The blog post that (I think) started off this chain of posts, at https://blog.plover.com/misc/half-baked.html , has the most concise and lucid explanation I've found, so I'm just going to straight-up copy it: 0.666...7 is an “an object... said to ‘have order type ω+1’, and is completely legitimate.”
It's not very useful – it's exactly equal to 0.666... ! – but it's legitimate and well-defined.
My absolute favorite construction of objects like this is Conway's surreal numbers. These things appear perfectly naturally in the surreal numbers, and are completely well-defined, if (again) not very useful.
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1. we assert that 0.666...7 is a sequence of digits.
2. we assert that each digit in a sequence can be assigned an integer index corresponding to its position in the sequence. I.e. we can defined each digit's index as "the number of digits that precede this digit".
3. from (1) and (2) it follows that the index for 7 must be an integer.
4. the ellipses represents an infinite number of digits (infinitely repeating the repeated digit pattern preceding it).
5. from (2) and (4) it follows that the index for 7 must be the integer value "infinity", because it has an infinite number of digits preceding it.
6. (3) and (5) cannot both be true, because infinity is not an integer.
7. from (6) it follows that (1), and/or (2), and/or (4) must be false
8. (4) is, by definition, true.
9. from (7) and (8) it follows that (1) and/or (2) must be false.
10. (1) is our fundamental assertion. If (1) was false then there is wouldn't even be a sequence of digits for us to reason about. So (1) is true.
11. from (6), (8), and (10) it follows that (2) must be false for there to be no contradiction.
QED
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Now, certainly, for finite length numbers the assumption that each digit in a sequence has an integer index holds true, but it turns out we have mathematical notation that lets us write down numbers for which that property does not hold.
Infinities are fun. And difficult. But also fun.
A sequence in the mathematical sense is a function whose domain is the natural numbers. Please define that function for the creature you're working with here. Otherwise you're trying to prove things about an object with no definition. You will end up in trouble.
As it currently stands, assumption (1) is similar in nature to me saying "gnarfgnarf is an imaginary number". It's completely meaningles unless I define what I mean by gnarfgnarf.
So: what do you mean by 0.666…7?
Same here: we have a number written as 0.666...7 using conventional mathematical notation. The comment that is being replied to asserts that this can be treated as a sequence, and so we start the proof with that definition: "0.666...7 is a sequence", and now we're done. You, as reader of the proof, have been informed that those nine symbols, in that order, for the rest the proof, represent a sequence. Not "a specific sequence", but "any sequence", and it must follow all the rules that sequences follow.
We then show that simply by being "a sequence", due to the properties of sequences, we get a contradiction. Our first assertion is the definition for the purpose of this proof, and is sufficient.
Ill-defined. Try again.