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This reasoning is understandable, but also incorrect, and we can point to where it breaks down: the idea that the 7 is "at a specific decimal" doesn't hold true
by TheRealPomax 7y ago
This reasoning is understandable, but also incorrect, and we can point to where it breaks down: the idea that the 7 is "at a specific decimal" doesn't hold true, due to that pesky ellipsis. The crazy bit about the infinite repetition is that the 7 in 0.666...7 is not at a specific decimal. It's not even at "the last decimal" because there is no last. So, let's show this via a proof by contradiction:
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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.
- gspr 7y ago> 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.
- TheRealPomax 7y agoWhy? https://news.ycombinator.com/item?id=23009160 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.
- gspr 7y agoMy 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. So: what do you mean by 0.666…7?
- TheRealPomax 7y agoNo, 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.
- gspr 7y agoI'm really sorry, I jumped to conclusions as to what you were saying without properly reading the comment you linked to!