I need to work out an answer to this, frankly rubbish question. Now time has moved on and so has my command of the problem. I can rewrite this question with a precise mathematical description and in a way that is actually answerable. I think the transinfinite induction technique used to solve it is likely similar to a Hilbert Hotel type problem and will be useful to other mathematicians.

My first thought is simply to completely rewrite the text of the question but a) the question will be essentially unrecognisable other than its title, and b) I imagine that old question will simply languish in the unanswered question queue for all eternity or until removed/abandoned.

Should I ask anew, or rewrite the original?

I'm happy to edit that question and give it a bounty to attract some attention. I feel like sacrificing a bit of reputation would be the most "ethical" thing to do for having asked so badly in the first place. But I also think to simply delete the question and ask in its new form makes more sense. It's not my intention to ask this question here in Meta, Just for background, so it's apparent how different the wordings of the question are, here's the induction better described:

Consider the function:

$f(X)=\frac{1}{3}\{2^nx-1:n\in\Bbb{N_{>0}},x\in X\}$

Let $g_0=e=1$

Let $g_{n+1}=f(g_{n})$

so e.g. $g_1=f(1)=\frac{1}{3}\{1,3,7,15,31,\ldots\}$

$g_2=\frac{1}{p}\{-1,1,5,13,29,\ldots\}\\ \cup\frac{1}{9}\{3,9,21,45,93,\ldots\}\\ \cup\{11,25,53,109,221,\ldots\}\\ \cup\ldots$

so $g_n$ is in a sense an n-dimensional set.

Note that $1\in f(1)\implies$ by induction: $g_n\subset g_{m}\forall m\geq n$

So we can think of $g_{n+1}$ as supersets of $g_{n}$ or supergroups, given an appropriate group operation.

Let $G$ be the completion of these under the superset operation. In other words, define the reverse map $f^{-1}(x)=(3x+1)\lvert3x+1\rvert_2$ and then:

$G=\{x:(\exists n\in\Bbb{N}:f^{-n}(x)=1)\}$

So the construciton of $G$ is an infinite-dimensional induction

a) can we fully define the set $G$, and if so, how?

I think it looks "something like" $\dfrac{1}{3^{\infty}}\Bbb{Z}_2^{\times}$ - whatever that is.

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    $\begingroup$ My two cents' worth: write a new question, with links both ways between the two questions. $\endgroup$ – Gerry Myerson Apr 12 '18 at 12:41
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    $\begingroup$ It depends on how much has transpired at the original question. If there are valid helpful solutions to the rubbish question, then it is bad form to edit the question. It's also bad form to repost a question without transferring helpful content from comments, as it looks like you are simply ignoring the responses. But there is a path between: e.g. linking, or even better, including the best bits in the new post. But if there really were no solutions to the question, it seems like it would be a lot easier to edit it improving it with anything helpful that happened. $\endgroup$ – rschwieb Apr 12 '18 at 13:28

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