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> You can. The proof still works if you do that. > What you’re refusing to accept here is the definition of infinity. I'm refusing to accept the definition of

> You can. The proof still works if you do that. > What you’re refusing to accept here is the definition of infinity. I'm refusing to accept the definition of infinity? I have no idea what you mean by that. Would you make the same statement were you aware that I do in fact have a PhD in mathematics in the field of analysis? That I have in fact studied logic? Just as a hypothetical scenario.

dahart 7y ago
The proof hinges on the definition of infinity. 0.9bar7 is a completely nonsensical number precisely because of the definition of infinity. Yes, I think you’re failing to accept the definition of infinity. You’re rejecting a proof that many other PhDs in math accept along with some notable mathematicians like Euler. I reject your appeal to authority; having an advanced degree doesn’t mean you’re somehow automatically right, lots of PhDs in math have failed the Monty Hall problem, lots of people with advanced degrees are wrong all the time. I myself am a walking example on occasion. I haven’t heard a reason yet why the logic of the proof doesn’t work, I’ve only heard that you don’t accept it. What is the reason it doesn’t work?
- CogitoCogito 7y ago
  > The proof hinges on the definition of infinity. 0.9bar7 is a completely nonsensical number precisely because of the definition of infinity. Yes, I think you’re failing to accept the definition of infinity. You’re rejecting a proof that many other PhDs in math accept along with some notable mathematicians like Euler. I reject your appeal to authority; having an advanced degree doesn’t mean you’re somehow automatically right, lots of PhDs in math have failed the Monty Hall problem, lots of people with advanced degrees are wrong all the time. I myself am a walking example on occasion. I wasn't appealing to authority. I never said that implied that I was correct. (Had I wanted to do that, I would have brought that up much earlier in this comment thread.) I was really just curious if you still believed that I was "failing to accept the definition of infinity" even if you knew that about me. Apparently you do. > I haven’t heard a reason yet why the logic of the proof doesn’t work, I’ve only heard that you don’t accept it. What is the reason it doesn’t work? https://news.ycombinator.com/item?id=23007600 https://news.ycombinator.com/item?id=23007600
  - dahart 7y ago
    Okay, I understand what you’re trying to say. I accept that the proof does not define what “...” means formally, and that is the problem you have with it. The infinity is understood implicitly to have the property that the 9s never end. In that sense, I think the proof does make use of limits, it just relies on definitions not written as part of the proof. Isn’t that okay, doesn’t that actually happen very often? The proof also doesn’t define what addition, multiplication, and equality mean either, but other kinds of mathematicians might complain on those grounds. How complete is complete, and what is the purpose of a proof if not to demonstrate a truth economically, relying on, rather than restating, the already laid foundation? How could this particular proof be shown to have weakness or fail before adding rigorous definitions of limits? Would it be any more apparently true to a wide variety of mathematicians and students if it had the type of rigor you’re advocating?
    - CogitoCogito 7y ago
      See my comment here since it's related to this: https://news.ycombinator.com/item?id=23008366 https://news.ycombinator.com/item?id=23008366 Assuming that all my suspicions in that comment are correct and these proofs actually are invalid proofs (not the results which are true), then the question might become: does it matter if the proof of a fact is incorrect if the fact itself is correct? That is a philosophical question and I'm honestly not sure how I'd answer it...
      - dahart 7y ago
        I don't necessarily understand your use of the word "invalid", when what it seems you mean is incomplete and/or too informal for your taste. > does it matter if the proof of a fact is incorrect if the fact itself is correct? Your language isn't allowing for a notion of precision, or for multiple forms of correctness, and it's not considering audience, communication or level of expertise either. I don't think it's a question of correct vs incorrect, I think you're asking for more precision, and/or for a form that meets your own higher standard. It does matter if a proof is wrong, if there is a step in the proof that can be shown to be false. But that's not the case here, what you want is additional definition. BTW, reading the blog post you linked to on surreals, the "proof" looks to me to be more hand-wavy than Euler's proof that .9bar = 1. The proof begins by stating there are a finite number of 9s, in direct contradiction to the hypothesis. 10^-inf = 0, so from this blog post I don't yet see any reason why surreals clarify anything here, it feels like the opposite, it feels like obfuscation. This could be an argument over representation and not the values of numbers. If you start by defining 0.9bar to be a different number than 1 for the specific reason that it's written down a different way, then fine. That's what the surreal "proof" tells me. Euler's proof is talking about the value of 0.9bar in the limit, not the representation. (Even if that's stated without rigorous definitions of limits.) The proof is saying the values of .9bar and 1 are the same. If the surreal number .9bar were strictly less than 1, that must mean there's another surreal number closer to 1, but there isn't, so I don't accept the surreal argument as valid logic, other than playing a semantic trick by saying 'look I defined they way we write a number to be meaningful, therefore 0.999... is by definition different than 1.' By the way, that blog post claims "The set of real numbers contains no infinitesimals." Wikipedia claims: "the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers..."
- ubercow13 7y ago
  >if you shift the decimal point by an infinite number of places, then there are still an infinite number of 9s to the right >0.9bar7 is a completely nonsensical number For the same reason that 0.9...7 isn't a meaningful number, you cannot move the decimal 'an infinite number of times' and then after this, look at what number you have left and see it still has infinite 9s left. It's like you're trying to perform transfinite induction on the set of numbers generated by moving the decimal point. You can only move the point a countable number of times, so there is no sense in which the property can still be true after infinity many times.
  - dahart 7y ago
    > For the same reason that 0.9...7 isn't a meaningful number, you cannot move the decimal 'an infinite number of times' and then after this, look at what number you have left and see it still has infinite 9s left. Yes you absolutely can, for exactly the same reason. 0.9bar7 is nonsensical precisely because you can move the decimal to the right an infinite number of times, and still have an infinite number of 9s before the 7. > You can only move the point a countable number of times Not true, the implicit definition of “...”, the very statement that there are an “infinite” number of 9s, means exactly the opposite of what you claim, it means you can move the decimal an infinite number of times.
    - ubercow13 7y ago
      In your proof you say >And here is the logical (induction) step: if you shift the decimal point by an infinite number of places, then there are still an infinite number of 9s to the right This is not how induction works. The induction shows that you can shift the decimal point any finite number of steps to the right and there will still be infinite 9's after it. If you want to show something is still true after infinity steps, you require transfinite induction, but this doesn't make sense because the '...' decimal representation only represents a countably infinte number of 9's. This is the same reason 0.9bar7 doesn't make sense - because decimal representations only have countably many digits.
      - ubercow13 7y ago

dahart 7y ago

The proof hinges on the definition of infinity. 0.9bar7 is a completely nonsensical number precisely because of the definition of infinity. Yes, I think you’re failing to accept the definition of infinity. You’re rejecting a proof that many other PhDs in math accept along with some notable mathematicians like Euler. I reject your appeal to authority; having an advanced degree doesn’t mean you’re somehow automatically right, lots of PhDs in math have failed the Monty Hall problem, lots of people with advanced degrees are wrong all the time. I myself am a walking example on occasion. I haven’t heard a reason yet why the logic of the proof doesn’t work, I’ve only heard that you don’t accept it. What is the reason it doesn’t work?

CogitoCogito 7y ago
> The proof hinges on the definition of infinity. 0.9bar7 is a completely nonsensical number precisely because of the definition of infinity. Yes, I think you’re failing to accept the definition of infinity. You’re rejecting a proof that many other PhDs in math accept along with some notable mathematicians like Euler. I reject your appeal to authority; having an advanced degree doesn’t mean you’re somehow automatically right, lots of PhDs in math have failed the Monty Hall problem, lots of people with advanced degrees are wrong all the time. I myself am a walking example on occasion. I wasn't appealing to authority. I never said that implied that I was correct. (Had I wanted to do that, I would have brought that up much earlier in this comment thread.) I was really just curious if you still believed that I was "failing to accept the definition of infinity" even if you knew that about me. Apparently you do. > I haven’t heard a reason yet why the logic of the proof doesn’t work, I’ve only heard that you don’t accept it. What is the reason it doesn’t work? https://news.ycombinator.com/item?id=23007600 https://news.ycombinator.com/item?id=23007600
- dahart 7y ago
  Okay, I understand what you’re trying to say. I accept that the proof does not define what “...” means formally, and that is the problem you have with it. The infinity is understood implicitly to have the property that the 9s never end. In that sense, I think the proof does make use of limits, it just relies on definitions not written as part of the proof. Isn’t that okay, doesn’t that actually happen very often? The proof also doesn’t define what addition, multiplication, and equality mean either, but other kinds of mathematicians might complain on those grounds. How complete is complete, and what is the purpose of a proof if not to demonstrate a truth economically, relying on, rather than restating, the already laid foundation? How could this particular proof be shown to have weakness or fail before adding rigorous definitions of limits? Would it be any more apparently true to a wide variety of mathematicians and students if it had the type of rigor you’re advocating?
  - CogitoCogito 7y ago
    See my comment here since it's related to this: https://news.ycombinator.com/item?id=23008366 https://news.ycombinator.com/item?id=23008366 Assuming that all my suspicions in that comment are correct and these proofs actually are invalid proofs (not the results which are true), then the question might become: does it matter if the proof of a fact is incorrect if the fact itself is correct? That is a philosophical question and I'm honestly not sure how I'd answer it...
    - dahart 7y ago
      I don't necessarily understand your use of the word "invalid", when what it seems you mean is incomplete and/or too informal for your taste. > does it matter if the proof of a fact is incorrect if the fact itself is correct? Your language isn't allowing for a notion of precision, or for multiple forms of correctness, and it's not considering audience, communication or level of expertise either. I don't think it's a question of correct vs incorrect, I think you're asking for more precision, and/or for a form that meets your own higher standard. It does matter if a proof is wrong, if there is a step in the proof that can be shown to be false. But that's not the case here, what you want is additional definition. BTW, reading the blog post you linked to on surreals, the "proof" looks to me to be more hand-wavy than Euler's proof that .9bar = 1. The proof begins by stating there are a finite number of 9s, in direct contradiction to the hypothesis. 10^-inf = 0, so from this blog post I don't yet see any reason why surreals clarify anything here, it feels like the opposite, it feels like obfuscation. This could be an argument over representation and not the values of numbers. If you start by defining 0.9bar to be a different number than 1 for the specific reason that it's written down a different way, then fine. That's what the surreal "proof" tells me. Euler's proof is talking about the value of 0.9bar in the limit, not the representation. (Even if that's stated without rigorous definitions of limits.) The proof is saying the values of .9bar and 1 are the same. If the surreal number .9bar were strictly less than 1, that must mean there's another surreal number closer to 1, but there isn't, so I don't accept the surreal argument as valid logic, other than playing a semantic trick by saying 'look I defined they way we write a number to be meaningful, therefore 0.999... is by definition different than 1.' By the way, that blog post claims "The set of real numbers contains no infinitesimals." Wikipedia claims: "the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers..."
ubercow13 7y ago
>if you shift the decimal point by an infinite number of places, then there are still an infinite number of 9s to the right >0.9bar7 is a completely nonsensical number For the same reason that 0.9...7 isn't a meaningful number, you cannot move the decimal 'an infinite number of times' and then after this, look at what number you have left and see it still has infinite 9s left. It's like you're trying to perform transfinite induction on the set of numbers generated by moving the decimal point. You can only move the point a countable number of times, so there is no sense in which the property can still be true after infinity many times.
- dahart 7y ago
  > For the same reason that 0.9...7 isn't a meaningful number, you cannot move the decimal 'an infinite number of times' and then after this, look at what number you have left and see it still has infinite 9s left. Yes you absolutely can, for exactly the same reason. 0.9bar7 is nonsensical precisely because you can move the decimal to the right an infinite number of times, and still have an infinite number of 9s before the 7. > You can only move the point a countable number of times Not true, the implicit definition of “...”, the very statement that there are an “infinite” number of 9s, means exactly the opposite of what you claim, it means you can move the decimal an infinite number of times.
  - ubercow13 7y ago
    In your proof you say >And here is the logical (induction) step: if you shift the decimal point by an infinite number of places, then there are still an infinite number of 9s to the right This is not how induction works. The induction shows that you can shift the decimal point any finite number of steps to the right and there will still be infinite 9's after it. If you want to show something is still true after infinity steps, you require transfinite induction, but this doesn't make sense because the '...' decimal representation only represents a countably infinte number of 9's. This is the same reason 0.9bar7 doesn't make sense - because decimal representations only have countably many digits.
    - ubercow13 7y ago