On 11/17/2012 12:10 AM, Andy Latto wrote:
On Fri, Nov 16, 2012 at 11:57 PM, David Wilson <davidwwilson@comcast.net> wrote:
It occurred to me that the standard proof of 0.9999... = 1 depends on the continuity of the reals numbers, which I believe the surreals lack. I don't know what you mean by "the continuity of the reals"; do you mean completeness? I know what it means for a function to be continuous, but not a set. My terminology is not always correct. It is difficult to look up the terminology by mathemetical concept, even online, and I haven't done any analysis to speak of 30 years.
But yes, I think I meant completeness in the sense that limits of sequences of set elements are in the set.
But I don't understand why you think the proof that .9999... = 1 requires continuity. It seems to me that if we write this as a two-way-infinite series,
... + 0 + 0 + 0 + .9 + .09 + .009 + ..., with a_1 = .9
Then all we need to assume about such sums to prove that if it has a sum, that sum must be 1 is exactly the three properties you describe below:
If m in R and b_i = m a_i for all i in Z, then sum(b) = m sum(a). If c_i = a_i + b_i for all i in Z, then sum(c) = sum(a) + sum(b).
We would want sums to be preserved by shifting the sequence left or right, e.g:
If k in Z and b_i = a_{i+k} for all i in Z, then sum(b) = sum(a).
If the sum is S, the first property tells us that the sum of ... + 0 + 0 + 0 + 9 + .9 + .09 +... with a_1 = 9, is 10 * S
The third property says that the sum of this is still 10 * S when we shift it over by 1, so that a_0 = 9 and a_1 = .9.
Now the second property (subtracting rather than adding, but we can use the first property to multiply the first sequence by -1) says that the sum of the sequence
...0 + 0 + 0 + 9 + 0 + 0 + 0 + ...
So any notion of summability that assigns the sum 9 to this series, and satisfies your three properties, must assign the sum 0 to the sum 1 you mean sequence ... 0 + 0 + 0 + .9 + .09 + .009 + ....., with no assumption of "continuity" needed.
Andy What I said was: the *standard* proof that 0.9999... = 1 does require completeness.
The standard proof does not conclude that "if 0.9999... has a value, it is 1", it proves "0.9999... = 1". It does so first establishing a recursive definition for finite decimals which does not involve a limit. It then establishes a value for infinite decimals as a limit of finite decimals, the latter definition requiring completeness to ensure that this limit is indeed a real number in all cases. But in this case, I could allow you are right. You can show that the particular decimal 0.9999... has real value 1 without having to show that all infinite decimals have real limit values (which I think would require at least a weakened version of completeness). However, in our double-ended formalism above, I have not actually defined a sum value for any of the double-ended sums. I have only said things like "if this double-ended sum has value a, then that double ended sum have value b". From these sorts of statements, the best you can prove is "if this double-ended sum has a value, it is a". Clearly we want sum({0}) = 0, but even in conjunction with the other formal properties, I don't that good enough to establish ... + 0 + 0 + 0 + 9 + 0 + 0 + 0 + ... = 9. At any rate, I expect this whole house of cards to collapse, which is to say, I expect that the sum() function cannot be consistently defined so as to give the intuitive values in the simple cases and also conform to the sum, scalar multiplication, and shifting rules.
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