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comment Who is the original author of this simple paradoxical decomposition?
@ Anton Petrunin: Of course I read Hausdorff in the German orginal: wwwuser.gwdg.de/~subtypo3/gdz/pdf/PPN235181684_0075/… Mazurkiewics, born in 1888, was a student when he found the paradox. So I guess it was about simultaneously with Hausdorff 1914, As I said: Hausdorff and BT need AC. This is a big difference.
Jan
9
comment Who is the original author of this simple paradoxical decomposition?
Thank you, I also read Stan Wagon's book which is certainly the standard source for this topic, in particular if the reader is not fluent in French. And I looked into a lot of other literature, but could not find the crucial idea $A\cup B$ = $\mathbb{Z}$ and simultaneously, after applying operators according to the idea of Sierpinski and Mazurkiewicz, $A' = \mathbb{Z}$ and $B' = \mathbb{Z}$. Therefore I asked here. But I will not yet give up. Perhaps there will be an expert on history here around who has come across the source on some old conference proceedings or lessons.
Jan
8
comment Who is the original author of this simple paradoxical decomposition?
@Anton Patrunin: Hausdorff's disadvantage: His proof requires AC. His advantage: His set, the surface of the unit sphere, is spatially bounded. Same with Banach-Tarski. Sierpinski and Mazurkiewicz get by without AC, alas their set is not spatially bounded. The idea in my question is of the same kind: Unbounded set, no AC required. I don't know whether this idea merits mentioning in literature. That's why I ask whether someone has seen it.
Jan
8
comment Who is the original author of this simple paradoxical decomposition?
@ YCor: Cantor was fully aware of the bijection between integers and even integers, already before he invented the term cardinal number in 1886, but he did not use this for paradoxical decompositions of point sets. Nonmeasurable sets were first shown by Vitali, then by Hausdorff.
Jan
8
comment Who is the original author of this simple paradoxical decomposition?
@Anton Patrunin: See the first sentence of my question, please.
Jan
8
comment Who is the original author of this simple paradoxical decomposition?
Hausdorff in his paper (Math. Ann. 75, 1914) did not hint to the AC (introduced in 1904), but it is necessary for this kind of decomposition.
Jan
8
comment Who is the original author of this simple paradoxical decomposition?
Thank you. Of course I know Hilbert's hotel, but I miss the idea of a paradoxical decomposition of a set.
Jan
8
asked Who is the original author of this simple paradoxical decomposition?
Oct
23
comment Is the analysis as taught in universities in fact the analysis of definable numbers?
@ Andreas Blass: Your last comment is based on the erroneous assumption that it is relevant whether one can define how to pass from a definition to the thing it defines. That is irrelevant because every thing defined by a single definition is unique (even if it is a variable or set). Taking into account that all languages belong to a countable set (because languages must be known by sender and receiver at least) we see that all things defined belong to a countable set too.
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