User blog:Fejfo/bitstring lexographical notation

This is an idea I have for a notation, but I have trouble defining it. The idea is to consider all lexographically/alphabetically orderd bitstrings (strings of zeros and ones). But this isn't a well order, so to make this a well-order you find the first point at which there is an infinite decending sequence and choose a next standard bitsting form it.

Ex at the start an infnite decending sequence 1, 01, 001, 0001, ... occurs after \( \omega \) bitstrings 0, 00, 000, ...

So we may choose 1 as the next bitstring, and representation of \( \omega \)

You can say "in the n-th system choose the n-th element from the first infinte deceding sequence" but that obviously runs into the problem that there is no such thing as "the first infinite decending sequence".

You can find some more info, adhoc examples (of how it is supposed to work) and a formalisation attempt here: https://docs.google.com/spreadsheets/d/1xpbWtM7TwqxT_-ZzEyTkaaLYnf95ehvd3kC2_xML6wU/edit#gid=107710322

While "analysing" I noticed a simmularity between inifnite decending sequences and fundamental sequences. 1, 01, 001, 0001, ... corresponds to the FS for \( \omega \) empty string, 0, 00, 000, ... (ie the ending ones removed)

I think the strength of the second system might be because it hyjacks on the fundamental sequences I know, and thus would be impossible to formalise, but there may also be a definition with out it.

You could also consider what the effect of random choices would be. You could choose the googoled element of a random infnite decending sequence, and possibly be quite confident it is "a small step".

These are just some ideas, please tell me if you know how you can make anything well-defined out of them.