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This problem has exactly the wrong setup for using th the axiom of choice: First, the axiom of choice requires that you have a countable number of sets that yo

This problem has exactly the wrong setup for using th the axiom of choice: First, the axiom of choice requires that you have a countable number of sets that you are choosing elements from, but there are undoubtably many of the equivalence classes that he described [0]. So the author is using something stronger than the axiom of choice to arrive at his paradox. Second, if you actually are in a situation where you have to choose from a countably infinite number of sets, you only need the axiom of choice if there is no selection rule for choosing an element available. In this case there is a rule you can use, namely: select the sequence in which the "finite prefix" is all zeros. [0]: the number of equivalence classes is uncountable because there is a 1:1 relation between the equivelance class and an the infinite sequence that is common to all the sequences in the equivalence class once theirs uncommon prefixes have been truncated.

n4r9 10y ago
> the axiom of choice requires that you have a countable number of sets that you are choosing elements from You're referring to the "axiom of countable choice", which is a different axiom. The Wikipedia entry for the axiom of choice makes it clear that the number of sets can be uncountable. > select the sequence in which the "finite prefix" is all zeros This doesn't really make sense as a selection rule. The size of the prefix can vary between pairs of members from the same equivalence class.
yequalsx 10y ago
The axiom of choice applies to index sets of arbitrary size. Indeed there is an axiom of countable choice for those who don't like the axiom of choice.
TimonKnigge 10y ago
> First, the axiom of voice requires that you have a countable number of sets that you are choosing elements from, but there are undoubtably many of the equivalence classes that he described [0]. No it does not? The axiom of countable choice [0] is a strictly weaker axiom. [0] https://en.wikipedia.org/wiki/Axiom_of_countable_choice https://en.wikipedia.org/wiki/Axiom_of_countable_choice

n4r9 10y ago

> the axiom of choice requires that you have a countable number of sets that you are choosing elements from You're referring to the "axiom of countable choice", which is a different axiom. The Wikipedia entry for the axiom of choice makes it clear that the number of sets can be uncountable. > select the sequence in which the "finite prefix" is all zeros This doesn't really make sense as a selection rule. The size of the prefix can vary between pairs of members from the same equivalence class.

yequalsx 10y ago

The axiom of choice applies to index sets of arbitrary size. Indeed there is an axiom of countable choice for those who don't like the axiom of choice.

TimonKnigge 10y ago

> First, the axiom of voice requires that you have a countable number of sets that you are choosing elements from, but there are undoubtably many of the equivalence classes that he described [0]. No it does not? The axiom of countable choice [0] is a strictly weaker axiom. [0] https://en.wikipedia.org/wiki/Axiom_of_countable_choice https://en.wikipedia.org/wiki/Axiom_of_countable_choice