3 ms·

It has little to do with the Hilbert Hotel. It is sufficient to decompose the original ball into 5 (!) pieces: https://mathworld.wolfram.com/Banach-TarskiParad

It has little to do with the Hilbert Hotel. It is sufficient to decompose the original ball into 5 (!) pieces: https://mathworld.wolfram.com/Banach-TarskiParadox.html https://mathworld.wolfram.com/Banach-TarskiParadox.html It is an existence proof that relies on the Axiom of Choice, of course there is no algorithm to actually construct those pieces. This is one of the reasons that people distinguish between ZFC and ZF (Zermelo-Fraenkel with and without AC). ZFC is not regarded as sound by some because of these Paradoxes.

IngoBlechschmid 6y ago
> ZFC is not regarded as sound by some because of these Paradoxes. Can you give relevant pointers? I'm curious because while I do know a couple of arguments for the inconsistency of ZFC, none of these are related to the Banach–Tarski paradox or similar nonintuitive results. Also I'd like to stress that ZFC and ZF are exactly as consistent: If ZF is consistent, then so is ZFC (and vice versa). This meta-result is proven in a very weak finitary logical meta-system ("PRA"), hence can be trusted even if one is wary of set-theoretic infinities. The keyword is "Gödel's constructible hierarchy". In this line, there are even more astonishing meta-theorems, all provable in PRA: * From any ZFC-proof of a purely number-theoretical statement (a statement which refers only to natural numbers), any usage of the axiom of choice can be mechanically eliminated. That is, any ZFC-proof of such a result can be transformed into a ZF-proof. * From any ZFC-proof of a statement of the form "for all numbers n, there exists a number m such that ...", where in "..." no more quantifiers ("for all", "there exist") appear, any usage of the law of excluded middle ("any statement is either true or not") can be mechanically eliminated. That is, any such ZFC-proof can be transformed into a proof in IZF ("Intuitionistic Zermelo–Fraenkel"). The keyword is "double negation translation". These meta-theorems indicate that the axiom of choice and the law of excluded middle can be regarded as "mathematical phantoms". Just as the complex numbers work wonders for us but can be compiled away to pairs of real numbers, these set-theoretic principles promise to work wonders for us and their usage can be compiled away if we wish so.
- deleted 6y ago
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sukilot 6y ago
The construction involves a rotation of an infnitely detailed object in a way which is analogous to Hilbert Hotel. (Rotating it makes it fill empty space) B-T is Hilbert Hotel plus Axiom of Choice.

IngoBlechschmid 6y ago

> ZFC is not regarded as sound by some because of these Paradoxes. Can you give relevant pointers? I'm curious because while I do know a couple of arguments for the inconsistency of ZFC, none of these are related to the Banach–Tarski paradox or similar nonintuitive results. Also I'd like to stress that ZFC and ZF are exactly as consistent: If ZF is consistent, then so is ZFC (and vice versa). This meta-result is proven in a very weak finitary logical meta-system ("PRA"), hence can be trusted even if one is wary of set-theoretic infinities. The keyword is "Gödel's constructible hierarchy". In this line, there are even more astonishing meta-theorems, all provable in PRA: * From any ZFC-proof of a purely number-theoretical statement (a statement which refers only to natural numbers), any usage of the axiom of choice can be mechanically eliminated. That is, any ZFC-proof of such a result can be transformed into a ZF-proof. * From any ZFC-proof of a statement of the form "for all numbers n, there exists a number m such that ...", where in "..." no more quantifiers ("for all", "there exist") appear, any usage of the law of excluded middle ("any statement is either true or not") can be mechanically eliminated. That is, any such ZFC-proof can be transformed into a proof in IZF ("Intuitionistic Zermelo–Fraenkel"). The keyword is "double negation translation". These meta-theorems indicate that the axiom of choice and the law of excluded middle can be regarded as "mathematical phantoms". Just as the complex numbers work wonders for us but can be compiled away to pairs of real numbers, these set-theoretic principles promise to work wonders for us and their usage can be compiled away if we wish so.

deleted 6y ago
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sukilot 6y ago

The construction involves a rotation of an infnitely detailed object in a way which is analogous to Hilbert Hotel. (Rotating it makes it fill empty space) B-T is Hilbert Hotel plus Axiom of Choice.