> 1. we assert that 0.666...7 is a sequence of digits.
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.
Why? https://news.ycombinator.com/item?id=23009160 posited that we were in a situation where (1) holds, so that's where we're starting. If we assume (1) holds, then (2) cannot hold. We can abandon (1) but then we're no longer replying to that specific comment, now we're trying to prove something else.
My point is that it cannot be clear what assuming (1) entails when you aren't properly defining the quantities involved. You have to answer in clear and mathematical language what the quantity in (1) is defined as. What is the definition of "0.666…7"?
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.
No, it isn't. The idea that saying "gnarfgnarf is an imaginary number is completely meaningless" is the opposite of true: if you assert that gnarfgnarf is an imaginary number, that is the definition we'll be using for the remainder of whatever proof we use that in. Anywhere the proof now talks about gnarfgnarf, we're talking about something that is an imaginary number, and has to follow all the rules that imaginary numbers have to follow, without ever having to say which imaginary number it is, or further define it. It's "any" imaginary number, we just call it "gnarfgnarf" instead of "x" or "a + bi" or the like.
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.
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.