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Predicting Real Numbers

Regarding the above question, the solutions require creating classes of sequences with representative sequences.

How are those sequences constructed?

How is it possible to generate a "list" of the classes and representatives if the sequences are made of real numbers which implies the set of sequences is uncountable?

Thanks in advance for any answers.

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It is adequate for the question to define sequences as functions with domain $\mathbb{N}$. So a sequence of reals is just a function $\mathbb{N}\rightarrow\mathbb{R}$.

You are correct that there are uncountably many of them, but if you accept constructing $\mathbb{R}$ then why should the collection $\mathbb{R}^{\mathbb{N}} = \{f \ : \ f:\mathbb{N}\rightarrow\mathbb{R}\}$ be a problem?

They then go on to define an equivalence relation on $\mathbb{R}^{\mathbb{N}}$ by $f \sim g$ iff $\exists n \forall m >n f(m)=g(m)$ (this is called tail equivalence). They then pick a representative from each class, and this is where choice is used, it wasn't used before.

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    $\begingroup$ Hi James, thanks for your response. Could you elaborate on your last paragraph regarding how the equivalence relation is defined (not very familiar with the process) and how they pick the representative? I have only a very weak grasp of the axiom of choice. Any help would be greatly appreciated! $\endgroup$ – Complimentarity Aug 13 '15 at 9:50
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    $\begingroup$ So the right hand side of the iff is the definition of the relation. You should check it is one. More informally it says two sequences are equivalent of they are eventually the same. As for picking the representatives, it is strongly non-constructive. Choice says (basically) given a collection of non-empty bins, you can pick precisely one thing out of each bin, all at once. It doesn't explain how to pick, as the bins can be filled with weird things so that there can be no general procedure. Sometimes you don't need choice to show you can make the choices... $\endgroup$ – James Aug 13 '15 at 14:32
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    $\begingroup$ ... For instance, if the number of bins is finite, or if the only contain 1 thing, or if there is some process that allows you to describe how to do the picking (e.g. You have a global well ordering around). $\endgroup$ – James Aug 13 '15 at 14:33
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    $\begingroup$ Thank you so much, I think I understand it now. $\endgroup$ – Complimentarity Aug 13 '15 at 23:13

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