinality
1138:paradoxical
592:are called
502:is a group
348:, the sets
168:, based on
135:uncountable
6390:Categories
6247:: 73–116.
6099:References
6052:2203.11158
5956:: 247–267.
5888:Fund. Math
5855:Fund. Math
5787:Fund. Math
5710:: 141–145.
5647:: 141–145.
5607:: 73–116.
5588:On p. 85.
5416:Fund. Math
5298:(Report).
5135:economics.
5129:Lie groups
5037:such that
4867:\ {(0,0)}.
4830:Kenzi Satô
4828:. In 2003
4803:are SL(2,
4762:amenable.
4725:dense sets
4658:free group
4494:free group
4285:, i.e., ρ(
2797:about the
2773:axis, and
2769:about the
2701:isomorphic
2697:free group
1773:The group
1656:generators
1546:isomorphic
1539:generators
1535:free group
1495:Stan Wagon
1443:Paul Cohen
1439:set theory
1306:free group
1248:similarity
1234:set theory
1136:is called
1100:such that
973:such that
730:such that
536:isometries
450:free group
441:, but for
6166:122711859
6039:Math. Ann
6024:209936338
5989:126151042
5975:: 91–95.
5950:Ens. Math
5836:125603157
5762:: 96–99.
5571:122711859
5508:: 21–22.
5472:: 13–19.
5305:1303.5631
5251:CiteSeerX
5090:△
5063:△
4950:ℵ
4892:ℵ
4735:. If the
4548:ℵ
4528:Lie group
4526:analytic
4524:connected
4167:countable
4077:∪
4064:∪
4007:∪
3994:∪
3924:…
3921:∪
3910:−
3902:∪
3891:−
3848:∈
3827:∈
3748:−
3651:∖
3637:−
3589:∪
3583:∪
3466:dense set
3367:≠
3364:ω
3336:ω
3308:ω
3285:≠
3211:∈
3197:∈
3031:ω
3008:≥
2993:≠
2977:…
2858:…
2852:ω
2824:ω
2785:θ
2739:
2730:θ
2574:−
2544:−
2491:−
2452:−
2422:−
2409:−
2376:−
2343:−
2284:−
2231:−
2189:−
2116:∪
2105:−
2037:∪
2026:−
1988:but also
1965:−
1951:∪
1936:∪
1925:−
1911:∪
1896:∪
1509:plus the
1477:, nor of
1226:bijective
1175:interiors
1170:with non-
1037:≤
1031:≤
958:∈
919:≤
907:≤
895:∅
879:∩
853:∩
789:⋃
748:⋃
705:…
659:…
563:congruent
506:on a set
372:congruent
306:∪
303:⋯
300:∪
254:∪
251:⋯
248:∪
100:rotations
6368:Vsauce.
6316:(2016).
6265:(1994).
6230:(1929).
6112:(1924).
5410:(1947).
5369:(1924).
5281:15825008
5273:27588401
5205:(2011).
5155:See also
5078:, where
4615:(2) and
4609:solvable
4484:≥ 3 and
4291:disjoint
4277:with no
4265:α. Then
4253:, where
1398:polygons
439:solvable
378:Now let
338:between
210:interior
196:subsets
74:disjoint
58:geometry
6374:YouTube
6354:), the
6206:2321514
4586:In the
4408:, then
4325:) over
4317:be the
4313:). Let
3472:.) The
3448:by the
3356:, then
2816:. Let
1711:yields
1689:and no
1537:in two
1382:Zermelo
1209:, then
1124:, are
573:subsets
528:), and
194:bounded
145:locales
96:paradox
77:subsets
51:theorem
6324:
6298:
6275:
6204:
6164:
6022:
5987:
5834:
5569:
5279:
5271:
5253:
4874:a set
4345:) ~ ρ(
4212:Proof.
4103:Step 4
3450:action
3446:orbits
3433:Step 3
3002:
2948:
2736:arccos
2691:Step 2
2614:Group
2162:) and
1855:) and
1588:) and
1562:Step 1
1304:, the
518:is an
504:acting
492:and a
435:= 1, 2
112:volume
81:points
6235:(PDF)
6220:(PDF)
6202:JSTOR
6162:S2CID
6117:(PDF)
6047:arXiv
6020:S2CID
5985:S2CID
5832:S2CID
5595:(PDF)
5567:S2CID
5498:(PDF)
5462:(PDF)
5374:(PDF)
5300:arXiv
5277:S2CID
5269:JSTOR
5219:(PDF)
5212:(PDF)
5189:Notes
4601:SO(3)
4452:into
4436:into
4428:into
4420:into
4387:SO(3)
4365:) ∪ (
4357:) = (
4349:) ∪ (
4321:of ρ(
4293:from
4289:) is
4194:Claim
4165:, is
2921:with
1750:abaab
1493:. As
1172:empty
1153:sharp
610:, if
601:, or
482:group
108:ought
49:is a
6322:ISBN
6296:ISBN
6273:ISBN
6082:72.
5702:)".
5639:)".
5112:and
4924:and
4863:and
4795:(2,
4731:and
4671:(2,
4400:and
4373:) =
4305:, ρ(
4301:<
4120:N.B.
3954:}.)
3437:The
3400:and
3352:and
3254:and
2939:>
2808:and
2714:and
2561:and
2067:and
1733:abab
1719:abab
1713:abab
1706:abab
1699:abab
1677:and
1661:and
1354:and
1215:and
1197:and
1118:and
1106:and
1094:and
1070:are
1064:and
913:<
616:and
580:and
370:are
359:and
342:and
218:and
202:and
160:and
102:and
62:ball
45:The
6249:doi
6194:doi
6154:doi
6131:doi
6084:doi
6080:218
6057:doi
6043:389
6012:doi
5977:doi
5973:197
5929:doi
5925:145
5896:doi
5892:222
5863:doi
5859:132
5824:doi
5795:doi
5764:doi
5735:doi
5731:125
5672:doi
5668:180
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5559:doi
5510:doi
5506:138
5474:doi
5470:138
5424:doi
5388:doi
5343:hdl
5333:doi
5329:163
5261:doi
5031:of
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4978:of
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4760:not
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4634:of
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4337:∪ (
4281:in
4237:in
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1847:),
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1614:by
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1467:not
1465:is
1384:'s
1232:'s
586:of
538:of
484:of
446:≥ 3
426:of
172:by
91:".
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693:B
670:k
666:A
662:,
656:,
651:1
647:A
635:X
625:G
619:B
613:A
606:G
596:G
589:X
583:B
577:A
568:G
559:)
557:n
555:(
553:E
547:X
541:X
531:G
526:n
521:n
515:X
509:X
499:G
444:n
433:n
424:)
422:n
420:(
418:E
405:A
399:B
393:A
387:B
381:A
374:.
366:i
362:B
355:i
351:A
345:k
340:1
335:i
330:k
314:k
310:B
295:1
291:B
287:=
284:B
262:k
258:A
243:1
239:A
235:=
232:A
221:B
215:A
205:B
199:A
87:"
34:.
20:)
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