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Infinities are difficult and your explanation is wrong. This notation .666...7 is meaningless. The notation .666... indicates a decimal representation that does

Infinities are difficult and your explanation is wrong. This notation .666...7 is meaningless. The notation .666... indicates a decimal representation that does not end. The representation has infinitely many sixes and does not end. If there is a 7 in the representation then it must be at a specific decimal. If it was at the end of the representation then it would be a finite decimal representation. The notation is meaningless with respect to the real numbers.

enedil 6y ago
Yes, exactly, .666...7 means a number that has 6 at each decimal position and 7 for index k, where k is greater than any natural number. This exactly is .666... (just as parent commenter explained).
- gspr 6y ago
  > Yes, exactly, .666...7 means a number that has 6 at each decimal position and 7 for index k, where k is greater than any natural number. This exactly is .666... (just as parent commenter explained). Ill-defined. Try again.
ping_pong 6y ago
It's as much nonsense as someone saying "infinity plus one is bigger than infinity!"
exmadscientist 6y ago
I don't agree with you, or with the sibling comment that claims that "infinity plus one" is "nonsense". It is in fact precisely well-defined, to a mathematician working in a framework that admits it and makes it worth discussing. The blog post that (I think) started off this chain of posts, at https://blog.plover.com/misc/half-baked.html 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.
TheRealPomax 6y 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: --- 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 --- 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 6y 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 6y ago
    Why? 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.

enedil 6y ago

Yes, exactly, .666...7 means a number that has 6 at each decimal position and 7 for index k, where k is greater than any natural number. This exactly is .666... (just as parent commenter explained).

gspr 6y ago
> Yes, exactly, .666...7 means a number that has 6 at each decimal position and 7 for index k, where k is greater than any natural number. This exactly is .666... (just as parent commenter explained). Ill-defined. Try again.

ping_pong 6y ago

It's as much nonsense as someone saying "infinity plus one is bigger than infinity!"

exmadscientist 6y ago

I don't agree with you, or with the sibling comment that claims that "infinity plus one" is "nonsense". It is in fact precisely well-defined, to a mathematician working in a framework that admits it and makes it worth discussing. The blog post that (I think) started off this chain of posts, at https://blog.plover.com/misc/half-baked.html 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.

TheRealPomax 6y 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: --- 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 --- 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 6y 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 6y ago
  Why? 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.